# Electron Scattering at a Potential
Temporal Step Discontinuity

Furkan Ok  Department of Electrical Engineering, KU Leuven, Leuven, 3000,
Belgium  Amir Bahrami  Department of Electrical Engineering, KU Leuven,
Leuven, 3000, Belgium  Christophe Caloz  Department of Electrical Engineering,
KU Leuven, Leuven, 3000, Belgium<EMAIL_ADDRESS>
###### Abstract

We solve the problem of electron scattering at a potential temporal step
discontinuity. For this purpose, instead of the Schrödinger equation, we use
the Dirac equation, for access to back-scattering and relativistic solutions.
We show that back-scattering, which is associated with gauge symmetry
breaking, requires a vector potential, whereas a scalar potential induces only
Aharonov–Bohm type energy transitions. We derive the scattering probabilities,
which are found to be of later-forward and later-backward nature, with the
later-backward wave being a relativistic effect, and compare the results with
those for the spatial step and classical electromagnetic counterparts of the
problem. Given the unrealizability of an infinitely sharp temporal
discontinuity – which is of the same nature as its spatial counterpart! – we
also provide solutions for a smooth potential step and demonstrate that the
same physics as for the infinitely sharp case is obtained when the duration of
the potential transition is sufficiently smaller than the de Broglie period of
the electron (or deeply sub-period).

## Introduction

Electron scattering at a potential _spatial_ step is a canonical problem that
is treated in the introductory section of most textbooks on quantum mechanics
[1, 2, 3, 4, 5, 6] and that underpins uncountable phenomena (e.g., quantum
reflection, transmission and interference, quantum tunneling, quantum wells
and scattering resonances, quantum coherent transport) and applications (e.g.,
p-n junction diodes, transistors, semiconductor lasers, and detectors,
scanning tunneling microscopy, quantum computing, particle accelerators). The
problem is typically addressed by resolving the Schrödinger equation [7] for
non-relativistic particles, but requires promotion to the Klein-Gordon
equation [8, 9] or to the Dirac equation [10] for relativistic particles, of
spin 0 or $1/2$, respectively.

The problem of electron scattering at a potential _temporal_ step is arguably,
from space-time duality, as fundamental as that of the spatial step. A number
of works on quantum phenomena occurring at step discontinuities have been
reported in the literature [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,
23], but the specific problem of electron scattering at a potential temporal
step has surprisingly not been explicitly resolved yet. Such a gap needs to be
filled. This is all the more obvious when considering the promising
opportunities of transposing to the quantum realm recent concepts developed in
the booming field of _classical_ electromagnetic modulation-based time-varying
[24, 25, 26, 27] and space-time varying [28, 29, 30, 31, 32] metamaterials
[33, 34, 29, 35], which have already led to a wealth of novel effects and
applications, including the inverse prism [36], linear-time invariance bound
breaking [37], temporal aiming [38], extreme energy transformation [39],
temporal antireflection coating [40], temporal polarization conversion [41],
temporal analog computing [42, 43], static-to-dynamic field conversion [44],
temporal Faraday rotation [45, 46, 47], arbitrary transfer function emulation
[48], optimization-free filter and matched-filter [49], broadband parametric
amplification [50, 51], wave deflection and shifted refocusing [52] and
nonreciprocity and optical isolation [53, 54, 55, 56, 57].

We present here an exact and comprehensive resolution of the problem of
electron scattering at a potential temporal step discontinuity. We first show
that the Schrödinger equation cannot account for electron back-scattering for
that problem and therefore decide to resort to the (more general) Dirac
equation. We then demonstrate that a _scalar_ potential temporal step does not
produce back-scattering, whereas a _vector_ potential temporal step does (see
Supplementary Sec. 1), and explain this fact in terms of related gauge
symmetry and symmetry breaking. We next derive formulas for the scattering
coefficients, probabilities, and energy transitions of the electronic wave.
Finally, we demonstrate that the corresponding scattering is a relativistic
effect. Throughout the report, we systematically compare the problem with its
spatial counterpart, and also point out some similarities and differences with
corresponding electromagnetic problems [33, 34, 35]. Finally, we also provide
solutions for a smooth step potential and investigate the related physics
versus the transition duration of the step with respect to the de Broglie
period of the electron.

## Spatial and Temporal Sharp Potential Step Discontinuities

Figure 1 represents the problem of electron scattering at a potential step
discontinuity, with the discontinuity being spatial in Fig. 1(a) and temporal
in Fig. 1(b). The latter is the problem at hand in the report while the former
is considered as its dual reference. In both cases, the changing parameter is
a component of the four-vector potential $A^{\mu}=(V,\mathbf{A})$ and we shall
later see why, as indicated in the figure, $V$ and $\mathbf{A}$ are the most
relevant components for the spatial and temporal cases, respectively.

Figure 1: Electron scattering at a potential (a) spatial and (b) temporal step
discontinuity, in spacetime (top panels) and space/time-transverse coordinates
(bottom panels). The subscripts i, r, t, b, and f stand for incident,
reflected, transmitted, later-backward, and later-forward, while the
subscripts p and g stand for phase and group (velocity), respectively.

As known from textbooks, the scattered electronic waves in the spatial problem
[Fig. 1(a)] are reflected and transmitted waves, with conserved energy
($\Delta E=0$) and transformed momentum ($\Delta{p}\neq{0}$), as in classical
electromagnetics. We shall show that scattering is quite different in the
counterpart temporal problem [Fig. 1(b)]. First, the scattered electronic
waves are generally _later-backward_ and _later-forward_ waves, as opposed to
reflected and transmitted waves, where the term _later_ means “after the
temporal discontinuity”, contrasting with the term _earlier_ , which refers to
the wave launched before the temporal discontinuity, while the terms _forward_
and _backward_ denote corresponding propagation directions in space [33].
Second, it is now the momentum that is conserved ($\Delta p=0$), while the
energy is transformed ($\Delta E\neq{0}$). These two aspects parallel the
situation that prevails in classical electromagnetics [24, 34], but with a
number of differences, such as the fact the phase and group velocities are
generally distinct, as represented in the figure, and the fact that the later-
backward wave exists only in the relativistic regime, whereas it is always
present in classical electromagnetics. Throughout the paper, we shall restrict
our attention to the 1+1-dimensional case, with one dimension of space ($z$)
and the dimension of time ($t$). Moreover, we shall use natural units
($\hbar=c=1$) and the Minkowski metric
$\eta^{\mu\nu}=\mathrm{diag}(1,-1,-1,-1)$ throughout the report.

## Choice of an Appropriate Equation

### Limitations of the Schrödinger Equation

One may first be tempted to address the problem of the sharp temporal
potential step [Fig. 1(b)] with the Schrödinger equation, as typically done
for the sharp spatial potential step [Fig. 1(a)] in the non-relativistic
regime. The Schrödinger equation reads
$i\partial_{t}\psi=-{\nabla}^{2}\psi/(2m)$ [1, 2, 3, 4, 5, 6], where $m$ is
the mass of the particle, which we shall consider from now on as being the
electron. This equation has unpaired spatial and temporal derivatives, with
the former (${\nabla}^{2}$) being of the second order and the latter
($\partial_{t}$) of the first order.

In the case of the sharp spatial step, where one typically assumes the
monochromatic ansatz $\psi\propto\textrm{e}^{-iEt}$, the Schrödinger equation
reduces to $\psi=-{\nabla}^{2}\psi/(2mE)$. Therefore, ${\nabla}^{2}\psi$ must
be finite to ensure finite $\psi$, and hence $\gradient\psi$ must be
continuous. In addition, $\psi$ must also be continuous, for otherwise,
$\gradient\psi$ would be singular, and so would then also be
$\psi\propto{\nabla}^{2}\psi$, and hence $\psi$. We have then the double
boundary condition that both $\psi$ and ${\gradient}\psi$ must be continuous
at the spatial discontinuity. Thus, the second-order derivative operator
${\nabla}^{2}$ provides two boundary conditions, viz., the continuity of both
$\psi$ and of ${\gradient}\psi$ at the spatial discontinuity, which leads to a
fully determined problem whose resolution provides the usual reflected and
transmitted scattered electronic waves [Fig. 1(a)].

In contrast, the _first-order derivative_ $\partial_{t}$ provides only one
boundary condition in the sharp temporal potential step problem [Fig. 1(b)].
Assuming the plane-wave ansatz $\psi\propto\textrm{e}^{ipz}$, the Schrödinger
equation reduces to $\psi=i(2m/p^{2})\partial_{t}\psi$ and $\partial_{t}\psi$
must therefore be finite to ensure finite $\psi$, which entails that $\psi$
must be continuous at the temporal discontinuity. However, this is indeed the
_the only_ boundary condition, due to the absence of a higher-order temporal
derivative. Consequently, the Schrödinger equation does not readily include
sufficient information to account for more than one scattered wave, which
would involve more unknowns, specifically two unknowns in the possible case of
later-forward and later-backward waves. Moreover, the Schrödinger equation is
not relativistic, and would hence miss related solutions.

### Selection of the Dirac Equation

The Dirac equation, which reads for the free-electron case
$i\partial_{t}\psi=-(i\alpha^{i}\partial_{i}-\gamma^{0}m)\psi$, where $\psi$
is a ($4\times{}1$) spinor and where $\alpha^{i}$ and $\gamma^{0}$ are
($4\times{}4$) matrices [10, 3, 58, 59] (see Supplementary Sec. 2), seems to
represent a safer approach for obtaining a complete solution to our temporal
step problem. It also has, as the Schrödinger equation, a first-order temporal
derivative order ($\partial_{t}$), but it involves multiple sub-equations that
might together support sufficient information to account for more than one
scattered wave, including relativistic ones.

Let us then try to address the problem with the Dirac equation. In order to
account for the potentials in Fig. 1, we extend the free-electron Dirac
equation to its minimal-coupling form [10, 3, 58, 59] (see Supplementary Sec.
2)

$[\gamma^{\mu}\left(i\partial_{\mu}-qA_{\mu}\right)-m]\psi=0,$ (1a) where
$\gamma^{\mu}$ are the matrices (Dirac-Pauli representation)
$\gamma^{0}=\begin{pmatrix}I&0\\\ 0&-I\end{pmatrix}\quad\text{ and
}\quad\gamma^{i}=\begin{pmatrix}0&\sigma^{i}\\\ -\sigma^{i}&0\end{pmatrix},$
(1b) with $I$ and $\sigma_{i}$ being the ($2\times{2}$) unit and Pauli
matrices, respectively.

Inserting the positive-energy monochromatic plane traveling wave ansatz
$\psi=\begin{pmatrix}\varphi\\\ \vartheta\end{pmatrix}\mathrm{e}^{-i(Et-pz)}$
into Eqs. (1) yields the general solution form (see Supplementary Sec. 3.1)

$\psi=\begin{pmatrix}1\\\ 0\\\ \frac{E-qV-m}{p-qA}\\\
0\end{pmatrix}\mathrm{e}^{-i(Et-pz)},$ (2)

where $q=-e$ ($e>0$) is the charge of the electron. We make here the choice of
a non-localized, continuous-wave ansatz because it is both the simplest and
most appropriate regime to reveal the fundamental physics of the problem. The
localized, wave-packet regime would be the next interesting regime to
consider, with expected interesting novel time-delay physics, such as the
quantum analog of the temporal Goos-Hänchen shift [60].

## Scalar Potential Discontinuity

One may first attempt to apply the general solution (2) to the case of a pure-
scalar potential, i.e., $A^{\mu}=(V,0)$, as typically done for the spatial
step [Fig. 1(a)], which corresponds in the problem at hand to a temporal
scalar potential step $V(t)$, with $V(t<t_{0})=V_{1}$ and
$V(t>t_{0})=V_{2}=V_{1}+\Delta{V}$, with $\Delta{V}=V_{2}-V_{1}$, where
$t_{0}$ is the switching time. However, it may be easily verified (see
Supplementary Sec. 3.2.1) that, although providing the expected energy shift
(from $E_{\mathrm{i}}=\sqrt{p^{2}+m^{2}}+qV_{1}$ to
$E_{\mathrm{f}}=\sqrt{p^{2}+m^{2}}+qV_{2}$), of potential interest for
amplification applications [6], such a potential does not produce any later-
backward wave scattering! This result, which might a priori appear surprising,
may be explained in terms of gauge invariance symmetry.

The (external) electric and magnetic fields, $\mathbf{E}$ and $\mathbf{B}$,
associated with the potential modulation, are generally related to the
potentials as $\mathbf{E}=-{\gradient}V-\partial_{t}\mathbf{A}$ and
$\mathbf{B}=\curl\mathbf{A}$, which are invariant under the gauge
transformation [61, 62]

$V\rightarrow
V^{\prime}=V-\partialderivative{\Lambda}{t}\quad\textrm{and}\quad\mathbf{A}\rightarrow\mathbf{A}^{\prime}=\mathbf{A}+{\gradient}\Lambda,$
(3)

where $\Lambda$ is an arbitrary scalar function. The sharp temporal potential
step $V(t)$ considered in the previous paragraph is equivalent to the
transformation $V^{\prime}=V_{1}+\Delta{V}\theta(t-t_{0})$, where
$\theta(t-t_{0})$ is the Heaviside step function, and $\mathbf{A}^{\prime}=0$,
which is a particular case of the gauge transformation
$\eqref{eq:gauge_transf}$ with $V=V_{1}$,
$-\partial_{t}\Lambda=\Delta{}V\theta(t-t_{0})$, $\mathbf{A}=0$ and
${\gradient}\Lambda=0$, corresponding to $\Lambda=-\Delta{}Vt\theta(t-t_{0})$.
Therefore, this sharp potential step does not involve any change in the
external fields, which explains why we found that it produces no later-
backward scattering. [In fact, a similar result – unchanged external fields
and the consequent absence of back-scattering – is found in the case of the
sharp spatial step for the potential $A(z)$ (see Supplementary Secs. 4.2 and
3.2.3)]. The external fields are actually zero, since $\mathbf{A}=0$ and
${\gradient}V={\gradient}V(t)=0$; the related energy transition due to
potential without field is therefore somewhat akin to the Aharonov–Bohm effect
[63]. This absence of back-scattering contrasts with the situation of the
pure-scalar sharp spatial potential step $V(z)$ [Fig. 1(a)], whose (reflected
wave) back-scattering results from the breaking of the gauge condition (3)
(see Supplementary Sec. 4.1).

## Vector Potential Discontinuity

We may suspect at this point that, since $V(t)$ fails to break the gauge
symmetry (3), its pure-vector potential counterpart $\mathbf{A}(t)$ should
break it, and hence bring about back-scattering, as the familiar (reflection)
sharp spatial step $V(z)$. That this is indeed the case is shown as follows.
Assuming $\mathbf{A}(t)=A(t)\mathbf{\hat{z}}$, the step function reads now
$A(t<t_{0})=A_{1}$ and $A(t>t_{0})=A_{2}=A_{1}+\Delta{A}$, with
$\Delta{A}=A_{2}-A_{1}$. The corresponding transformation is
$A^{\prime}=A_{1}+\Delta{}A$$\theta(t-t_{0})$ and $V^{\prime}=V_{1}+\Delta{V}$
with $V_{1}=\Delta{V}=0$. Consistency with the gauge (3), given the mapping
$A=A_{1}$, $\nabla\Lambda=\Delta\mathbf{A}(t)$ or
$\partial_{z}\Lambda=\Delta{A}\theta(t-t_{0})$, $V=V_{1}=0$ and
$-\partial_{t}\Lambda=0$, would now demand that
$\Lambda=\Delta{}A\theta(t-t_{0})z=\Lambda(z,t)$ along with
$\Lambda\neq\Lambda(t)$. The incompatibility between the last two conditions
on $\Lambda$ indicates that the transformation indeed breaks the symmetry of
the gauge (3), which entails transformed external fields and which may hence
lead to electron back-scattering.

We can now solve the problem of interest, for the potential, $A(t)$, using the
later-forward and later-backward ansätze corresponding to the related
temporal-step classical electromagnetic solutions [24, 34] [Fig. 1(b)].
According to Noether’s theorem [64], for such a potential, momentum is
conserved ($\Delta p=0$) due to spatial translational symmetry, viz.,
$p_{\mathrm{i}}=p_{\mathrm{f}}=p_{\mathrm{b}}=p$, while broken temporal
translational symmetry leads to energy transitions, which are given by the
dispersion relation (see Supplementary Secs. 3.1 and 3.2.4)

$E_{1,2}^{2}=\left(p-qA_{1,2}\right)^{2}+m^{2},$ (4)

where the subscript labels 1 and 2 refer to the earlier and later potential
regions, respectively. Equation (4) leads to the energy relations

$E_{\mathrm{i}}=\sqrt{(p-qA_{1})^{2}+m^{2}},$ (5a)
$E_{\mathrm{f}}=\sqrt{(p-qA_{2})^{2}+m^{2}}\quad\textrm{and}\quad
E_{\mathrm{b}}=-E_{\mathrm{f}},$ (5b)

where, assuming $E_{\mathrm{f}}>0$, the apparent negative energy
$E_{\mathrm{b}}<0$ in the last relation simply represents propagation in the
negative $z$ direction ($v_{\mathrm{g,b}}<0$), with positive energy
($|E_{\mathrm{b}}|>0$) (see Supplementary Sec. 5).

## Dispersion and Transition Diagrams

Figure 2 plots the dispersion relations and electronic transitions for the two
problems in Fig. 1, with Figs. 2(a) and 2(b) corresponding to the (reference)
spatial step and temporal step problems in Figs. 1(a) and (b), respectively,
and with indications of the phase and group velocities (see Supplementary Sec.
5), corresponding to those in Fig. 1.

Figure 2: Dispersion relations and electronic transitions corresponding to
Fig. 1 for (a) the spatial step $V(z)$ [Fig. 1(a)], with (horizontal) momentum
transitions, and (b) the temporal step $A(t)$ [Fig. 1(b)], with (vertical)
energy transitions, as well as corresponding (c) momenta and (d) energies
versus potential steps.

The spatial step [Fig. 2(a)] features the well-known vertical dispersion
shifting with horizontal (momentum) transitions from the incident to the
reflected and transmitted states (see Supplementary Sec. 3.2.2). The temporal
step [Figs. 2(b)] [Eqs. (4) and (5)] exhibits _perfectly dual_
characteristics, with horizontal dispersion shifting and vertical (energy)
transitions from the earlier to the later-backward and later-forward states,
whose energy levels versus $E_{\mathrm{i}}$ are obtained by solving Eq. (5a)
for $p$ and inserting the result into Eqs. (5b), which yields

$E_{\mathrm{f}}=-E_{\mathrm{b}}=\sqrt{\left(\sqrt{E_{\mathrm{i}}^{2}-m^{2}}-q\Delta{A}\right)^{2}+m^{2}}.$
(6)

Note that the orthogonal dispersion shifting in electronic scattering in Fig.
2 is a feature that does not exist in the classical electromagnetic
counterparts of these problems, which rather involve (refractive index)
dispersion curves that are rotated with respect to each other and that do not
differ between the space and time cases [34] (see Supplementary Sec. 6).

## Scattering Coefficients

Upon the basis of the energy relations (5), the scattering amplitudes and
probabilities may be easily found by inserting the expression for the vector
potential step function $A(t)$ into the general solution form (2) and
enforcing the continuity condition
$\left.\psi_{1}\right|_{t=t_{0}}=\left.\psi_{2}\right|_{t=t_{0}}$. The
resulting later-backward and later-forward amplitude coefficients are (see
Supplementary Sec. 3.2.4)

$b=\frac{\Gamma_{\mathrm{t}}-1}{2\Gamma_{\mathrm{t}}}\quad\text{ and }\quad
f=\frac{1+\Gamma_{\mathrm{t}}}{2\Gamma_{\mathrm{t}}},$ (7a) corresponding to
the probabilities
$B=\left|b\right|^{2}\frac{2\Gamma_{\mathrm{t}}^{2}}{1+\Gamma_{\mathrm{t}}^{2}}\quad\text{
and }\quad
F=\left|f\right|^{2}\frac{2\Gamma_{\mathrm{t}}^{2}}{1+\Gamma_{\mathrm{t}}^{2}},$
(7b) where
$\Gamma_{\mathrm{t}}=\frac{\sqrt{\left(\sqrt{E_{\mathrm{i}}^{2}-m^{2}}-(qA_{2}-qA_{1})\right)^{2}+m^{2}}}{\left(\sqrt{E_{\mathrm{i}}^{2}-m^{2}}-(qA_{2}-qA_{1})\right)\left(\dfrac{E_{\mathrm{i}}-m}{\sqrt{E_{\mathrm{i}}^{2}-m^{2}}}\right)+m}.$
(7c)

Interestingly, the amplitude coefficients in Eqs. (7a) are formally identical
to those for classical electromagnetic scattering at a refractive index
temporal step discontinuity [24, 34], with the parameter $\Gamma_{\mathrm{t}}$
in Eq. (7c) replacing the refractive index contrast $N=n_{2}/n_{1}$ (see
Supplementary Sec. 6).

Figure 3 plots the electron scattering probabilities versus potential strength
for the two problems in Fig. 1, with Figs. 3(a) and 3(b) corresponding to the
(reference) spatial step and temporal step problems in Figs. 1(a) and (b),
respectively. The probabilities for the spatial step [Fig. 3(a)], also
computed here from the Dirac equation (see Supplementary Sec. 3.2.2), show the
well-known Klein paradox [65, 58], corresponding to the transmission gap in
the range $qV=[E-m,E+m]$ and increasing transmission with increasing potential
beyond the gap. In contrast, the probabilities for the temporal step [Fig.
3(b)] do not exhibit such a gap; they follow a monotonic trend of exchange
from forward propagation at low potentials to backward propagation at high
potentials. These observations interestingly suggest that a shifted Klein gap
may be expected in the case of a space-time (traveling) step. The asymptotic
response at high potentials ($qV/m,qA/m\gtrsim{5}$) is another fundamental
difference: while the temporal step is mostly “reflective” (backward-wave)
there, the spatial discontinuity is mostly transmissive, as a result of the
double reflection-transmission crossing due to the Klein effect. Otherwise,
the temporal step supports quasi-total forward transmission up to energies
($qA/m\approx{2}$) more than twice the cutoff of the quasi-total transmission
in the spatial case ($qV/m<1$) and a forward-backward crossing point
($qA/m\approx{3.4}$) almost identical to the transmission-reflection crossing
point in the spatial case ($qV/m\approx{3.2}$); these two observations
correspond to trends that are generally valid when the (incident) energy is
sufficient to produce a transition to the backward state, as understandable
from the dispersion diagram in Fig. 2(b).

Figure 3: Electron scattering probabilities versus potential strength
corresponding to Fig. 1 for (a) reflection and transmission at the spatial
step $V(z)$ [Fig. 1(a)] with $(V_{1},V_{2})=(0,V)$ and (b) later-backward and
later-forward propagation at the temporal step $A(t)$ [Fig. 1(b)] with
$(A_{1},A_{2})=(0,A)$, for the (incident) energy to rest mass ratio $E/m=2$.

## Smooth Temporal Potential Step

The infinitely sharp temporal potential step discussed so far is a canonical
structure, but it is not practically realizable, given its instantaneous
transition from $A_{1}$ to $A_{2}$ and corresponding electric field
singularity [$\mathbf{E}=-\partial\mathbf{A}/\partial{t}=-\delta(t-t_{0})$].
Note that the same unrealizability issue occurs in the infinitely sharp
spatial (scalar) potential discontinuity, given the dual contiguous transition
from $V_{1}$ to $V_{2}$ and corresponding electric field singularity
[$\mathbf{E}=-\nabla{V}=-\delta(z-z_{0})$]. What really matters then is to
determine whether the interesting physics predicted for the infinitely sharp
(unphysical) discontinuity survives as its transition is replaced by a smooth
one.
For this purpose, we choose a smooth-transition potential corresponding to the
hyperbolic tangent function

$\mathbf{A}(t)=\left[A_{1}+\frac{A_{2}-A_{1}}{2}\left(1+\tanh{\frac{t-t_{0}}{\eta}}\right)\right]\mathbf{\hat{z}},$
(8)

where the $\eta$ parameter is proportional to the transition time and whose
exact Dirac solution is derived in Supplementary Sec. 7. The corresponding
results are provided in Fig. 4, with Fig. 4(a) plotting the hyperbolic-tangent
potential and Fig. 4(b) plotting the scattering probabilities for three
representative transition times in terms of the “de Broglie period” of the
electron, $T_{\mathrm{dB}}$. In the sharpest case, $\eta=T_{\mathrm{dB}}/40$,
the scattering probabilities are indistinguishable from those for the
infinitely sharp discontinuity in Fig. 3(b), because the transition is _deeply
sub-period_ , the temporal dual regime of deep sub-wavelength. At
$\eta=T_{\mathrm{dB}}/4$, which may be considered as the temporal dual of the
(spatial) _Fabry-Pérot condition_ , the back-scattering level is less than
half of that in the former case. Finally, in the smoothest case,
$\eta=2T_{\mathrm{dB}}$, the transition has become so slow with respect to the
period, that the electron does not “see” it anymore, which results in zero
back-scattering.

Figure 4: Alternative problem of a smooth temporal potential step for three
different transition durations in terms of the de Broglie period,
$T_{\mathrm{dB}}$, for an electron with energy to rest mass ratio of $E/m=2$,
as in Fig. 3. (a) Hyperbolic tangent function [Eq. (8)] of the related
transition, between the potentials $A_{1}$ and $A_{2}$, over the time $\eta$.
(b) Corresponding later-backward and later-forward probabilities, $B$ and $F$.

The deeply sub-period regime ($\eta<T_{\mathrm{dB}}/10$) can unfortunately not
readily be attained in current technologies, because of the extremely short de
Broglie period, $T_{\mathrm{dB}}=h/E=h/(2mc^{2})\sim{4}\times{10}^{-21}$ s,
but it might be reachable soon, given recent spectacular progress in
attosecond lasers. Moreover, the present investigation can be easily extended
to Dirac-type materials, such as graphene, whose de Broglie period is much
smaller (e.g.,
$T_{\mathrm{dB}}^{\mathrm{graphene}}=(3a/2)/v_{\textrm{F}}\approx{3.7}\times{10}^{-16}$
s, where $a=0.246$ nm is the lattice constant and
$v_{\textrm{F}}\approx{10}^{6}$ m/s is the Fermi velocity).

## Discussion

### Relativistic Nature of Back-Scattering

The Dirac solutions in Eqs. (7) and Fig. 3(b) confirm the validity of the
later-forward wave and later-backward wave ansatz for the scattering potential
$\mathbf{A}=A(t)\mathbf{\hat{z}}$. In the non-relativistic regime, where
$\Gamma_{\textrm{t}}=1$ (see Supplementary Sec. 8), these solutions reduce to
$b=0$ and $f=1$, actually corresponding to the purely later-forward solution
of the Schrödinger equation. This fact reveals that later-backward scattering
is a _relativistic effect_. Note that the temporal step problem, which may be
seen as the infinite-velocity limit of a superluminal space-time modulation
medium [34], does not seem to be relativistic per se. Indeed, the
corresponding Lorentz factor is $\gamma=1/\sqrt{1-(v_{\mathrm{f}}/c)^{2}}$
with $v_{\mathrm{f}}=c^{2}/v_{\mathrm{m}}$, where $v_{\mathrm{f}}$ is the
velocity of the (instantaneous) frame and $v_{\mathrm{m}}$ is the velocity of
the modulation [66], so that $\gamma\stackrel{{\scriptstyle
v_{\mathrm{m}}\rightarrow\infty}}{{=}}1$ (no boost); it is really the speed of
the electron ($v$) (not that of the modulation ($v_{\mathrm{m}}$) in which it
propagates) that may be relativistic in our problem. The conditional
(relativistic) nature of the later-backward wave may a priori seem
contradictory, given that the electromagnetic-counterpart problem
unconditionally supports back-scattering [24, 34]. However, considering that
the particle (photon) in the latter case is inherently relativistic
($v_{\mathrm{photon}}=c$), whereas it is not necessarily in the former case
($v_{\mathrm{electron}}<c$), makes the finding a posteriori much less
surprising.

### Experimental Perspectives

#### Potential Generation

It is well-known that producing a magnetic vector potential ($\mathbf{A}$) to
our liking might be a difficult task. This is at least the case when
$\mathbf{A}$ is produced from a $\mathbf{B}$-field source, according to the
relation $\mathbf{B}=\nabla\times\mathbf{A}$, as in the experiment originally
proposed by Aharonov and Bohm [63] and later realized by Tonomura et al. [67,
68], where $\mathbf{A}$ is a distant effect of an enclosed $\mathbf{B}$ field,
theoretically requiring an infinite solenoid and producing an inconvenient
curved potential [63] or requiring a toroid with cumbersome superconductor
shielding [67, 68]. Fortunately, our interest here is not to produce a
potential in a field-free region, as in the Aharonov-Bohm effect, but just a
temporal step potential with a short transition, without any further specific
restriction. This might be realized from an $\mathbf{E}$-field source,
according to the relation $\mathbf{E}=-\partial\mathbf{A}/\partial{t}$.
Indeed, inserting Eq. (8) into this relation leads to the pulse function

$\mathbf{E}(t)=-\frac{(A_{2}-A_{1})}{2\eta}\sech^{2}\left(\frac{t-t_{0}}{\eta}\right)\mathbf{\hat{z}},$
(9)

which may be produced by an ultrashort-pulse laser [69] to provide the desired
potential step function [Eq. (8)] – collocated and aligned with the electric
field. Achieving maximal back-scattering as in Fig. 3(b) requires, as shown in
the previous section, a sub-period ($\eta<T_{\mathrm{dB}}/10$) pulse, but a
broader (larger $\eta$) pulse, simply generating a smoother step, may still
produce some back-scattering, as shown in Fig. 4.

#### Measurement

Once the potential has been generated, as just described, electrons should be
shot by an electron gun parallel to the electric field (perpendicular to the
laser beam axis) and an appropriate detection procedure should be used to
measure the scattering probabilities predicted by Eqs. (7b) [Fig. 3(b)]. That
detection procedure might be delicate because the physical interpretation of
Dirac spinors [Eq. (2)] and related quantities is not trivial: while the
operators (position, momentum, energy, spin, etc.) associated with the
Schrödinger equation directly correspond to observables, those associated with
the Dirac equation do not! However, the _scattering probabilities_ to measure
here are not problematic. Their magnitudes can be obtained by placing an
electron counter in the forward region of the setup, measuring the related
electron count, $N_{\mathrm{F}}$, and deducing the corresponding electron
count in the backward direction as
$N_{\mathrm{B}}=N_{\mathrm{gun}}-N_{\mathrm{F}}$, to obtain
$F_{\mathrm{meas}}=N_{\mathrm{F}}/N_{\mathrm{gun}}$ and
$B_{\mathrm{meas}}=N_{\mathrm{B}}/N_{\mathrm{gun}}$. Moreover, their energy
levels, predicted by Eqs. (6) [Fig. 2(d)], can be obtained via the phase
differences
$\Delta\phi_{\mathrm{f,b}}=-\left(E_{\mathrm{i}}-E_{\mathrm{f,b}}\right)t/\hbar$
measured by an electron interferometer.

### Summary and Outlook

In this report, we have resolved the fundamental problem of electron
scattering at a potential temporal step discontinuity, with a systematic
comparison to the spatial counterpart of the problem and mention of
similarities and differences with the classical electromagnetic counterparts
of the two problems. The related effects described in this report might lead
to a wide range of new concepts and applications in semiconductor electronics,
quantum computing and information processing, and attosecond physics. A simple
application would be a versatile (aligned-outputs) beam splitter, with tunable
splitting ratio and splitting angle, consisting of a rotatable laser with
varying intensity, where the latter controls the splitting ratio, according to
Fig. 3(b), and the former controls the output splitting direction.

## Data availability

All data generated or analyzed during this study are included in this
published article and its supplementary information files.

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## Acknowledgements

This work is supported by FWO under grant G0B0623N.

## Author contributions

F.O. conducted the calculations. F.O. and A.B. prepared the figures. C.C.
wrote the manuscript, integrating scientific insights from F.O., A.B., and
C.C. All authors reviewed the manuscript. C.C. supervised the overall process.

## Additional information

The authors declare no competing interests.

## Supplementary Material for “Electron Scattering at a Temporal Step
Discontinuity"

###### Contents

  1. 1 Types of Scalar and Vector Potential Spatial and Temporal Steps
  2. 2 The Dirac Equation
  3. 3 Solution to the Dirac Equations
    1. 3.1 General Solution in the Presence of Both Scalar and Vector Potentials
    2. 3.2 Particular Solutions in the Presence of Specific Potentials
      1. 3.2.1 Scalar Potential Temporal Step, $V(t)$
      2. 3.2.2 Scalar Potential Spatial Step, $V(z)$
      3. 3.2.3 Vector Potential Spatial Step, $A(z)$
      4. 3.2.4 Vector Potential Temporal Step, $A(t)$
  4. 4 Gauge Transformations and Symmetries
    1. 4.1 Scalar Potential Spatial Step $V(z)$
    2. 4.2 Vector Potential Spatial Step $A(z)$
  5. 5 Phase and Group Velocities
  6. 6 Spatial and Temporal Step Electromagnetic Problems
    1. 6.1 Spatial Step Problem
    2. 6.2 Temporal Step Problem
  7. 7 Smooth Temporal Step
    1. 7.1 Dirac Equation for a Time-Varying Vector Potential
    2. 7.2 Solution for the Earlier Medium ($t<t_{0}$)
    3. 7.3 Solution for the Later Medium ($t>t_{0}$)
    4. 7.4 Asymptotic Forms
      1. 7.4.1 Asymptotic Forms for the Earlier Medium ($t<t_{0}$)
      2. 7.4.2 Asymptotic Forms for the Later Medium ($t>t_{0}$)
    5. 7.5 Weyl to Dirac Spinor Transformation
    6. 7.6 Boundary Conditions
    7. 7.7 Asymptotic Spinor Wavefunctions and Scattering Probabilities
    8. 7.8 Dimensional Analysis for Time Constant $\tau$
  8. 8 Non-relativistic Regime
    1. 8.1 Non-relativistic Limit of the Dirac Equation
    2. 8.2 Numerical Example for the Relativistic Regime
    3. 8.3 Spatial Scattering Coefficients in the Non-relativistic Regime
    4. 8.4 Temporal Scattering Coefficients in the Non-relativistic Regime

##  1 Types of Scalar and Vector Potential Spatial and Temporal Steps

Figure 5 depicts the different scalar and vector potential spatial and
temporal steps considered in the paper.

Figure 5: Types of potential steps considered. (a) Scalar potential spatial
step, $V(z)$. (b) Scalar potential temporal step, $V(t)$. (c) Vector potential
spatial step, $A(z)$. (d) Vector potential temporal step, $A(t)$.

##  2 The Dirac Equation

The Dirac equation, in natural units ($\hbar=c=1$), reads [10, 58]

$i\partialderivative{t}\psi=\mathcal{\hat{H}}\psi,$ (10a) where $\psi$ is the
($4\times{}1$) Dirac spinor wavefunction and $\mathcal{\hat{H}}$ is the
($4\times{}4$) Hamiltonian, which is in the free (fermion) particle case
$\mathcal{\hat{H}}=-i\alpha^{i}\partial_{i}+\gamma^{0}m.$ (10b) This
Hamiltonian is composed of the following elements:

  * •

the ($4\times{}4$) $\alpha^{i}$ matrices

$\alpha^{i}=\begin{pmatrix}0&\sigma^{i}\\\
\sigma^{i}&0\end{pmatrix},\quad\text{with}\quad i=1,2,3,$ (10c)

whose components are the Pauli matrices

$\sigma^{1}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\quad\sigma^{2}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix}\quad\text{and}\quad\sigma^{3}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix},$ (10d)

  * •

the four-gradient

$\partial_{\mu}\equiv\left(\partial_{0},\partial_{i}\right),\quad\text{with}\quad
i=1,2,3,$ (10e)

  * •

the ($4\times{}4$) $\gamma^{0}$ matrix (Dirac-Pauli representation)

$\gamma^{0}=\begin{pmatrix}I&0\\\ 0&-I\end{pmatrix},\quad\text{where}\quad
I=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix},$ (10f)

  * •

the particle mass, $m$.

Inserting Eq. (10b) into Eq. (10a) multiplied by $\gamma^{0}$ yields

$i\gamma^{0}\partial_{0}\psi=\left(-i\gamma^{0}\alpha^{i}\partial_{i}+\gamma^{0}\gamma^{0}m\right)\psi.$
(11)

Then combining Eqs. (10f) and (10c), which gives

$\gamma^{0}\alpha^{i}=\begin{pmatrix}0&\sigma^{i}\\\
-\sigma^{i}&0\end{pmatrix}=\gamma^{i},$ (12)

and noting that $\gamma^{0}\gamma^{0}=1$, simplifies Eq. (11) to

$i\gamma^{0}\partial_{0}\psi=\left(-i\gamma^{i}\partial_{i}+m\right)\psi.$
(13)

Finally, moving the right-hand side terms of this equation to its left-hand
side provides the explicit form of the free-particle Dirac equation,

$\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi=0,\quad\text{with}\quad\mu=0,1,2,3,$
(14)

which assumes the Einstein summation convention, whereby the repetition of an
index in a given term implies summation over that index. Note that Eq. (14)
represents four scalar equations via the $\gamma^{\mu}$ matrices.

When the particle is subjected to an electromagnetic field, expressed in terms
of the four-potential [61]

$A_{\mu}=\left(A_{0},-\mathbf{A}\right),$ (15)

the Dirac equation (14) transforms according to the minimal-coupling
prescription [59]

$\partial_{\mu}\rightarrow\partial_{\mu}+iqA_{\mu},$ (16)

viz.,

$\left[\gamma^{\mu}(i\partial_{\mu}-qA_{\mu})-m\right]\psi=0,$ (17)

where $q$ is the charge of the particle.

The Dirac equation (17) may then be solved with the general ansatz

$\psi=\begin{pmatrix}\varphi\\\ \vartheta\end{pmatrix}\mathrm{e}^{\mp{}ip\cdot
x},$ (18)

where $\varphi$ and $\vartheta$ are two-component ($2\times{}1$) spinors and

$p\cdot x=p^{\mu}x_{\mu}=\eta^{\mu\sigma}p_{\sigma}x_{\mu},$ (19)

with

$\eta^{\mu\sigma}=\begin{pmatrix}1&0&0&0\\\ 0&-1&0&0\\\ 0&0&-1&0\\\
0&0&0&-1\end{pmatrix}$ (20)

being the Minkowski metric (West Coast convention). That the ansatz (18) is
correct will be verified in Sec. 3. In Eq. (19),

$p_{\mu}=(E,-\mathbf{p})$ (21)

is the momentum four-vector and

$x_{\mu}=(t,-\mathbf{x})$ (22)

is the position four-vector. The expression
$p\cdot{x}=Et-\mathbf{p}\cdot\mathbf{x}$ is then identified as the phase of a
plane wave, which reveals that the solution form in Eq. (18) corresponds to a
plane-wave solution. We see then that two signs in that equation [Eq. (18)]
correspond to positive and negative energies, which Dirac identified as
corresponding to the particle (e.g., electron) and its antiparticle (e.g.,
positron). Assuming that we are exclusively dealing with the former type, we
shall ignore negative energies and hence drop the positive sign in the spinor
waveform (18). Thus, further restricting our attention to the $1+1$
dimensional case, with the spatial dimension being $z$, we find that the
ansatz (18) reduces to the specific form

$\psi=\begin{pmatrix}\varphi\\\ \vartheta\end{pmatrix}\mathrm{e}^{-i(Et-pz)}.$
(23)

##  3 Solution to the Dirac Equations

###  3.1 General Solution in the Presence of Both Scalar and Vector Potentials

Substituting the ansatz (23) into the Dirac equation (17) gives

$\left[\gamma^{\mu}(i\partial_{\mu}-qA_{\mu})-m\right]\begin{pmatrix}\varphi\\\
\vartheta\end{pmatrix}\mathrm{e}^{-i(Et-pz)}=0.$ (24a) Expanding this relation
according to the Einstein summation convention yields
$\left[\gamma^{0}(i\partial_{0}-qA_{0})+\gamma^{3}(i\partial_{3}+qA_{3})-m\right]\begin{pmatrix}\varphi\\\
\vartheta\end{pmatrix}\mathrm{e}^{-i(Et-pz)}=0,$ (24b) which becomes after
applying the derivative operators to the exponential term
$\left[\gamma^{0}(E-qA_{0})-\gamma^{3}(p-qA_{3})-m\right]\begin{pmatrix}\varphi\\\
\vartheta\end{pmatrix}\mathrm{e}^{-i(Et-pz)}=0.$ (24c) Inserting now the gamma
matrices $\gamma^{0}$ and $\gamma^{3}$ from Eqs. (10f) and (12), respectively,
into this relation and dropping the exponential function, leads to
$\left[\begin{pmatrix}I&0\\\
0&-I\end{pmatrix}(E-qA_{0})-\begin{pmatrix}0&\sigma^{3}\\\
-\sigma^{3}&0\end{pmatrix}(p-qA_{3})-m\right]\begin{pmatrix}\varphi\\\
\vartheta\end{pmatrix}=0,$ (24d)

which splits into the two equations

$(E-qA_{0})\varphi-(p-qA_{3})\sigma^{3}\vartheta-m\varphi=0$ (25a) and
$-(E-qA_{0})\vartheta+(p-qA_{3})\sigma^{3}\varphi-m\vartheta=0.$ (25b)

Separately solving these equations for $\varphi$ yields then

$\varphi=\frac{(p-qA_{3})\sigma^{3}}{E-qA_{0}-m}\vartheta$ (26a) and
$\varphi=\frac{E-qA_{0}+m}{(p-qA_{3})\sigma^{3}}\vartheta.$ (26b)

Equating these two relations and using the identity $(\sigma^{3})^{2}=I$ that
follows from Eq. (10d), entails the Dirac energy-momentum or dispersion
relation

$(E-qA_{0})^{2}=(p-qA_{3})^{2}+m^{2},$ (27)

which reduces for $A_{0}=A_{3}=0$ to Einstein’s equation
$E^{2}=p^{2}+m^{2}=(\gamma m)^{2}$, where $\gamma=(1-v^{2})^{-1/2}$ is the
Lorentz factor, with $v$ begin the velocity of the particle.

At this point, we may select either the spin-up solution or the spin-down
solution, which correspond to the spinors

$\varphi_{\uparrow}=\begin{pmatrix}1\\\ 0\end{pmatrix}$ (28a) and
$\varphi_{\downarrow}=\begin{pmatrix}0\\\ 1\end{pmatrix},$ (28b)

respectively. Choosing the spin-up solution and correspondingly inserting Eq.
(28a) into Eq. (26a), yields

$\begin{pmatrix}1\\\
0\end{pmatrix}=\frac{p-qA_{3}}{E-qA_{0}-m}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\begin{pmatrix}\vartheta_{1}\\\ \vartheta_{2}\end{pmatrix}$
(29a) or $\begin{pmatrix}1\\\
0\end{pmatrix}=\frac{p-qA_{3}}{E-qA_{0}-m}\begin{pmatrix}\vartheta_{1}\\\
-\vartheta_{2}\end{pmatrix},$ (29b)

which determines the spinor as $\vartheta$ as

$\vartheta=\begin{pmatrix}\vartheta_{1}\\\
\vartheta_{2}\end{pmatrix}=\begin{pmatrix}\frac{E-qA_{0}-m}{p-qA_{3}}\\\
0\end{pmatrix}.$ (30)

Finally, substituting Eq. (28a) and Eq. (30) into (23) yields the solution
form

$\psi=\begin{pmatrix}1\\\ 0\\\ \frac{E-qA_{0}-m}{p-qA_{3}}\\\
0\end{pmatrix}\mathrm{e}^{-i(Et-pz)}.$ (31)

Note that the alternative choice of Eq. (26b) would have led to the third
element of this spinor being replaced by $\frac{p-qA_{3}}{E-qA_{0}+m}$, which
can easily be shown to be equal to $\frac{E-qA_{0}-m}{p-qA_{3}}$. Further note
that the alternative choice of the spin-down solution [Eq. (28b)] would have
led to the entries $1$ and $\frac{E-qA_{0}-m}{p-qA_{3}}$ being replaced by
zero and shifted to the second and fourth slots of the spinor, respectively,
with the sign of the latter changed.

###  3.2 Particular Solutions in the Presence of Specific Potentials

We shall now apply the general solution in Eq. (31) to the different step
potential introduced in Sec. 1.

####  3.2.1 Scalar Potential Temporal Step, $V(t)$

The scalar potential temporal step, shown in Fig. 5(b), may be written

$A_{0}=V(t)=\begin{cases}V_{1}&\;\textrm{for}\;t<t_{0},\\\
V_{2}&\;\textrm{for}\;t>t_{0},\end{cases}\quad\textrm{and}\quad A_{3}=0.$ (32)

Substituting this potential into the solution form (31) and using ansätze
corresponding to the related temporal-step electromagnetic solutions [24, 34],
we assume the incident, or earlier ($t<t_{0}$), spinor wavefunction

$\psi_{1}=\begin{pmatrix}1\\\ 0\\\
\frac{E_{\mathrm{i}}-qV_{1}-m}{p_{\mathrm{i}}}\\\
0\end{pmatrix}\mathrm{e}^{-iE_{\mathrm{i}}t}\mathrm{e}^{ip_{\mathrm{i}}z}$
(33a) and the later ($t>t_{0}$) spinor wavefunction
$\psi_{2}=f\begin{pmatrix}1\\\ 0\\\
\frac{E_{\mathrm{f}}-qV_{2}-m}{p_{\mathrm{f}}}\\\
0\end{pmatrix}\mathrm{e}^{-iE_{\mathrm{f}}t}\mathrm{e}^{ip_{\mathrm{f}}z}+b\begin{pmatrix}1\\\
0\\\ \frac{E_{\mathrm{b}}-qV_{2}-m}{p_{\mathrm{b}}}\\\
0\end{pmatrix}\mathrm{e}^{-iE_{\mathrm{b}}t}\mathrm{e}^{ip_{\mathrm{b}}z},$
(33b) whose first and second terms correspond to later-forward and later-
backward waves, respectively, with corresponding amplitude coefficients $f$
and $b$.

According to Noether’s theorem [64], momentum is conserved ($\Delta{p}=0$) due
to spatial translational symmetry, viz.,
$p_{\mathrm{i}}=p_{\mathrm{f}}=p_{\mathrm{b}}=p,$ (34a) whereas the breaking
of temporal symmetry entails energy transformations, which are found from Eq.
(27) with $A_{0}=V(t)$ and $A_{3}=0$ to be
$E_{\mathrm{i}}=\sqrt{p^{2}+m^{2}}+qV_{1},$ (34b)
$E_{\mathrm{f}}=\sqrt{p^{2}+m^{2}}+qV_{2}$ (34c) and
$E_{\mathrm{b}}=-E_{\mathrm{f}}+2qV_{2},$ (34d) where the apparent negative
energy in the last relation simply represents propagation in the negative $z$
direction, with positive energy.

According to Eq. (17), the spinor wavefunction must be continuous at the
temporal discontinuity, viz.,

$\left.\psi_{1}\right|_{t=t_{0}}=\left.\psi_{2}\right|_{t=t_{0}}.$ (35)

Inserting Eqs. (33) into this relation yields

$1=f+b$ (36a) and
$\frac{E_{\mathrm{i}}-qV_{1}-m}{p_{\mathrm{i}}}=f\frac{E_{\mathrm{f}}-qV_{2}-m}{p_{\mathrm{f}}}+b\frac{E_{\mathrm{b}}-qV_{2}-m}{p_{\mathrm{b}}}.$
(36b)

Substituting then Eqs. (34) into Eq. (36b), and solving the system formed by
the resulting equation and Eq. (36a), finally leads, after some algebraic
manipulations, to

$f=1\quad\text{ and }\quad b=0,$ (37)

which reveals that the scalar potential temporal step, $V(t)$, does not
produce any back-scattering.

####  3.2.2 Scalar Potential Spatial Step, $V(z)$

The scalar potential spatial step, shown in Fig. 5(a), may be written

$A_{0}=V(z)=\begin{cases}V_{1}&\;\textrm{for}\;z<z_{0},\\\
V_{2}&\;\textrm{for}\;z>z_{0},\end{cases}\quad\textrm{and}\quad A_{3}=0.$ (38)

Substituting this potential into the solution form (31) and using ansätze
corresponding to the related spatial-step electromagnetic solutions [61, 34],
we assume the left ($z<z_{0}$) spinor wavefunction

$\psi_{1}=\begin{pmatrix}1\\\ 0\\\
\frac{E_{\mathrm{i}}-qV_{1}-m}{p_{\mathrm{i}}}\\\
0\end{pmatrix}\mathrm{e}^{-iE_{\mathrm{i}}t}\mathrm{e}^{ip_{\mathrm{i}}z}+r\begin{pmatrix}1\\\
0\\\ \frac{E_{\mathrm{r}}-qV_{1}-m}{p_{\mathrm{r}}}\\\
0\end{pmatrix}\mathrm{e}^{-iE_{\mathrm{r}}t}\mathrm{e}^{ip_{\mathrm{r}}z},$
(39a) whose first and second terms correspond to incident and reflected waves,
respectively, with reflection amplitude coefficient $r$, and the transmitted,
or right ($z>z_{0}$), spinor wavefunction $\psi_{2}=t\begin{pmatrix}1\\\ 0\\\
\frac{E_{\mathrm{t}}-qV_{2}-m}{p_{\mathrm{t}}}\\\
0\end{pmatrix}\mathrm{e}^{-iE_{\mathrm{t}}t}\mathrm{e}^{ip_{\mathrm{t}}z},$
(39b)

with transmission amplitude coefficient $t$.

According to Noether’s theorem [64], energy is conserved ($\Delta{E}=0$) due
to temporal translational symmetry, viz.,
$E_{\mathrm{i}}=E_{\mathrm{r}}=E_{\mathrm{t}}=E,$ (40a) whereas the breaking
of spatial symmetry implies momentum transformations, which are found from Eq.
(27) with $A_{0}=V(z)$ and $A_{3}=0$ to be
$p_{\mathrm{i}}=\sqrt{(E-qV_{1})^{2}-m^{2}},$ (40b)
$p_{\mathrm{r}}=-p_{\mathrm{i}}$ (40c) and
$p_{\mathrm{t}}=\sqrt{(E-qV_{2})^{2}-m^{2}},$ (40d) where the apparent
negative momentum in Eq. (40c) simply represents propagation in the negative
$z$ direction, with positive momentum, whose momentum versus $p_{\mathrm{i}}$
are obtained by solving Eq. (40b) for $E$ and inserting the result into Eqs.
(40c) and (40d), which yields
$p_{\mathrm{t}}=\sqrt{\left(\sqrt{p_{\mathrm{i}}^{2}+m^{2}}-q\Delta{V}\right)^{2}-m^{2}}.$
(40e)

According to Eq. (17), the spinor wavefunction must be continuous at the
spatial discontinuity, viz.,

$\left.\psi_{1}\right|_{z=z_{0}}=\left.\psi_{2}\right|_{z=z_{0}}.$ (41)

Inserting Eqs. (39) in this relation yields

$1+r=t$ (42a) and
$\frac{E_{\mathrm{i}}-qV_{1}-m}{p_{\mathrm{i}}}+r\frac{E_{\mathrm{r}}-qV_{1}-m}{p_{\mathrm{r}}}=t\frac{E_{\mathrm{t}}-qV_{2}-m}{p_{\mathrm{t}}}.$
(42b)

Substituting then Eqs. (40) into Eq. (42b), and solving the system formed by
the resulting equation and Eq. (42a), finally leads, after some algebraic
manipulations, to

$r=\frac{1-\Gamma_{\mathrm{s}}}{1+\Gamma_{\mathrm{s}}}\quad\text{ and }\quad
t=\frac{2}{1+\Gamma_{\mathrm{s}}},$ (43a) where
$\Gamma_{\mathrm{s}}=\frac{{(E-qV_{2}-m)}{p_{\mathrm{i}}}}{{{(E-qV_{1}-m)}{p_{\mathrm{t}}}}}.$
(43b)

The probabilities associated with the reflected and transmitted waves may then
be found from the ratios of the corresponding components of the conserved
Dirac current. The Dirac conserved current is

$j^{\mu}=\bar{\psi}\gamma^{\mu}\psi,$ (44)

where $\bar{\psi}$ is the Dirac adjoint,

$\bar{\psi}=\psi^{\dagger}\gamma^{0},$ (45)

with $\psi^{\dagger}$ being the Hermitian conjugate of $\psi$. In our problem
[Fig. 5(a)], $j^{\mu}$ has only a $z$ spatial component, and takes then the
forms

$j^{z}_{1}=\psi^{\dagger}_{1}\gamma^{0}\gamma^{3}\psi_{1}$ (46a) and
$j^{z}_{2}=\psi^{\dagger}_{2}\gamma^{0}\gamma^{3}\psi_{2},$ (46b)

in the left and right regions, respectively. Substituting Eqs. (39), (10f) and
(12) into these relations yields

$j^{z}_{1}=j^{z}_{\mathrm{i}}-j^{z}_{\mathrm{r}},$ (47a) with
$j^{z}_{\mathrm{i}}=2\frac{E-qV_{1}-m}{p_{\mathrm{i}}}\quad\text{and}\quad
j^{z}_{\mathrm{r}}=2|r|^{2}\frac{E-qV_{1}-m}{p_{\mathrm{i}}},$ (47b) and
$j^{z}_{2}=j^{z}_{\mathrm{t}},$ (47c) with
$j^{z}_{\mathrm{t}}=2|t|^{2}\frac{E-qV_{2}-m}{p_{\mathrm{t}}}.$ (47d)

The reflection and the transmission probabilities are then obtained as

$R=\left|\frac{j^{z}_{\mathrm{r}}}{j^{z}_{\mathrm{i}}}\right|=|r|^{2}=\left|\frac{1-\Gamma_{\mathrm{s}}}{1+\Gamma_{\mathrm{s}}}\right|^{2}$
(48a) and
$T=\left|\frac{j^{z}_{\mathrm{t}}}{j^{z}_{\mathrm{i}}}\right|=|t|^{2}\Gamma_{\mathrm{s}}=\left|\frac{2}{1+\Gamma_{\mathrm{s}}}\right|^{2}\Gamma_{\mathrm{s}},$
(48b)

where $\Gamma_{\mathrm{s}}$ was defined in Eqs. (43b). Note that these
probabilities verify the probability conservation formula

$R+T=1.$ (49)

An alternative, pragmatic way to determine the probabilities (48) is to write

$R=|r|^{2}\mathcal{C}_{R}\quad\text{and}\quad T=|t|^{2}\mathcal{C}_{T},$ (50)

where the parameters $\mathcal{C}_{R}$ and $\mathcal{C}_{T}$ are “momentum-
transition” coefficients, associated with change of region (from $V_{1}$ to
$V_{2}$). Therefore, we must have $\mathcal{C}_{R}=1$ and $\mathcal{C}_{T}\neq
1$. The coefficient $\mathcal{C}_{T}$ may then be determined from the
probability conservation, specifically by inserting $R=|r|^{2}$ and
$T=|t|^{2}\mathcal{C}_{T}$ with Eq. (43) into Eq. (49), which leads to
$\mathcal{C}_{T}=\Gamma_{\mathrm{s}}$ and hence retrieves the results in Eqs.
(48).

One may distinguish three potential regions in plotting $R$ and $T$, assuming
$(E-qV_{1})>m$ so that $p_{\mathrm{i}}$ [Eq. (40b)] is real, depending on
$p_{\mathrm{t}}$ [Eq. (40d)] [58]:

  1. 1.

$qV_{2}<E-m$: $p_{\mathrm{t}}=$ real and hence
$\Gamma_{\mathrm{s}}=\frac{{(E-qV_{2}-m)}{\sqrt{(E-qV_{1})^{2}-m^{2}}}}{{{(E-qV_{1}-m)}{\sqrt{(E-qV_{2})^{2}-m^{2}}}}}$;

  2. 2.

$E-m<qV_{2}<E+m$: $p_{\mathrm{t}}=$ is imaginary and hence
$\Gamma_{\mathrm{s}}=\frac{{(E-qV_{2}-m)}{\sqrt{(E-qV_{1})^{2}-m^{2}}}}{{{(E-qV_{1}-m)}{(i)\sqrt{-(E-qV_{2})^{2}+m^{2}}}}}$;

  3. 3.

$E+m<qV_{2}$: $p_{\mathrm{t}}=$ real. However, if we choose
$p_{\mathrm{t}}>0$, then, according to Eq. (40d),
$v_{\mathrm{g}}=dE/dp_{\mathrm{t}}=\frac{p_{\mathrm{t}}}{E-qV_{2}}<0$ since
$E-qV_{2}<0$, which is contradictory to the assumption of positive-$z$
propagation for the transmitted wave. Therefore, in order to maintain a
positive group velocity, we must assign a negative sign to $p_{\mathrm{t}}$,
viz., use
$\Gamma_{\mathrm{s}}=\frac{{(E-qV_{2}-m)}{\sqrt{(E-qV_{1})^{2}-m^{2}}}}{{{(E-qV_{1}-m)}{(-)\sqrt{(E-qV_{2})^{2}-m^{2}}}}}$.

####  3.2.3 Vector Potential Spatial Step, $A(z)$

The vector potential spatial step, shown in Fig. 5(c), may be written

$A_{3}=A(z)=\begin{cases}A_{1}&\;\textrm{for}\;z<z_{0},\\\
A_{2}&\;\textrm{for}\;z>z_{0},\end{cases}\quad\textrm{and}\quad A_{0}=0.$ (51)

Substituting this potential into the solution form (31) and using ansätze
corresponding to the related spatial-step electromagnetic solutions [61, 34],
we assume the left ($z<z_{0}$) spinor wavefunction

$\psi_{1}=\begin{pmatrix}1\\\ 0\\\
\frac{E_{\mathrm{i}}-m}{p_{\mathrm{i}}-qA_{1}}\\\
0\end{pmatrix}\mathrm{e}^{-iE_{\mathrm{i}}t}\mathrm{e}^{ip_{\mathrm{i}}z}+r\begin{pmatrix}1\\\
0\\\ \frac{E_{\mathrm{r}}-m}{p_{\mathrm{r}}-qA_{1}}\\\
0\end{pmatrix}\mathrm{e}^{-iE_{\mathrm{r}}t}\mathrm{e}^{ip_{\mathrm{r}}z},$
(52a) whose first and second terms correspond to incident and reflected waves,
respectively, with reflection amplitude coefficient $r$, and the transmitted,
or right ($z>z_{0}$), spinor wavefunction $\psi_{2}=t\begin{pmatrix}1\\\ 0\\\
\frac{E_{\mathrm{t}}-m}{p_{\mathrm{t}}-qA_{2}}\\\
0\end{pmatrix}\mathrm{e}^{-iE_{\mathrm{t}}t}\mathrm{e}^{ip_{\mathrm{t}}z},$
(52b)

with transmission amplitude coefficient $t$.

According to Noether’s theorem [64], energy is conserved ($\Delta{E}=0$) due
to temporal translational symmetry, viz.,
$E_{\mathrm{i}}=E_{\mathrm{r}}=E_{\mathrm{t}}=E,$ (53a) whereas the breaking
of spatial symmetry implies momentum transformations, which are found from Eq.
(27) with $A_{3}=A(z)$ and $A_{0}=V=0$ to be
$p_{\mathrm{i}}=\sqrt{E^{2}-m^{2}}+qA_{1},$ (53b)
$p_{\mathrm{r}}=-p_{\mathrm{i}}+2qA_{1}$ (53c) and
$p_{\mathrm{t}}=\sqrt{E^{2}-m^{2}}+qA_{2},$ (53d) where the apparent negative
momentum in Eq. (53c) simply represents propagation in the negative $z$
direction, with positive momentum.

According to Eq. (17), the spinor wavefunction must be continuous at the
spatial discontinuity, viz.,

$\left.\psi_{1}\right|_{z=z_{0}}=\left.\psi_{2}\right|_{z=z_{0}}.$ (54)

Inserting Eqs. (52) in this relation yields

$1+r=t$ (55a) and
$\frac{E_{\mathrm{i}}-m}{p_{\mathrm{i}}-qA_{1}}+r\frac{E_{\mathrm{r}}-m}{p_{\mathrm{r}}-qA_{1}}=t\frac{E_{\mathrm{t}}-m}{p_{\mathrm{t}}-qA_{2}}.$
(55b)

Substituting then Eqs. (53) into Eq. (55b), and solving the system formed by
the resulting equation and Eq. (55a), finally leads, after some algebraic
manipulations, to

$r=0\quad\text{ and }\quad t=1,$ (56)

which reveals that the vector potential spatial step, $A(z)$, does not produce
any back-scattering.

####  3.2.4 Vector Potential Temporal Step, $A(t)$

The vector potential temporal step, shown in Fig. 5(d), may be written

$A_{3}=A(t)=\begin{cases}A_{1}&\;\textrm{for}\;t<t_{0},\\\
A_{2}&\;\textrm{for}\;t>t_{0},\end{cases}\quad\textrm{and}\quad A_{0}=V=0.$
(57)

Substituting this potential into the solution form (31) and using ansätze
corresponding to the related temporal-step electromagnetic solutions [24, 34],
we assume the incident, or earlier ($t<t_{0}$), spinor wavefunction

$\psi_{1}=\begin{pmatrix}1\\\ 0\\\
\frac{E_{\mathrm{i}}-m}{p_{\mathrm{i}}-qA_{1}}\\\
0\end{pmatrix}\mathrm{e}^{-iE_{\mathrm{i}}t}\mathrm{e}^{ip_{\mathrm{i}}z}$
(58a) and the later ($t>t_{0}$) spinor wavefunction
$\psi_{2}=f\begin{pmatrix}1\\\ 0\\\
\frac{E_{\mathrm{f}}-m}{p_{\mathrm{f}}-qA_{2}}\\\
0\end{pmatrix}\mathrm{e}^{-iE_{\mathrm{f}}t}\mathrm{e}^{ip_{\mathrm{f}}z}+b\begin{pmatrix}1\\\
0\\\ \frac{E_{\mathrm{b}}-m}{p_{\mathrm{b}}-qA_{2}}\\\
0\end{pmatrix}\mathrm{e}^{-iE_{\mathrm{b}}t}\mathrm{e}^{ip_{\mathrm{b}}z},$
(58b) whose first and second terms correspond to later-forward and later-
backward waves, respectively, with corresponding amplitude coefficients $f$
and $b$.

According to Noether’s theorem [64], momentum is conserved ($\Delta{p}=0$) due
to spatial translational symmetry, viz.,
$p_{\mathrm{i}}=p_{\mathrm{f}}=p_{\mathrm{b}}=p,$ (59a) whereas the breaking
of spatial symmetry implies energy transformations, which are found from Eq.
(27) with $A_{3}=A(t)$ and $A_{0}=V=0$ to be
$E_{\mathrm{i}}=\sqrt{(p-qA_{1})^{2}+m^{2}},$ (59b)
$E_{\mathrm{f}}=\sqrt{(p-qA_{2})^{2}+m^{2}}$ (59c) and
$E_{\mathrm{b}}=-E_{\mathrm{f}},$ (59d) where the apparent negative energy in
the last relation simply represents propagation in the negative $z$ direction,
with positive energy.

According to Eq. (17), the spinor wavefunction must be continuous at the
temporal discontinuity, viz.,

$\left.\psi_{1}\right|_{t=t_{0}}=\left.\psi_{2}\right|_{t=t_{0}}.$ (60)

Inserting Eqs. (33) into this relation yields

$1=f+b$ (61a) and
$\frac{E_{\mathrm{i}}-m}{p_{\mathrm{i}}-qA_{1}}=f\frac{E_{\mathrm{f}}-m}{p_{\mathrm{f}}-qA_{2}}+b\frac{E_{\mathrm{b}}-m}{p_{\mathrm{b}}-qA_{2}}.$
(61b)

Substituting then Eqs. (59) into Eq. (61b), and solving the system formed by
the resulting equation and Eq. (61a), finally leads, after some algebraic
manipulations, to

$f=\frac{1+\Gamma_{\mathrm{t}}}{2\Gamma_{\mathrm{t}}}\quad\text{ and }\quad
b=\frac{\Gamma_{\mathrm{t}}-1}{2\Gamma_{\mathrm{t}}}$ (62a) where
$\Gamma_{\mathrm{t}}=\dfrac{\dfrac{E_{\mathrm{f}}}{p-qA_{2}}}{\dfrac{E_{\mathrm{i}}-m}{p-qA_{1}}+\dfrac{m}{p-qA_{2}}}.$
(62b) This expression may be alternatively written in terms of
$E_{\mathrm{i}}$ and $A_{1,2}$ only upon first using Eq. (59c) to eliminate
$E_{\mathrm{f}}$ and then substituting in the resulting expression
$p=\sqrt{E_{\mathrm{i}}^{2}-m^{2}}+qA_{1},$ (62c) which was obtained from Eq.
(59c), which yields
$\Gamma_{\mathrm{t}}=\dfrac{\sqrt{\left(\sqrt{E_{\mathrm{i}}^{2}-m^{2}}-(qA_{2}-qA_{1})\right)^{2}+m^{2}}}{\left(\sqrt{E_{\mathrm{i}}^{2}-m^{2}}-(qA_{2}-qA_{1})\right)\left(\dfrac{E_{\mathrm{i}}-m}{\sqrt{E_{\mathrm{i}}^{2}-m^{2}}}\right)+m}.$
(62d)

The probabilities associated with the later-forward and later-backward waves
cannot be found from the ratios of the corresponding Dirac currents, contrary
to the case of the $V(z)$ problem in Sec. 3.2.1, because the Dirac current is
_not_ conserved here, due to the non-conservation of energy. However, we may
resort to an alternative approach similar to that also used in Sec. 3.2.1,
writing

$F=|f|^{2}\mathcal{C}_{F}\quad\text{and}\quad B=|b|^{2}\mathcal{C}_{B},$ (63)

where the parameters $\mathcal{C}_{F}$ and $\mathcal{C}_{B}$ are now “energy-
transition” coefficients, associated with change of region (from $A_{1}$ to
$A_{2}$). Therefore, we must have
$\mathcal{C}_{F}=\mathcal{C}_{B}=\mathcal{C}$, since the two probabilities
correspond to the same change of region, from the earlier region ($A_{1}$) to
the later region ($A_{2}$), so that

$F=|f|^{2}\mathcal{C}\quad\text{and}\quad B=|b|^{2}\mathcal{C}.$ (64)

At the same, probability must be conserved, since the particle can only either
keep moving forward or move backward, viz.,

$F+B=1.$ (65)

Substituting then Eqs. (64) with Eqs. (62a) into Eq. (65) yields then

$\mathcal{C}=\frac{2\Gamma_{\mathrm{t}}^{2}}{1+\Gamma_{\mathrm{t}}^{2}},$ (66)

so that the later forward and backward probabilities are finally obtained from
Eq. (64) as

$F=|f|^{2}\frac{2\Gamma_{\mathrm{t}}^{2}}{1+\Gamma_{\mathrm{t}}^{2}}=\left|\frac{1+\Gamma_{\mathrm{t}}}{2\Gamma_{\mathrm{t}}}\right|^{2}\frac{2\Gamma_{\mathrm{t}}^{2}}{1+\Gamma_{\mathrm{t}}^{2}}$
(67a) and
$B=|b|^{2}\frac{2\Gamma_{\mathrm{t}}^{2}}{1+\Gamma_{\mathrm{t}}^{2}}=\left|\frac{\Gamma_{\mathrm{t}}-1}{2\Gamma_{\mathrm{t}}}\right|^{2}\frac{2\Gamma_{\mathrm{t}}^{2}}{1+\Gamma_{\mathrm{t}}^{2}},$
(67b)

where Eq. (62a) was used in the second equalities and where
$\Gamma_{\mathrm{t}}$ was defined in Eq. (62d).

##  4 Gauge Transformations and Symmetries

###  4.1 Scalar Potential Spatial Step $V(z)$

The electric and magnetic fields, $\mathbf{E}$ and $\mathbf{B}$, are generally
related to the potentials as

$\mathbf{E}=-\gradient V-\partialderivative{\mathbf{A}}{t}$ (68a) and
$\mathbf{B}=\curl\mathbf{A},$ (68b)

which are invariant under the gauge transformation [61, 62]

$V\rightarrow V^{\prime}=V-\partialderivative{\Lambda}{t}$ (69a) and
$\mathbf{A}\rightarrow\mathbf{A}^{\prime}=\mathbf{A}+\gradient\Lambda,$ (69b)

where $\Lambda$ is an arbitrary scalar function. The potential spatial step
$V(z)$ is equivalent to the transformation

$V^{\prime}=V_{1}+\Delta
V\theta(z-z_{0})\quad\textrm{and}\quad\mathbf{A}^{\prime}=0,$ (70)

Consistency with the gauge (69) would demand that

$\partialderivative{\Lambda}{t}=-\Delta{}V\theta(z-z_{0})\quad\textrm{and}\quad\gradient\Lambda=0.$
(71)

which imply

$\Lambda=-\Delta{}Vt\theta(z-z_{0})\quad\textrm{and}\quad\Lambda\neq\Lambda(z).$
(72)

respectively. The incompatibility between these two conditions on $\Lambda$
indicates that the transformation indeed breaks the symmetry of the gauge
(69), which explains why the corresponding potential leads to electron back-
scattering [1] (Sec. 4.1).

###  4.2 Vector Potential Spatial Step $A(z)$

The vector potential spatial step $\mathbf{A}(z)=A(z)\mathbf{\hat{z}}$ is
equivalent to the transformation

$V^{\prime}=0\quad\textrm{and}\quad\mathbf{A}^{\prime}=A_{1}+\Delta{}A\theta(z-z_{0}),$
(73)

which is a particular case of the gauge transformation
$\eqref{eq:s_gauge_transf}$ with

$\partialderivative{\Lambda}{t}=0\quad\textrm{and}\quad\gradient\Lambda=\Delta{}A\theta(z-z_{0}),$
(74)

corresponding to

$\Lambda=\Delta{}A\theta(z-z_{0})z=\Lambda(z)$ (75)

Therefore, the step $A(z)$ does not involve any change in the fields, and
hence also in the impedance, which explains why we found that it produces no
(reflected wave) back-scattering (Sec. 3.2.3).

##  5 Phase and Group Velocities

The phase and group velocities may be computed from the dispersion relation
(27), i.e.,

$(E-qA_{0})^{2}=(p-qA_{3})^{2}+m^{2}.$ (76)

The phase velocity is defined as

$v_{\mathrm{p}}=\frac{E}{p}.$ (77)

In general, it may be found by solving Eq. (76) for $p$ and substituting the
result into Eq. (77), which yields

$v_{\mathrm{p}}=\frac{E}{\sqrt{(E-qA_{0})^{2}-m^{2}}+qA_{3}}.$ (78)

For the cases of the scalar potential spatial step [$V(z)$] (Sec. 3.2.2) and
the vector potential temporal step [$A(t)$] (Sec. 3.2.4), the scattered phase
velocities may be directly obtained from Eq. (77) as

$v_{\mathrm{p,t}}=\frac{E}{p_{\mathrm{t}}}\quad\textrm{and}\quad
v_{\mathrm{p.r}}=\frac{E}{p_{\mathrm{r}}},$ (79)

with $p_{\mathrm{t}}$ and $p_{\mathrm{r}}$ given by Eq. (40d) and Eq. (40b),
and

$v_{\mathrm{p,f}}=\frac{E_{\mathrm{f}}}{p}\quad\textrm{and}\quad
v_{\mathrm{p,b}}=\frac{E_{\mathrm{b}}}{p},$ (80)

with $E_{\mathrm{f}}$ and $E_{\mathrm{b}}$ given by Eq. (59c) and Eq. (59d),
respectively.

The group velocity is defined as

$v_{\mathrm{g}}=\partialderivative{E}{p}.$ (81)

Its general expression may be found by taking the derivative of Eq. (76)
versus $p$ and isolating $\partial E/\partial p$, which results in

$v_{\mathrm{g}}=\frac{p-qA_{3}}{E-qA_{0}}.$ (82)

For the cases of the scalar potential spatial step [$V(z)$] (Sec. 3.2.2) and
the vector potential temporal step [$A(t)$] (Sec. 3.2.4), the scattered group
velocities may be obtained from Eq. (82) as

$v_{\mathrm{g,t}}=\frac{p_{\mathrm{t}}}{E-qV}\quad\textrm{and}\quad
v_{\mathrm{g,r}}=\frac{p_{\mathrm{r}}}{E-qV},$ (83)

with $p_{\mathrm{t}}$ and $p_{\mathrm{r}}$ given by Eq. (40d) and Eq. (40b),
and

$v_{\mathrm{g,f}}=\frac{p-qA}{E_{\mathrm{f}}}\quad\textrm{and}\quad
v_{\mathrm{g,b}}=\frac{p-qA}{E_{\mathrm{b}}},$ (84)

with $E_{\mathrm{f}}$ and $E_{\mathrm{b}}$ given by Eq. (59c) and Eq. (59d),
respectively.

In the problem of the scalar potential spatial step [$V(z)$] (Sec. 3.2.2),
assuming $qV_{1}<E_{\mathrm{i}}-m$ and $qV_{1}<qV_{2}$, we find, using Eqs.
(79) and Eq. (83)

  1. 1.

$qV_{2}<E-m$ $\rightarrow$ $v_{\mathrm{g},2}<v_{\mathrm{g},1}$ and
$v_{\mathrm{p},1}<v_{\mathrm{p},2}$,

  2. 2.

$E-m<qV_{2}<E+m$ $\rightarrow$ $v_{\mathrm{g},2}=0$ and
$v_{\mathrm{p},2}=\infty$ (Klein region),

  3. 3.

$E+m<qV_{2}<2E-V_{1}$ $\rightarrow$ $v_{\mathrm{g},2}<v_{\mathrm{g},1}$ and
$v_{\mathrm{p},1}<v_{\mathrm{p},2}$,

  4. 4.

$2E-V_{1}<V_{2}$ $\rightarrow$ $v_{\mathrm{g},1}<v_{\mathrm{g},2}$ and
$v_{\mathrm{p},2}<v_{\mathrm{p},1}$,

while in the problem of the vector potential temporal step [$A(t)$] (Sec.
3.2.4), assuming $qA_{1}<p_{\mathrm{i}}$ and $qA_{1}<qA_{2}$, we find, using
Eqs. (80) and Eq. (84),

  1. 1.

$qA_{2}<p_{\mathrm{i}}$ $\rightarrow$ $v_{\mathrm{g},2}<v_{\mathrm{g},1}$ and
$v_{\mathrm{p},2}<v_{\mathrm{p},1}$,

  2. 2.

$p_{\mathrm{i}}<qA_{2}<2p_{\mathrm{i}}-A_{1}$ $\rightarrow$
$v_{\mathrm{g},2}<v_{\mathrm{g},1}$ and $v_{\mathrm{p},2}<v_{\mathrm{p},1}$,

  3. 3.

$2p_{\mathrm{i}}-A_{1}<qA_{2}$ $\rightarrow$
$v_{\mathrm{g},1}<v_{\mathrm{g},2}$ and $v_{\mathrm{p},1}<v_{\mathrm{p},2}$,

where $1$ and $2$ refer to the two regions.

##  6 Spatial and Temporal Step Electromagnetic Problems

We provide here the main results pertaining to the 1+1D spatial and temporal
step electromagnetic problems for the sake of comparison. Related details are
available in [33, 34].

###  6.1 Spatial Step Problem

The wave equation may be written as

$\left[\frac{1}{n^{2}}\partialderivative[2]{z}-\partialderivative[2]{t}\right]\left\\{\begin{array}[]{l}E(z,t)\\\
H(z,t)\end{array}\right\\}=0.$ (85)

The $E$ and $H$ fields before ($z<z_{0}$, region 1) and after ($z>z_{0}$,
region 2) the step are

$E_{1}=\mathrm{e}^{-i\omega_{\mathrm{i}}t}\mathrm{e}^{ik_{\mathrm{i}}z}+r\mathrm{e}^{-i\omega_{\mathrm{r}}t}\mathrm{e}^{ik_{\mathrm{r}}z},$
(86a)
$H_{1}=\left(\mathrm{e}^{-i\omega_{\mathrm{i}}t}\mathrm{e}^{ik_{\mathrm{i}}z}-r\mathrm{e}^{-i\omega_{\mathrm{r}}t}\mathrm{e}^{ik_{\mathrm{r}}z}\right)/\eta_{1},$
(86b)
$E_{2}=t\mathrm{e}^{-i\omega_{\mathrm{t}}t}\mathrm{e}^{ik_{\mathrm{t}}z},$
(86c)
$H_{2}=t\mathrm{e}^{-i\omega_{\mathrm{t}}t}\mathrm{e}^{ik_{\mathrm{t}}z}/\eta_{2},$
(86d)

where

$\eta_{1,2}=\sqrt{\frac{\mu_{1,2}}{\epsilon_{1,2}}}$ (87)

is the intrinsic medium impedance, and

$\omega_{\mathrm{i}}=\omega_{\mathrm{r}}=\omega_{\mathrm{t}},$ (88a)
$k_{\mathrm{i}}=-k_{\mathrm{r}}=n_{1}\frac{\omega_{\mathrm{i}}}{c},\quad\text{and}\quad
k_{\mathrm{t}}=n_{2}\frac{\omega_{\mathrm{i}}}{c}.$ (88b)

Inserting Eqs. (86) to (88) into the boundary condition relation

$\left.E_{1}\right|_{z=z_{0}}=\left.E_{2}\right|_{z=z_{0}}\quad\text{and}\quad\left.H_{1}\right|_{z=z_{0}}=\left.H_{2}\right|_{z=z_{0}}$
(89)

yields then the scattering amplitude coefficients

$r=\frac{1-N}{1+N}\quad\text{and}\quad t=\frac{2}{1+N},$ (90)

while the reflectance and transmittance are found, using the Poynting vector
definitions

$R=\left|\frac{\mathbf{E_{\mathrm{r}}}\times\mathbf{H_{\mathrm{r}}}}{\mathbf{E_{\mathrm{i}}}\times\mathbf{H_{\mathrm{i}}}}\right|\quad\text{and}\quad
T=\left|\frac{\mathbf{E_{\mathrm{t}}}\times\mathbf{H_{\mathrm{t}}}}{\mathbf{E_{\mathrm{i}}}\times\mathbf{H_{\mathrm{i}}}}\right|,$
(91)

and substituting Eqs. (86) in these relations, as

$R=|r|^{2}=\left|\frac{1-N}{1+N}\right|^{2}\quad\text{and}\quad
T=|t|^{2}N=\left|\frac{2}{1+N}\right|^{2}N,$ (92)

where $N=\dfrac{n_{2}}{n_{1}}=\dfrac{\eta_{1}}{\eta_{2}}$, assuming
nonmagnetic materials.

The space-time and dispersion diagrams corresponding to these results are
provided in Fig. 6(a).

###  6.2 Temporal Step Problem

The wave equation may be written as

$\left[\partialderivative[2]{z}-n^{2}\partialderivative[2]{t}\right]\left\\{\begin{array}[]{l}D(z,t)\\\
B(z,t)\end{array}\right\\}=0.$ (93)

The $D$ and $B$ fields before ($t<t_{0}$, region 1) and after ($t>t_{0}$,
region 2) the step are

$D_{1}=\mathrm{e}^{-i\omega_{\mathrm{i}}t}\mathrm{e}^{ik_{\mathrm{i}}z},$
(94a)
$B_{1}=\left(\mathrm{e}^{-i\omega_{\mathrm{i}}t}\mathrm{e}^{ik_{\mathrm{i}}z}\right)/\eta_{1},$
(94b)
$D_{2}=f\mathrm{e}^{-i\omega_{\mathrm{f}}t}\mathrm{e}^{ik_{\mathrm{f}}z}+b\mathrm{e}^{-i\omega_{\mathrm{b}}t}\mathrm{e}^{ik_{\mathrm{b}}z},$
(94c)
$B_{2}=\left(f\mathrm{e}^{-i\omega_{\mathrm{f}}t}\mathrm{e}^{ik_{\mathrm{f}}z}-b\mathrm{e}^{-i\omega_{\mathrm{b}}t}\mathrm{e}^{ik_{\mathrm{b}}z}\right)/\eta_{2},$
(94d)

and

$k_{\mathrm{i}}=k_{\mathrm{f}}=k_{\mathrm{b}},$ (95a)
$\omega_{\mathrm{i}}=\frac{c}{n_{1}}k_{\mathrm{i}}\quad\text{and}\quad\omega_{\mathrm{f}}=-\omega_{\mathrm{b}}=\frac{c}{n_{2}}k_{\mathrm{i}}.$
(95b)

Inserting Eqs. (94) to (95) into the boundary condition relation

$\left.D_{1}\right|_{t=t_{0}}=\left.D_{2}\right|_{t=t_{0}}\quad\text{and}\quad\left.B_{1}\right|_{t=t_{0}}=\left.B_{2}\right|_{t=t_{0}},$
(96)

yields then the scattering amplitude coefficients

$f=\frac{1+N}{2N}\quad\text{and}\quad b=\frac{N-1}{2N},$ (97)

while the reflectance and transmittance are found, using the Poynting vector
definitions

$F=\left|\frac{\mathbf{E_{\mathrm{f}}}\times\mathbf{H_{\mathrm{f}}}}{\mathbf{E_{\mathrm{i}}}\times\mathbf{H_{\mathrm{i}}}}\right|=\left|\frac{\mathbf{D_{\mathrm{f}}}/\epsilon_{2}\times\mathbf{B_{\mathrm{f}}}/\mu_{2}}{\mathbf{D_{\mathrm{i}}}/\epsilon_{1}\times\mathbf{B_{\mathrm{i}}}/\mu_{1}}\right|=\left|\frac{\mathbf{D_{\mathrm{f}}}\times\mathbf{B_{\mathrm{f}}}}{\mathbf{D_{\mathrm{i}}}\times\mathbf{B_{\mathrm{i}}}}\right|\left(\frac{n_{1}}{n_{2}}\right)^{2}$
(98a)
$B=\left|\frac{\mathbf{E_{\mathrm{b}}}\times\mathbf{H_{\mathrm{b}}}}{\mathbf{E_{\mathrm{i}}}\times\mathbf{H_{\mathrm{i}}}}\right|=\left|\frac{\mathbf{D_{\mathrm{b}}}/\epsilon_{2}\times\mathbf{B_{\mathrm{b}}}/\mu_{2}}{\mathbf{D_{\mathrm{i}}}/\epsilon_{1}\times\mathbf{B_{\mathrm{i}}}/\mu_{1}}\right|=\left|\frac{\mathbf{D_{\mathrm{b}}}\times\mathbf{B_{\mathrm{b}}}}{\mathbf{D_{\mathrm{i}}}\times\mathbf{B_{\mathrm{i}}}}\right|\left(\frac{n_{1}}{n_{2}}\right)^{2},$
(98b)

and substituting Eqs. (94) in these relations, as

$F=|f|^{2}\frac{1}{N}=\left|\frac{1+N}{2N}\right|^{2}\frac{1}{N}\quad\text{and}\quad
B=|b|^{2}\frac{1}{N}=\left|\frac{N-1}{2N}\right|^{2}\frac{1}{N},$ (99)

where $N=\dfrac{n_{2}}{n_{1}}=\dfrac{\eta_{1}}{\eta_{2}}$, assuming
nonmagnetic materials.

The space-time and dispersion diagrams corresponding to these results are
provided in Fig. 6(b).

Figure 6: Electromagnetic wave scattering at (a) a spatial and (b) a temporal
refractive index step discontinuity, represented in space-time diagrams (left
panels) and dispersion diagrams with transitions (right panels).

##  7 Smooth Temporal Step

###  7.1 Dirac Equation for a Time-Varying Vector Potential

We start with the Dirac equation in its covariant form [58], viz.,

$\left[\gamma^{\mu}(i\partial_{\mu}-qA_{\mu})-m\right]\psi=0,$ (100a) which
involves the four-gradient
$\partial_{\mu}\equiv\left(\partial_{0},\partial_{i}\right),\quad\text{with}\quad
i=1,2,3,$ (100b) and the gamma matrices111Note that we have chosen here the
Weyl representation for these matrices [58]. Other choices would have been the
Dirac and Majorana representations [58]. The reason for that choice is that it
leads – as we found in painstaking derivations – to convenient hypergeometric
expressions and ultimately to closed-form solutions to the problem.
$\gamma^{0}=\begin{pmatrix}0&I\\\ I&0\end{pmatrix},\quad\text{with}\quad
I=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix},$ (100c) and
$\gamma^{i}=\begin{pmatrix}0&-\sigma^{i}\\\ \sigma^{i}&0\end{pmatrix},$ (100d)
with the Pauli matrices $\sigma^{1}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\quad\sigma^{2}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix}\quad\text{and}\quad\sigma^{3}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}.$ (100e)

In the sequel of this section, we assume a 1+1D ($t$ and $z$) spacetime system
and a $z$-directed magnetic vector potential, viz., $A_{0}=A_{1}=A_{2}=0$ and
$A_{3}=A_{z}(t)$. These assumption reduce Eq. (100a) to

$\left[\gamma^{0}i\partial_{0}+\gamma^{3}(i\partial_{3}+qA_{z}(t))-m\right]\psi=0,$
(101)

which, using the spinor ansatz

$\psi=\begin{pmatrix}\varphi\\\ \vartheta\end{pmatrix}e^{ipz},$ (102)

takes the more explicit form

$\left[\gamma^{0}i\partial_{0}-\gamma^{3}(p-qA_{z}(t))-m\right]\begin{pmatrix}\varphi\\\
\vartheta\end{pmatrix}=0.$ (103)

Inserting the gamma matrices (100c) and (100d) with the sigma matrices (100e)
into Eq. (103) yields then

$\left[\begin{pmatrix}0&1\\\
1&0\end{pmatrix}i\partial_{0}-\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}(p-qA_{z}(t))-m\right]\begin{pmatrix}\varphi\\\
\vartheta\end{pmatrix}=0,$ (104)

which splits into the two equations

$i\derivative{\vartheta}{t}+(p-qA_{z}(t))\vartheta-m\varphi=0$ (105a) and
$i\derivative{\varphi}{t}-(p-qA_{z}(t))\varphi-m\vartheta=0.$ (105b)

Isolating $\vartheta$ in Eq. (105b),

$\vartheta=\frac{1}{m}\left[i\derivative{\varphi}{t}-(p-qA_{z}(t))\varphi\right],$
(106)

and substituting this expression as well as its time derivative,

$\derivative{\vartheta}{t}=\frac{1}{m}\left[i\derivative[2]{\varphi}{t}+q\derivative{A_{z}(t)}{t}\varphi-(p-qA_{z}(t))\derivative{\varphi}{t}\right],$
(107)

into Eq. (105a), we get

$\derivative[2]{\varphi}{t}+\left[(p-qA_{z}(t))^{2}+m^{2}-iq\derivative{A_{z}(t)}{t}\right]\varphi=0.$
(108)

Now, we select for the potential $A_{z}(t)$ the smooth vector potential
corresponding to the hyperbolic tangent function

$A_{z}(t)=A_{1}+\frac{A_{2}-A_{1}}{2}\left[1+\tanh\left(\frac{t-t_{0}}{\tau}\right)\right]=\frac{A_{1}+A_{2}\mathrm{e}^{2\frac{t-t_{0}}{\tau}}}{1+\mathrm{e}^{2\frac{t-t_{0}}{\tau}}},$
(109)

which corresponds to the electric field

$E_{z}(t)=-\frac{\partial A_{z}(t)}{\partial
t}=-\frac{(A_{2}-A_{1})}{2\tau}\sech^{2}\left(\frac{t-t_{0}}{\tau}\right)=-\frac{2(A_{2}-A_{1})}{\tau}\frac{\mathrm{e}^{2\frac{t-t_{0}}{\tau}}}{\left(1+\mathrm{e}^{2\frac{t-t_{0}}{\tau}}\right)^{2}},$
(110)

where $\tau$ is a time constant.

Figure 7: Functions associated with the smooth potential investigated in Sec.
7 for different values of the time parameter $\tau$. (a) Potential function
itself, $A_{z}(t)$ [Eq. (109)]. (b) Corresponding electric field, $E_{z}(t)$
[Eq. (110)].

We shall next solve Eq. (108) for the earlier medium and later medium
separately so as to be able to subsequently apply boundary conditions between
them. For each case, we shall define a new variable that will simplify the
vector potential in Eq. (109) to the point that Eq. (108) will progressively
transform into a hypergeometric equation, which admits analytical solutions.

###  7.2 Solution for the Earlier Medium ($t<t_{0}$)

For the earlier (e) medium, we use the variable

$\zeta_{\mathrm{e}}=-\mathrm{e}^{2\frac{t-t_{0}}{\tau}},$ (111)

which simplifies Eq. (109) to

$A_{z}(\zeta_{\mathrm{e}})=\frac{A_{1}-A_{2}\zeta_{\mathrm{e}}}{1-\zeta_{\mathrm{e}}},$
(112a) and whose first and second time derivatives lead to
$\derivative{t}=\frac{2}{\tau}\zeta_{\mathrm{e}}\derivative{\zeta_{\mathrm{e}}},$
(112b) and
$\derivative[2]{t}=\frac{4}{\tau^{2}}\zeta_{\mathrm{e}}\derivative{\zeta_{\mathrm{e}}}+\frac{4}{\tau^{2}}\zeta_{\mathrm{e}}^{2}\derivative[2]{\zeta_{\mathrm{e}}},$
(112c)

respectively.

Inserting Eqs. (112) into Eq. (108) and multiplying the result by
$\frac{\tau^{2}}{4}\frac{1-\zeta_{\mathrm{e}}}{\zeta_{\mathrm{e}}}$, we get,
after some algebraic manipulations,

$\zeta_{\mathrm{e}}(1-\zeta_{\mathrm{e}})\derivative[2]{\varphi_{\mathrm{e}}}{\zeta_{\mathrm{e}}}+(1-\zeta_{\mathrm{e}})\derivative{\varphi_{\mathrm{e}}}{\zeta_{\mathrm{e}}}+\frac{1}{\zeta_{\mathrm{e}}(1-\zeta_{\mathrm{e}})}\left[\alpha_{2}\zeta_{\mathrm{e}}^{2}+\alpha_{0}\zeta_{\mathrm{e}}+\alpha_{1}\right]\varphi_{\mathrm{e}}=0,$
(113a) where
$\alpha_{0}=-\frac{\tau^{2}}{4}\left[((p-qA_{2})^{2}+m^{2})+((p-qA_{1})^{2}+m^{2})-(qA_{2}-qA_{1})^{2}-2i\frac{qA_{2}-qA_{1}}{\tau}\right],$
(113b)
$\alpha_{1}=\frac{\tau^{2}}{4}\left[(p-qA_{1})^{2}+m^{2}\right]=\frac{\tau^{2}}{4}E_{1}^{2},$
(113c) and
$\alpha_{2}=\frac{\tau^{2}}{4}\left[(p-qA_{2})^{2}+m^{2}\right]=\frac{\tau^{2}}{4}E_{2}^{2},$
(113d)

and whose singularities at $\zeta_{\mathrm{e}}=0$ and $1-\zeta_{\mathrm{e}}=0$
can be removed by inserting the expression

$\varphi_{\mathrm{e}}(\zeta_{\mathrm{e}})=\zeta_{\mathrm{e}}^{\mu}(1-\zeta_{\mathrm{e}})^{\nu}f(\zeta_{\mathrm{e}}),$
(114)

its derivative,

$\derivative{\varphi_{\mathrm{e}}}{\zeta_{\mathrm{e}}}=\zeta_{\mathrm{e}}^{\mu}(1-\zeta_{\mathrm{e}})^{\nu}\left[\left(\mu\frac{1}{\zeta_{\mathrm{e}}}-\nu\frac{1}{1-\zeta_{\mathrm{e}}}\right)f+\derivative{f}{\zeta_{\mathrm{e}}}\right],$
(115a) and its second derivative,
$\derivative[2]{\varphi_{\mathrm{e}}}{\zeta_{\mathrm{e}}}=\zeta_{\mathrm{e}}^{\mu}(1-\zeta_{\mathrm{e}})^{\nu}\left[\left(\mu(\mu-1)\frac{1}{\zeta_{\mathrm{e}}^{2}}+\nu(\nu-1)\frac{1}{(1-\zeta_{\mathrm{e}})^{2}}-2\mu\nu\frac{1}{\zeta_{\mathrm{e}}(1-\zeta_{\mathrm{e}})}\right)f+\left(2\mu\frac{1}{\zeta_{\mathrm{e}}}-2\nu\frac{1}{1-\zeta_{\mathrm{e}}}\right)\derivative{f}{\zeta_{\mathrm{e}}}+\derivative[2]{f}{\zeta_{\mathrm{e}}}\right].$
(115b)

Dividing the resulting equation by
$\zeta_{\mathrm{e}}^{\mu}(1-\zeta_{\mathrm{e}})^{\nu}$ finally yields the
hypergeometric-type equation

$\zeta_{\mathrm{e}}(1-\zeta_{\mathrm{e}})\derivative[2]{f}{\zeta_{\mathrm{e}}}+\left[(2\mu+1)-(2\mu+2\nu+1)\zeta_{\mathrm{e}}\right]\derivative{f}{\zeta_{\mathrm{e}}}-\left[(\mu+\nu+\lambda)(\mu+\nu-\lambda)\right]f=0,$
(116a) where $\mu=i\sqrt{\alpha_{1}}=i\frac{\tau}{2}E_{1},$ (116b)
$\nu=\frac{1-\sqrt{1-4(\alpha_{0}+\alpha_{1}+\alpha_{2})}}{2}=i\frac{\tau}{2}(qA_{2}-qA_{1}),$
(116c) and $\lambda=i\sqrt{\alpha_{2}}=i\frac{\tau}{2}E_{2}.$ (116d)

The solution to Eqs. (116) can be expressed in terms of Gauss hypergeometric
functions ${}_{2}F_{1}(\cdot)$ [70] as

$f(\zeta_{\mathrm{e}})=C_{1}{{}_{2}F_{1}}(\mu+\nu-\lambda,\mu+\nu+\lambda;1+2\mu;\zeta_{\mathrm{e}})+C_{2}\zeta_{\mathrm{e}}^{-2\mu}{{}_{2}F_{1}}(-\mu+\nu-\lambda,-\mu+\nu+\lambda;1-2\mu;\zeta_{\mathrm{e}}),$
(117)

and substituting this equation into Eq. (114) allows to determine the
wavefunction $\varphi_{\mathrm{e}}(\zeta_{\mathrm{e}})$ in terms of Gauss
hypergeometric functions as

$\begin{split}\varphi_{\mathrm{e}}(\zeta_{\mathrm{e}})&=C_{1}\zeta_{\mathrm{e}}^{\mu}(1-\zeta_{\mathrm{e}})^{\nu}{{}_{2}F_{1}}(\mu+\nu-\lambda,\mu+\nu+\lambda;1+2\mu;\zeta_{\mathrm{e}})\\\
&+C_{2}\zeta_{\mathrm{e}}^{-\mu}(1-\zeta_{\mathrm{e}})^{\nu}{{}_{2}F_{1}}(-\mu+\nu-\lambda,-\mu+\nu+\lambda;1-2\mu;\zeta_{\mathrm{e}}).\end{split}$
(118)

Since there is only an incident wave in the earlier medium, considering Eq.
(116b), the coefficient $C_{1}$ must be zero, which reduces Eq. (118) to

$\varphi_{\mathrm{e}}(\zeta_{\mathrm{e}})=C_{2}\zeta_{\mathrm{e}}^{-\mu}(1-\zeta_{\mathrm{e}})^{\nu}{{}_{2}F_{1}}(-\mu+\nu-\lambda,-\mu+\nu+\lambda;1-2\mu;\zeta_{\mathrm{e}}).$
(119)

To complete the spinor wavefunction, we also need to calculate
$\vartheta_{\mathrm{e}}$. This may be accomplished by inserting Eq. (119) into
Eq. (106) and using Eqs. (112a) and (112b). This includes the intermediate
step

$\begin{split}\derivative{\varphi_{\mathrm{e}}}{\zeta_{\mathrm{e}}}=&-C_{2}\mu\zeta_{\mathrm{e}}^{-\mu-1}(1-\zeta_{\mathrm{e}})^{\nu}{{}_{2}F_{1}}(-\mu+\nu-\lambda,-\mu+\nu+\lambda;1-2\mu;\zeta_{\mathrm{e}})\\\
&-C_{2}\zeta_{\mathrm{e}}^{-\mu}\nu(1-\zeta_{\mathrm{e}})^{\nu-1}{{}_{2}F_{1}}(-\mu+\nu-\lambda,-\mu+\nu+\lambda;1-2\mu;\zeta_{\mathrm{e}})\\\
&+C_{2}\zeta_{\mathrm{e}}^{-\mu}(1-\zeta_{\mathrm{e}})^{\nu}\frac{(-\mu+\nu-\lambda)(-\mu+\nu+\lambda)}{1-2\mu}{{}_{2}F_{1}}(-\mu+\nu-\lambda+1,-\mu+\nu+\lambda+1;1-2\mu+1;\zeta_{\mathrm{e}}),\end{split}$
(120)

where using the derivative property of hypergeometric functions [70],

$\derivative{\zeta}[{{}_{2}F_{1}}(a,b;c;\zeta)]=\frac{ab}{c}{{}_{2}F_{1}}(a+1,b+1;c+1,\zeta),$
(121)

leads to

$\begin{split}\vartheta_{\mathrm{e}}=\frac{C_{2}}{m}&\left[-i\frac{2}{\tau}\mu\zeta_{\mathrm{e}}^{-\mu}(1-\zeta_{\mathrm{e}})^{\nu}{{}_{2}F_{1}}(-\mu+\nu-\lambda,-\mu+\nu+\lambda;1-2\mu;\zeta_{\mathrm{e}})\right.\\\
&\left.-i\frac{2}{\tau}\zeta_{\mathrm{e}}^{-\mu+1}\nu(1-\zeta_{\mathrm{e}})^{\nu-1}{{}_{2}F_{1}}(-\mu+\nu-\lambda,-\mu+\nu+\lambda;1-2\mu;\zeta_{\mathrm{e}})\right.\\\
&\left.+i\frac{2}{\tau}\zeta_{\mathrm{e}}^{-\mu+1}(1-\zeta_{\mathrm{e}})^{\nu}\frac{(-\mu+\nu-\lambda)(-\mu+\nu+\lambda)}{1-2\mu}{{}_{2}F_{1}}(-\mu+\nu-\lambda+1,-\mu+\nu+\lambda+1;1-2\mu+1;\zeta_{\mathrm{e}})\right.\\\
&\left.-\left(p-\frac{qA_{1}-qA_{2}\zeta_{\mathrm{e}}}{1-\zeta_{\mathrm{e}}}\right)\zeta_{\mathrm{e}}^{-\mu}(1-\zeta_{\mathrm{e}})^{\nu}{{}_{2}F_{1}}(-\mu+\nu-\lambda,-\mu+\nu+\lambda;1-2\mu;\zeta_{\mathrm{e}})\right].\end{split}$
(122)

###  7.3 Solution for the Later Medium ($t>t_{0}$)

For the later (l) medium, we use the variable

$\zeta_{\mathrm{l}}=-\mathrm{e}^{-2\frac{t-t_{0}}{\tau}},$ (123)

which simplifies Eq. (109) to

$A_{z}(\zeta_{\mathrm{l}})=\frac{A_{2}-A_{1}\zeta_{\mathrm{l}}}{1-\zeta_{\mathrm{l}}},$
(124a) and whose first and second time derivatives lead to
$\derivative{t}=-\frac{2}{\tau}\zeta_{\mathrm{l}}\derivative{\zeta_{\mathrm{l}}}$
(124b) and
$\derivative[2]{t}=\frac{4}{\tau^{2}}\zeta_{\mathrm{l}}\derivative{\zeta_{\mathrm{l}}}+\frac{4}{\tau^{2}}\zeta_{\mathrm{l}}^{2}\derivative[2]{\zeta_{\mathrm{l}}}$
(124c)

respectively.

Inserting Eqs. (124) into Eq. (108) and multiplying the result by
$\frac{\tau^{2}}{4}\frac{1-\zeta_{\mathrm{l}}}{\zeta_{\mathrm{l}}}$, we get,
after some algebraic manipulations,

$\zeta_{\mathrm{l}}(1-\zeta_{\mathrm{l}})\derivative[2]{\varphi_{\mathrm{l}}}{\zeta_{\mathrm{l}}}+(1-\zeta_{\mathrm{l}})\derivative{\varphi_{\mathrm{l}}}{\zeta_{\mathrm{l}}}+\frac{1}{\zeta_{\mathrm{l}}(1-\zeta_{\mathrm{l}})}\left[\beta_{1}\zeta_{\mathrm{l}}^{2}+\beta_{0}\zeta_{\mathrm{l}}+\beta_{2}\right]\varphi_{\mathrm{l}}=0,$
(125a) where
$\beta_{0}=-\frac{\tau^{2}}{4}\left[((p-qA_{2})^{2}+m^{2})+((p-qA_{1})^{2}+m^{2})-(qA_{2}-qA_{1})^{2}-2i\frac{qA_{2}-qA_{1}}{\tau}\right],$
(125b)
$\beta_{1}=\frac{\tau^{2}}{4}\left[(p-qA_{1})^{2}+m^{2}\right]=\frac{\tau^{2}}{4}E_{1}^{2},$
(125c) and
$\beta_{2}=\frac{\tau^{2}}{4}\left[(p-qA_{2})^{2}+m^{2}\right]=\frac{\tau^{2}}{4}E_{2}^{2},$
(125d)

and whose singularities at $\zeta_{\mathrm{l}}=0$ and $1-\zeta_{\mathrm{l}}=0$
can be removed by inserting the expression

$\varphi_{\mathrm{l}}(\zeta_{\mathrm{l}})=\zeta_{\mathrm{l}}^{\sigma}(1-\zeta_{\mathrm{l}})^{\rho}f(\zeta_{\mathrm{l}}),$
(126)

its derivative,

$\derivative{\varphi_{\mathrm{l}}}{\zeta_{\mathrm{l}}}=\zeta_{\mathrm{l}}^{\sigma}(1-\zeta_{\mathrm{l}})^{\rho}\left[\left(\sigma\frac{1}{\zeta_{\mathrm{l}}}-\rho\frac{1}{1-\zeta_{\mathrm{l}}}\right)f+\derivative{f}{\zeta_{\mathrm{l}}}\right],$
(127a) and its second derivative,
$\derivative[2]{\varphi_{\mathrm{l}}}{\zeta_{\mathrm{l}}}=\zeta_{\mathrm{l}}^{\sigma}(1-\zeta_{\mathrm{l}})^{\rho}\left[\left(\sigma(\sigma-1)\frac{1}{\zeta_{\mathrm{l}}^{2}}+\rho(\rho-1)\frac{1}{(1-\zeta_{\mathrm{l}})^{2}}-2\sigma\rho\frac{1}{\zeta_{\mathrm{l}}(1-\zeta_{\mathrm{l}})}\right)f+\left(2\sigma\frac{1}{\zeta_{\mathrm{l}}}-2\rho\frac{1}{1-\zeta_{\mathrm{l}}}\right)\derivative{f}{\zeta_{\mathrm{l}}}+\derivative[2]{f}{\zeta_{\mathrm{l}}}\right].$
(127b)

Dividing the resulting equation by
$\zeta_{\mathrm{l}}^{\sigma}(1-\zeta_{\mathrm{l}})^{\rho}$ finally yields the
hypergeometric-type equation

$\zeta_{\mathrm{l}}(1-\zeta_{\mathrm{l}})\derivative[2]{f}{\zeta_{\mathrm{l}}}+\left[(2\sigma+1)-(2\sigma+2\rho+1)\zeta_{\mathrm{l}}\right]\derivative{f}{\zeta_{\mathrm{l}}}-\left[(\sigma+\rho+\eta)(\sigma+\rho-\eta)\right]f=0,$
(128a) where $\sigma=i\sqrt{\beta_{2}}=i\frac{\tau}{2}E_{2},$ (128b)
$\rho=\frac{1-\sqrt{1-4(\beta_{0}+\beta_{1}+\beta_{2})}}{2}=i\frac{\tau}{2}(qA_{2}-qA_{1}),$
(128c) and $\eta=i\sqrt{\beta_{1}}=i\frac{\tau}{2}E_{1}.$ (128d)

The solution to Eqs. (128) can be expressed in terms of Gauss hypergeometric
functions ${}_{2}F_{1}(\cdot)$ [70] as

$f(\zeta_{\mathrm{l}})=C_{3}{{}_{2}F_{1}}(\sigma+\rho-\eta,\sigma+\rho+\eta;1+2\sigma;\zeta_{\mathrm{l}})+C_{4}\zeta_{\mathrm{l}}^{-2\sigma}{{}_{2}F_{1}}(-\sigma+\rho-\eta,-\sigma+\rho+\eta;1-2\sigma;\zeta_{\mathrm{l}}).$
(129)

and substituting this equation into Eq. (126) allows to determine the
wavefunction $\varphi_{\mathrm{l}}(\zeta_{\mathrm{l}})$ in terms of Gauss
hypergeometric functions as

$\begin{split}\varphi_{\mathrm{l}}(\zeta_{\mathrm{l}})&=C_{3}\zeta_{\mathrm{l}}^{\sigma}(1-\zeta_{\mathrm{l}})^{\rho}{{}_{2}F_{1}}(\sigma+\rho-\eta,\sigma+\rho+\eta;1+2\sigma;\zeta_{\mathrm{l}})\\\
&+C_{4}\zeta_{\mathrm{l}}^{-\sigma}(1-\zeta_{\mathrm{l}})^{\rho}{{}_{2}F_{1}}(-\sigma+\rho-\eta,-\sigma+\rho+\eta;1-2\sigma;\zeta_{\mathrm{l}}).\end{split}$
(130)

To complete the spinor wavefunction, we also need to calculate
$\vartheta_{\mathrm{l}}$. This may be accomplished by inserting Eq. (130) into
Eq. (106) and using Eqs. (124a) and (124b). This includes the intermediate
step

$\begin{split}\derivative{\varphi_{\mathrm{l}}}{\zeta_{\mathrm{l}}}&=C_{3}\sigma\zeta_{\mathrm{l}}^{\sigma-1}(1-\zeta_{\mathrm{l}})^{\rho}{{}_{2}F_{1}}(\sigma+\rho-\eta,\sigma+\rho+\eta;1+2\sigma;\zeta_{\mathrm{l}})\\\
&-C_{3}\zeta_{\mathrm{l}}^{\sigma}\rho(1-\zeta_{\mathrm{l}})^{\rho-1}{{}_{2}F_{1}}(\sigma+\rho-\eta,\sigma+\rho+\eta;1+2\sigma;\zeta_{\mathrm{l}})\\\
&+C_{3}\zeta_{\mathrm{l}}^{\sigma}(1-\zeta_{\mathrm{l}})^{\rho}\frac{(\sigma+\rho-\eta)(\sigma+\rho+\eta)}{1-2\sigma}{{}_{2}F_{1}}(\sigma+\rho-\eta+1,\sigma+\rho+\eta+1;1+2\sigma+1;\zeta_{\mathrm{l}})\\\
&-C_{4}\sigma\zeta_{\mathrm{l}}^{-\sigma-1}(1-\zeta_{\mathrm{l}})^{\rho}{{}_{2}F_{1}}(-\sigma+\rho-\eta,-\sigma+\rho+\eta;1-2\sigma;\zeta_{\mathrm{l}})\\\
&-C_{4}\zeta_{\mathrm{l}}^{-\sigma}\rho(1-\zeta_{\mathrm{l}})^{\rho-1}{{}_{2}F_{1}}(-\sigma+\rho-\eta,-\sigma+\rho+\eta;1-2\sigma;\zeta_{\mathrm{l}})\\\
&+C_{4}\zeta_{\mathrm{l}}^{-\sigma}(1-\zeta_{\mathrm{l}})^{\rho}\frac{(-\sigma+\rho-\eta)(-\sigma+\rho+\eta)}{1-2\sigma}{{}_{2}F_{1}}(-\sigma+\rho-\eta+1,-\sigma+\rho+\eta+1;1-2\sigma+1;\zeta_{\mathrm{l}}),\end{split}$
(131)

where using the derivative property of hypergeometric functions given in Eq.
(121) leads to

$\begin{split}\vartheta_{\mathrm{l}}=\frac{C_{3}}{m}&\left[-i\frac{2}{\tau}\sigma\zeta_{\mathrm{l}}^{\sigma}(1-\zeta_{\mathrm{l}})^{\rho}{{}_{2}F_{1}}(\sigma+\rho-\eta,\sigma+\rho+\eta;1+2\sigma;\zeta_{\mathrm{l}})\right.\\\
&\left.+i\frac{2}{\tau}\zeta_{\mathrm{l}}^{\sigma+1}\rho(1-\zeta_{\mathrm{l}})^{\rho-1}{{}_{2}F_{1}}(\sigma+\rho-\eta,\sigma+\rho+\eta;1+2\sigma;\zeta_{\mathrm{l}})\right.\\\
&\left.-i\frac{2}{\tau}\zeta_{\mathrm{l}}^{\sigma+1}(1-\zeta_{\mathrm{l}})^{\rho}\frac{(\sigma+\rho-\eta)(\sigma+\rho+\eta)}{1-2\sigma}{{}_{2}F_{1}}(\sigma+\rho-\eta+1,\sigma+\rho+\eta+1;1+2\sigma+1;\zeta_{\mathrm{l}})\right.\\\
&\left.-\left(p-\frac{qA_{2}-qA_{1}\zeta_{\mathrm{l}}}{1-\zeta_{\mathrm{l}}}\right)\zeta_{\mathrm{l}}^{\sigma}(1-\zeta_{\mathrm{l}})^{\rho}{{}_{2}F_{1}}(\sigma+\rho-\eta,\sigma+\rho+\eta;1+2\sigma;\zeta_{\mathrm{l}})\right]\\\
+\frac{C_{4}}{m}&\left[i\frac{2}{\tau}\sigma\zeta_{\mathrm{l}}^{-\sigma}(1-\zeta_{\mathrm{l}})^{\rho}{{}_{2}F_{1}}(-\sigma+\rho-\eta,-\sigma+\rho+\eta;1-2\sigma;\zeta_{\mathrm{l}})\right.\\\
&\left.+i\frac{2}{\tau}\zeta_{\mathrm{l}}^{-\sigma+1}\rho(1-\zeta_{\mathrm{l}})^{\rho-1}{{}_{2}F_{1}}(-\sigma+\rho-\eta,-\sigma+\rho+\eta;1-2\sigma;\zeta_{\mathrm{l}})\right.\\\
&\left.-i\frac{2}{\tau}\zeta_{\mathrm{l}}^{-\sigma+1}(1-\zeta_{\mathrm{l}})^{\rho}\frac{(-\sigma+\rho-\eta)(-\sigma+\rho+\eta)}{1-2\sigma}{{}_{2}F_{1}}(-\sigma+\rho-\eta+1,-\sigma+\rho+\eta+1;1-2\sigma+1;\zeta_{\mathrm{l}})\right.\\\
&\left.-\left(p-\frac{qA_{2}-qA_{1}\zeta_{\mathrm{l}}}{1-\zeta_{\mathrm{l}}}\right)\zeta_{\mathrm{l}}^{-\sigma}(1-\zeta_{\mathrm{l}})^{\rho}{{}_{2}F_{1}}(-\sigma+\rho-\eta,-\sigma+\rho+\eta;1-2\sigma;\zeta_{\mathrm{l}})\right].\end{split}$
(132)

###  7.4 Asymptotic Forms

Being fundamentally different from its infinitely sharp counterpart in the
vicinity of its transition, the smooth potential [Fig. 7(a)] can obviously not
be compared with the infinitely sharp potential in that region. Instead, the
comparison must be carried out sufficiently far (in time) from the transition,
and we should therefore resort to an asymptotic evaluation of the above exact
solutions for the smooth potential step.

####  7.4.1 Asymptotic Forms for the Earlier Medium ($t<t_{0}$)

The asymptotic regime for the earlier medium corresponds to the limit
$t\rightarrow-\infty$, where $\zeta_{\mathrm{e}}\rightarrow{0}$ and
$(1-\zeta_{\mathrm{e}})\rightarrow 1$, and
$F(a,b;c;\zeta_{\mathrm{e}})\rightarrow 1$ [70]. Inserting these limits into
Eq. (118) yields the asymptotic form of $\varphi_{\mathrm{e}}$ in terms of
$\zeta_{\mathrm{e}}$,

$\varphi_{\mathrm{e}}(\zeta_{\mathrm{e}}\rightarrow
0)=C_{1}\zeta_{\mathrm{e}}^{\mu}+C_{2}\zeta_{\mathrm{e}}^{-\mu},$ (133)

and subsequently in terms of $t$

$\varphi_{\mathrm{e}}(t\rightarrow-\infty)=C_{1}(-\mathrm{e}^{2\frac{t-t_{0}}{\tau}})^{i\sqrt{\alpha_{1}}}+C_{2}(-\mathrm{e}^{2\frac{t-t_{0}}{\tau}})^{-i\sqrt{\alpha_{1}}},$
(134)

while inserting Eq. (134) and Eq. (109) into Eq. (106) yields

$\vartheta_{\mathrm{e}}(t\rightarrow-\infty)=\frac{C_{1}}{m}\left[-\frac{2}{\tau}\sqrt{\alpha_{1}}-(p-qA_{1})\right](-\mathrm{e}^{2\frac{t-t_{0}}{\tau}})^{i\sqrt{\alpha_{1}}}+\frac{C_{2}}{m}\left[\frac{2}{\tau}\sqrt{\alpha_{1}}-(p-qA_{1})\right](-\mathrm{e}^{2\frac{t-t_{0}}{\tau}})^{-i\sqrt{\alpha_{1}}}.$
(135)

Finally, since there is only the incident wave in the earlier medium, the
coefficient $C_{1}$ must be zero, which reduces Eq. (134) and (135) to

$\varphi_{\mathrm{e}}(t\rightarrow-\infty)=C_{2}(-\mathrm{e}^{2\frac{t-t_{0}}{\tau}})^{-i\sqrt{\alpha_{1}}}$
(136a) and
$\vartheta_{\mathrm{e}}(t\rightarrow-\infty)=\frac{C_{2}}{m}\left[\frac{2}{\tau}\sqrt{\alpha_{1}}-(p-qA_{1})\right](-\mathrm{e}^{2\frac{t-t_{0}}{\tau}})^{-i\sqrt{\alpha_{1}}}.$
(136b)

####  7.4.2 Asymptotic Forms for the Later Medium ($t>t_{0}$)

The asymptotic regime for the later medium corresponds to the limit
$t\rightarrow\infty$, where $\zeta_{\mathrm{l}}\rightarrow{0}$ and
$(1-\zeta_{\mathrm{e}})\rightarrow 1$, and
$F(a,b;c;\zeta_{\mathrm{l}})\rightarrow 1$ [70]. Inserting these limits into
Eq. (130) yields the asymptotic form of $\varphi_{\mathrm{l}}$ in terms of
$\zeta_{\mathrm{l}}$,

$\varphi_{\mathrm{l}}(\zeta_{\mathrm{l}}\rightarrow
0)=C_{3}\zeta_{\mathrm{l}}^{\sigma}+C_{4}\zeta_{\mathrm{l}}^{-\sigma},$ (137)

and subsequently in terms of $t$

$\varphi_{\mathrm{l}}(t\rightarrow+\infty)=C_{3}(-\mathrm{e}^{-2\frac{t-t_{0}}{\tau}})^{i\sqrt{\beta_{2}}}+C_{4}(-\mathrm{e}^{-2\frac{t-t_{0}}{\tau}})^{-i\sqrt{\beta_{2}}},$
(138)

while inserting Eq. (138) and Eq. (109) into Eq. (106) yields

$\vartheta_{\mathrm{l}}(t\rightarrow+\infty)=\frac{C_{3}}{m}\left[\frac{2}{\tau}\sqrt{\beta_{2}}-(p-qA_{2})\right](-\mathrm{e}^{-2\frac{t-t_{0}}{\tau}})^{i\sqrt{\beta_{2}}}+\frac{C_{4}}{m}\left[-\frac{2}{\tau}\sqrt{\beta_{2}}-(p-qA_{2})\right](-\mathrm{e}^{-2\frac{t-t_{0}}{\tau}})^{-i\sqrt{\beta_{2}}}.$
(139)

###  7.5 Weyl to Dirac Spinor Transformation

For consistency with the mathematical forms of the spinors used throughout
this paper, we transform our spinors from the Weyl representation to the Dirac
representation. This ensures that our calculations remain consistent and
comparable with the results established in the paper. This transformation [71]
may be written as

$\psi(\mathrm{Dirac})=U\psi(\mathrm{Weyl}),\quad\text{where}\quad
U=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\ 1&-1\end{pmatrix}.$ (140)

If we denote the Dirac and Weyl spinors as

$\psi(\mathrm{Dirac})=\begin{pmatrix}\varphi^{\mathrm{D}}\\\
\vartheta^{\mathrm{D}}\end{pmatrix}\quad\text{and}\quad\psi(\mathrm{Weyl})=\begin{pmatrix}\varphi\\\
\vartheta\end{pmatrix},$ (141)

we find using Eq. (140) that the spinor in the Dirac representation in terms
of the elements of the spinor in the Weyl representation read

$\begin{pmatrix}\varphi^{\mathrm{D}}\\\
\vartheta^{\mathrm{D}}\end{pmatrix}=\frac{1}{\sqrt{2}}\begin{pmatrix}\varphi+\vartheta\\\
\varphi-\vartheta\end{pmatrix}.$ (142)

For the earlier medium, inserting Eq. (136a) and Eq. (136b) into the first row
of Eq. (142), we find

$\varphi_{\mathrm{e}}^{\mathrm{D}}(t\rightarrow-\infty)=\frac{1}{\sqrt{2}}\left(C_{2}(-\mathrm{e}^{2\frac{t-t_{0}}{\tau}})^{-i\sqrt{\alpha_{1}}}+\frac{C_{2}}{m}\left[\frac{2}{\tau}\sqrt{\alpha_{1}}-(p-qA_{1})\right](-\mathrm{e}^{2\frac{t-t_{0}}{\tau}})^{-i\sqrt{\alpha_{1}}}\right),$
(143)

which, upon substituting Eq. (113c) and using $(-1)^{a}=\mathrm{e}^{i\pi a}$,
transforms to the the asymptotic form

$\varphi_{\mathrm{e}}^{\mathrm{D}}(t\rightarrow-\infty)=\frac{1}{\sqrt{2}}\frac{C_{2}}{m}\mathrm{e}^{\pi\frac{\tau}{2}E_{1}}\left(m+E_{1}-(p-qA_{1})\right)\mathrm{e}^{-iE_{1}(t-t_{0})}.$
(144)

Similarly, inserting Eq. (136a) and Eq. (136b) into the second row of Eq.
(142), then substituting Eq. (113c) into the resulting expression and using
$(-1)^{a}=\mathrm{e}^{i\pi a}$, we find

$\vartheta_{\mathrm{e}}^{\mathrm{D}}(t\rightarrow-\infty)=\frac{1}{\sqrt{2}}\frac{C_{2}}{m}\mathrm{e}^{\pi\frac{\tau}{2}E_{1}}\left(m-E_{1}+(p-qA_{1})\right)\mathrm{e}^{-iE_{1}(t-t_{0})}.$
(145)

This equation may be algebraically manipulated so as to take a form including
an expression that resembles the wave function for the infinitely sharp step
[Eq. (58a)], viz.,

$\vartheta_{\mathrm{e}}^{\mathrm{D}}(t\rightarrow-\infty)=\frac{1}{\sqrt{2}}\frac{C_{2}}{m}\mathrm{e}^{\pi\frac{\tau}{2}E_{1}}\left(m+E_{1}-(p-qA_{1})\right)\left(\frac{E_{1}-m}{p-qA_{1}}\right)\mathrm{e}^{-iE_{1}(t-t_{0})}.$
(146)

Thus, the Dirac spinor wavefunction for the earlier medium is

$\psi_{\mathrm{e}}^{\mathrm{D}}(t\rightarrow-\infty,z)=\begin{pmatrix}\varphi_{\mathrm{e}}^{\mathrm{D}}(t\rightarrow-\infty)\\\
\vartheta_{\mathrm{e}}^{\mathrm{D}}(t\rightarrow-\infty)\end{pmatrix}e^{ipz}=\frac{1}{\sqrt{2}}\frac{C_{2}}{m}\mathrm{e}^{\pi\frac{\tau}{2}E_{1}}\left(m+E_{1}-(p-qA_{1})\right)\begin{pmatrix}1\\\
\frac{E_{1}-m}{p-qA_{1}}\end{pmatrix}\mathrm{e}^{-iE_{1}(t-t_{0})}e^{ipz}.$
(147)

For the later medium, inserting Eq. (138) and Eq. (139) into the first row of
Eq. (142), we find

$\begin{split}\varphi_{\mathrm{l}}^{\mathrm{D}}(t\rightarrow+\infty)=\frac{1}{\sqrt{2}}&\left(C_{3}(-\mathrm{e}^{-2\frac{t-t_{0}}{\tau}})^{i\sqrt{\beta_{2}}}+C_{4}(-\mathrm{e}^{-2\frac{t-t_{0}}{\tau}})^{-i\sqrt{\beta_{2}}}\right.\\\
&\left.+\frac{C_{3}}{m}\left[\frac{2}{\tau}\sqrt{\beta_{2}}-(p-qA_{2})\right](-\mathrm{e}^{-2\frac{t-t_{0}}{\tau}})^{i\sqrt{\beta_{2}}}+\frac{C_{4}}{m}\left[-\frac{2}{\tau}\sqrt{\beta_{2}}-(p-qA_{2})\right](-\mathrm{e}^{-2\frac{t-t_{0}}{\tau}})^{-i\sqrt{\beta_{2}}}\right),\end{split}$
(148)

which, upon substituting Eq. (125d) and using $(-1)^{a}=\mathrm{e}^{i\pi a}$,
transforms to the the asymptotic form

$\begin{split}\varphi_{\mathrm{l}}^{\mathrm{D}}(t\rightarrow+\infty)=&\frac{1}{\sqrt{2}}\frac{C_{3}}{m}\mathrm{e}^{-\pi\frac{\tau}{2}E_{2}}\left(m+E_{2}-(p-qA_{2})\right)\mathrm{e}^{-iE_{2}(t-t_{0})}\\\
&+\frac{1}{\sqrt{2}}\frac{C_{4}}{m}\mathrm{e}^{\pi\frac{\tau}{2}E_{2}}\left(m-E_{2}-(p-qA_{2})\right)\mathrm{e}^{iE_{2}(t-t_{0})}.\end{split}$
(149)

Similarly, inserting Eq. (138) and Eq. (139) into the second row of Eq. (142),
then substituting Eq. (113c) into the resulting expression and using
$(-1)^{a}=\mathrm{e}^{i\pi a}$, we find

$\begin{split}\vartheta_{\mathrm{l}}^{\mathrm{D}}(t\rightarrow+\infty)=&\frac{1}{\sqrt{2}}\frac{C_{3}}{m}\mathrm{e}^{-\pi\frac{\tau}{2}E_{2}}\left(m-E_{2}+(p-qA_{2})\right)\mathrm{e}^{-iE_{2}(t-t_{0})}\\\
&+\frac{1}{\sqrt{2}}\frac{C_{4}}{m}\mathrm{e}^{\pi\frac{\tau}{2}E_{2}}\left(m+E_{2}+(p-qA_{2})\right)\mathrm{e}^{iE_{2}(t-t_{0})}.\end{split}$
(150)

This equation may be algebraically manipulated so as to take a form including
an expression that resembles the wave function for the infinitely sharp step
[Eq. (58b)], viz.,

$\begin{split}\vartheta_{\mathrm{l}}^{\mathrm{D}}(t\rightarrow+\infty)=&\frac{1}{\sqrt{2}}\frac{C_{3}}{m}\mathrm{e}^{-\pi\frac{\tau}{2}E_{2}}\left(m+E_{2}-(p-qA_{2})\right)\left(\frac{E_{2}-m}{p-qA_{2}}\right)\mathrm{e}^{-iE_{2}(t-t_{0})}\\\
&+\frac{1}{\sqrt{2}}\frac{C_{4}}{m}\mathrm{e}^{\pi\frac{\tau}{2}E_{2}}\left(m-E_{2}-(p-qA_{2})\right)\left(\frac{-E_{2}-m}{p-qA_{2}}\right)\mathrm{e}^{iE_{2}(t-t_{0})},\end{split}$
(151)

Thus, the Dirac spinor wavefunction for the later medium is

$\begin{split}\psi_{\mathrm{l}}^{\mathrm{D}}(t\rightarrow+\infty,z)=\begin{pmatrix}\varphi_{\mathrm{l}}^{\mathrm{D}}(t\rightarrow+\infty)\\\
\vartheta_{\mathrm{l}}^{\mathrm{D}}(t\rightarrow+\infty)\end{pmatrix}e^{ipz}&=\frac{1}{\sqrt{2}}\frac{C_{3}}{m}\mathrm{e}^{-\pi\frac{\tau}{2}E_{2}}\left(m+E_{2}-(p-qA_{2})\right)\begin{pmatrix}1\\\
\frac{E_{2}-m}{p-qA_{2}}\end{pmatrix}\mathrm{e}^{-iE_{2}(t-t_{0})}e^{ipz}\\\
&+\frac{1}{\sqrt{2}}\frac{C_{4}}{m}\mathrm{e}^{\pi\frac{\tau}{2}E_{2}}\left(m-E_{2}-(p-qA_{2})\right)\begin{pmatrix}1\\\
\frac{-E_{2}-m}{p-qA_{2}}\end{pmatrix}\mathrm{e}^{iE_{2}(t-t_{0})}e^{ipz}.\end{split}$
(152)

Inserting Eq. (119) and Eq. (122) into the first row of Eq. (142), we find the
_general form_ of $\varphi_{\mathrm{e}}$ in the Dirac representation as

$\begin{split}\varphi_{\mathrm{e}}^{\mathrm{D}}=\frac{1}{\sqrt{2}}\frac{C_{2}}{m}&\left[m\zeta_{\mathrm{e}}^{-\mu}(1-\zeta_{\mathrm{e}})^{\nu}{{}_{2}F_{1}}(-\mu+\nu-\lambda,-\mu+\nu+\lambda;1-2\mu;\zeta_{\mathrm{e}})\right.\\\
&\left.-i\frac{2}{\tau}\mu\zeta_{\mathrm{e}}^{-\mu}(1-\zeta_{\mathrm{e}})^{\nu}{{}_{2}F_{1}}(-\mu+\nu-\lambda,-\mu+\nu+\lambda;1-2\mu;\zeta_{\mathrm{e}})\right.\\\
&\left.-i\frac{2}{\tau}\zeta_{\mathrm{e}}^{-\mu+1}\nu(1-\zeta_{\mathrm{e}})^{\nu-1}{{}_{2}F_{1}}(-\mu+\nu-\lambda,-\mu+\nu+\lambda;1-2\mu;\zeta_{\mathrm{e}})\right.\\\
&\left.+i\frac{2}{\tau}\zeta_{\mathrm{e}}^{-\mu+1}(1-\zeta_{\mathrm{e}})^{\nu}\frac{(-\mu+\nu-\lambda)(-\mu+\nu+\lambda)}{1-2\mu}{{}_{2}F_{1}}(-\mu+\nu-\lambda+1,-\mu+\nu+\lambda+1;1-2\mu+1;\zeta_{\mathrm{e}})\right.\\\
&\left.-\left(p-\frac{qA_{1}-qA_{2}\zeta_{\mathrm{e}}}{1-\zeta_{\mathrm{e}}}\right)\zeta_{\mathrm{e}}^{-\mu}(1-\zeta_{\mathrm{e}})^{\nu}{{}_{2}F_{1}}(-\mu+\nu-\lambda,-\mu+\nu+\lambda;1-2\mu;\zeta_{\mathrm{e}})\right].\end{split}$
(153)

Similarly, inserting Eq. (119) and Eq. (122) into the second row of Eq. (142),
we find the general form of $\vartheta_{\mathrm{e}}$ in the Dirac
representation as

$\begin{split}\vartheta_{\mathrm{e}}^{\mathrm{D}}=\frac{1}{\sqrt{2}}\frac{C_{2}}{m}&\left[m\zeta_{\mathrm{e}}^{-\mu}(1-\zeta_{\mathrm{e}})^{\nu}{{}_{2}F_{1}}(-\mu+\nu-\lambda,-\mu+\nu+\lambda;1-2\mu;\zeta_{\mathrm{e}})\right.\\\
&\left.+i\frac{2}{\tau}\mu\zeta_{\mathrm{e}}^{-\mu}(1-\zeta_{\mathrm{e}})^{\nu}{{}_{2}F_{1}}(-\mu+\nu-\lambda,-\mu+\nu+\lambda;1-2\mu;\zeta_{\mathrm{e}})\right.\\\
&\left.+i\frac{2}{\tau}\zeta_{\mathrm{e}}^{-\mu+1}\nu(1-\zeta_{\mathrm{e}})^{\nu-1}{{}_{2}F_{1}}(-\mu+\nu-\lambda,-\mu+\nu+\lambda;1-2\mu;\zeta_{\mathrm{e}})\right.\\\
&\left.-i\frac{2}{\tau}\zeta_{\mathrm{e}}^{-\mu+1}(1-\zeta_{\mathrm{e}})^{\nu}\frac{(-\mu+\nu-\lambda)(-\mu+\nu+\lambda)}{1-2\mu}{{}_{2}F_{1}}(-\mu+\nu-\lambda+1,-\mu+\nu+\lambda+1;1-2\mu+1;\zeta_{\mathrm{e}})\right.\\\
&\left.+\left(p-\frac{qA_{1}-qA_{2}\zeta_{\mathrm{e}}}{1-\zeta_{\mathrm{e}}}\right)\zeta_{\mathrm{e}}^{-\mu}(1-\zeta_{\mathrm{e}})^{\nu}{{}_{2}F_{1}}(-\mu+\nu-\lambda,-\mu+\nu+\lambda;1-2\mu;\zeta_{\mathrm{e}})\right].\end{split}$
(154)

Still similarly, inserting Eq. (130) and Eq. (132) into the first row of Eq.
(142), we find the general form of $\varphi_{\mathrm{l}}$ in the Dirac
representation as

$\begin{split}\varphi_{\mathrm{l}}^{\mathrm{D}}=\frac{1}{\sqrt{2}}\frac{C_{3}}{m}&\left[m\zeta_{\mathrm{l}}^{\sigma}(1-\zeta_{\mathrm{l}})^{\rho}{{}_{2}F_{1}}(\sigma+\rho-\eta,\sigma+\rho+\eta;1+2\sigma;\zeta_{\mathrm{l}})\right.\\\
&\left.-i\frac{2}{\tau}\sigma\zeta_{\mathrm{l}}^{\sigma}(1-\zeta_{\mathrm{l}})^{\rho}{{}_{2}F_{1}}(\sigma+\rho-\eta,\sigma+\rho+\eta;1+2\sigma;\zeta_{\mathrm{l}})\right.\\\
&\left.+i\frac{2}{\tau}\zeta_{\mathrm{l}}^{\sigma+1}\rho(1-\zeta_{\mathrm{l}})^{\rho-1}{{}_{2}F_{1}}(\sigma+\rho-\eta,\sigma+\rho+\eta;1+2\sigma;\zeta_{\mathrm{l}})\right.\\\
&\left.-i\frac{2}{\tau}\zeta_{\mathrm{l}}^{\sigma+1}(1-\zeta_{\mathrm{l}})^{\rho}\frac{(\sigma+\rho-\eta)(\sigma+\rho+\eta)}{1-2\sigma}{{}_{2}F_{1}}(\sigma+\rho-\eta+1,\sigma+\rho+\eta+1;1+2\sigma+1;\zeta_{\mathrm{l}})\right.\\\
&\left.-\left(p-\frac{qA_{2}-qA_{1}\zeta_{\mathrm{l}}}{1-\zeta_{\mathrm{l}}}\right)\zeta_{\mathrm{l}}^{\sigma}(1-\zeta_{\mathrm{l}})^{\rho}{{}_{2}F_{1}}(\sigma+\rho-\eta,\sigma+\rho+\eta;1+2\sigma;\zeta_{\mathrm{l}})\right]\\\
+\frac{1}{\sqrt{2}}\frac{C_{4}}{m}&\left[m\zeta_{\mathrm{l}}^{-\sigma}(1-\zeta_{\mathrm{l}})^{\rho}{{}_{2}F_{1}}(-\sigma+\rho-\eta,-\sigma+\rho+\eta;1-2\sigma;\zeta_{\mathrm{l}})\right.\\\
&\left.+i\frac{2}{\tau}\sigma\zeta_{\mathrm{l}}^{-\sigma}(1-\zeta_{\mathrm{l}})^{\rho}{{}_{2}F_{1}}(-\sigma+\rho-\eta,-\sigma+\rho+\eta;1-2\sigma;\zeta_{\mathrm{l}})\right.\\\
&\left.+i\frac{2}{\tau}\zeta_{\mathrm{l}}^{-\sigma+1}\rho(1-\zeta_{\mathrm{l}})^{\rho-1}{{}_{2}F_{1}}(-\sigma+\rho-\eta,-\sigma+\rho+\eta;1-2\sigma;\zeta_{\mathrm{l}})\right.\\\
&\left.-i\frac{2}{\tau}\zeta_{\mathrm{l}}^{-\sigma+1}(1-\zeta_{\mathrm{l}})^{\rho}\frac{(-\sigma+\rho-\eta)(-\sigma+\rho+\eta)}{1-2\sigma}{{}_{2}F_{1}}(-\sigma+\rho-\eta+1,-\sigma+\rho+\eta+1;1-2\sigma+1;\zeta_{\mathrm{l}})\right.\\\
&\left.-\left(p-\frac{qA_{2}-qA_{1}\zeta_{\mathrm{l}}}{1-\zeta_{\mathrm{l}}}\right)\zeta_{\mathrm{l}}^{-\sigma}(1-\zeta_{\mathrm{l}})^{\rho}{{}_{2}F_{1}}(-\sigma+\rho-\eta,-\sigma+\rho+\eta;1-2\sigma;\zeta_{\mathrm{l}})\right],\end{split}$
(155)

and finally inserting Eq. (130) and Eq. (132) into the second row of Eq.
(142), we find the general form of $\vartheta_{\mathrm{l}}$ in the Dirac
representation as

$\begin{split}\vartheta_{\mathrm{l}}^{\mathrm{D}}=\frac{1}{\sqrt{2}}\frac{C_{3}}{m}&\left[m\zeta_{\mathrm{l}}^{\sigma}(1-\zeta_{\mathrm{l}})^{\rho}{{}_{2}F_{1}}(\sigma+\rho-\eta,\sigma+\rho+\eta;1+2\sigma;\zeta_{\mathrm{l}})\right.\\\
&\left.+i\frac{2}{\tau}\sigma\zeta_{\mathrm{l}}^{\sigma}(1-\zeta_{\mathrm{l}})^{\rho}{{}_{2}F_{1}}(\sigma+\rho-\eta,\sigma+\rho+\eta;1+2\sigma;\zeta_{\mathrm{l}})\right.\\\
&\left.-i\frac{2}{\tau}\zeta_{\mathrm{l}}^{\sigma+1}\rho(1-\zeta_{\mathrm{l}})^{\rho-1}{{}_{2}F_{1}}(\sigma+\rho-\eta,\sigma+\rho+\eta;1+2\sigma;\zeta_{\mathrm{l}})\right.\\\
&\left.+i\frac{2}{\tau}\zeta_{\mathrm{l}}^{\sigma+1}(1-\zeta_{\mathrm{l}})^{\rho}\frac{(\sigma+\rho-\eta)(\sigma+\rho+\eta)}{1-2\sigma}{{}_{2}F_{1}}(\sigma+\rho-\eta+1,\sigma+\rho+\eta+1;1+2\sigma+1;\zeta_{\mathrm{l}})\right.\\\
&\left.+\left(p-\frac{qA_{2}-qA_{1}\zeta_{\mathrm{l}}}{1-\zeta_{\mathrm{l}}}\right)\zeta_{\mathrm{l}}^{\sigma}(1-\zeta_{\mathrm{l}})^{\rho}{{}_{2}F_{1}}(\sigma+\rho-\eta,\sigma+\rho+\eta;1+2\sigma;\zeta_{\mathrm{l}})\right]\\\
+\frac{1}{\sqrt{2}}\frac{C_{4}}{m}&\left[m\zeta_{\mathrm{l}}^{-\sigma}(1-\zeta_{\mathrm{l}})^{\rho}{{}_{2}F_{1}}(-\sigma+\rho-\eta,-\sigma+\rho+\eta;1-2\sigma;\zeta_{\mathrm{l}})\right.\\\
&\left.-i\frac{2}{\tau}\sigma\zeta_{\mathrm{l}}^{-\sigma}(1-\zeta_{\mathrm{l}})^{\rho}{{}_{2}F_{1}}(-\sigma+\rho-\eta,-\sigma+\rho+\eta;1-2\sigma;\zeta_{\mathrm{l}})\right.\\\
&\left.-i\frac{2}{\tau}\zeta_{\mathrm{l}}^{-\sigma+1}\rho(1-\zeta_{\mathrm{l}})^{\rho-1}{{}_{2}F_{1}}(-\sigma+\rho-\eta,-\sigma+\rho+\eta;1-2\sigma;\zeta_{\mathrm{l}})\right.\\\
&\left.+i\frac{2}{\tau}\zeta_{\mathrm{l}}^{-\sigma+1}(1-\zeta_{\mathrm{l}})^{\rho}\frac{(-\sigma+\rho-\eta)(-\sigma+\rho+\eta)}{1-2\sigma}{{}_{2}F_{1}}(-\sigma+\rho-\eta+1,-\sigma+\rho+\eta+1;1-2\sigma+1;\zeta_{\mathrm{l}})\right.\\\
&\left.+\left(p-\frac{qA_{2}-qA_{1}\zeta_{\mathrm{l}}}{1-\zeta_{\mathrm{l}}}\right)\zeta_{\mathrm{l}}^{-\sigma}(1-\zeta_{\mathrm{l}})^{\rho}{{}_{2}F_{1}}(-\sigma+\rho-\eta,-\sigma+\rho+\eta;1-2\sigma;\zeta_{\mathrm{l}})\right].\end{split}$
(156)

###  7.6 Boundary Conditions

At this point, we need to determine the coefficient ratios $C_{3}/C_{2}$ and
$C_{4}/C_{2}$ to be able to calculate probability coefficients. For this
purpose, we apply the boundary conditions to the wavefunctions in the general
forms. Let us first write the Dirac wavefunctions in (Eqs. (153), (155),
(154), and (156)) at $t=t_{0}$, which will imply the use of the following
relations:

  * •

at $t=t_{0}$, $\zeta_{\mathrm{e}}=\zeta_{\mathrm{l}}=-1$ since
$\zeta_{\mathrm{e}}=-\mathrm{e}^{2\frac{t-t_{0}}{\tau}}$ and
$\zeta_{\mathrm{l}}=-\mathrm{e}^{-2\frac{t-t_{0}}{\tau}}$,

  * •

$(-1)^{a}=\mathrm{e}^{i\pi a}$ and $\mathrm{e}^{i\pi}=-1$,

  * •

$\rho=\nu$, $\eta=\mu$, and $\sigma=\lambda$, according to Eqs. (116b),
(116c), (116d), (128b), (128c) and (128d).

At $t=t_{0}$, the wavefunctions in Eqs. (153), (154), (155), and (156) may be
written as

$\varphi_{\mathrm{e}}^{\mathrm{D}}(t=t_{0})=\frac{2^{\nu-1}}{\sqrt{2}}\frac{C_{2}}{m}\mathrm{e}^{\pi\frac{\tau}{2}E_{1}}\left[\left(2m+\mathcal{D}_{1}\right)\mathcal{F}_{1}-\mathcal{D}_{2}\mathcal{F}_{2}\right],$
(157a)
$\varphi_{\mathrm{l}}^{\mathrm{D}}(t=t_{0})=\frac{2^{\nu-1}}{\sqrt{2}}\frac{C_{3}}{m}\mathrm{e}^{-\pi\frac{\tau}{2}E_{2}}\left[\left(2m+\mathcal{D}_{3}\right)\mathcal{F}_{3}+\mathcal{D}_{4}\mathcal{F}_{4}\right]+\frac{2^{\nu-1}}{\sqrt{2}}\frac{C_{4}}{m}\mathrm{e}^{\pi\frac{\tau}{2}E_{2}}\left[\left(2m-\mathcal{D}_{5}\right)\mathcal{F}_{5}+\mathcal{D}_{6}\mathcal{F}_{6}\right],$
(157b)
$\vartheta_{\mathrm{e}}^{\mathrm{D}}(t=t_{0})=\frac{2^{\nu-1}}{\sqrt{2}}\frac{C_{2}}{m}\mathrm{e}^{\pi\frac{\tau}{2}E_{1}}\left[\left(2m-\mathcal{D}_{1}\right)\mathcal{F}_{1}+\mathcal{D}_{2}\mathcal{F}_{2}\right],$
(157c)
$\vartheta_{\mathrm{l}}^{\mathrm{D}}(t=t_{0})=\frac{2^{\nu-1}}{\sqrt{2}}\frac{C_{3}}{m}\mathrm{e}^{-\pi\frac{\tau}{2}E_{2}}\left[\left(2m-\mathcal{D}_{3}\right)\mathcal{F}_{3}-\mathcal{D}_{4}\mathcal{F}_{4}\right]+\frac{2^{\nu-1}}{\sqrt{2}}\frac{C_{4}}{m}\mathrm{e}^{\pi\frac{\tau}{2}E_{2}}\left[\left(2m+\mathcal{D}_{5}\right)\mathcal{F}_{5}-\mathcal{D}_{6}\mathcal{F}_{6}\right],$
(157d)

where

$\mathcal{F}_{1}={{}_{2}F_{1}}(-\mu+\nu-\lambda,-\mu+\nu+\lambda;1-2\mu;-1),$
(158a)
$\mathcal{F}_{2}={{}_{2}F_{1}}(-\mu+\nu-\lambda+1,-\mu+\nu+\lambda+1;1-2\mu+1;-1),$
(158b)
$\mathcal{F}_{3}={{}_{2}F_{1}}(\lambda+\nu-\mu,\lambda+\nu+\mu;1+2\lambda;-1),$
(158c)
$\mathcal{F}_{4}={{}_{2}F_{1}}(\lambda+\nu-\mu+1,\lambda+\nu+\mu+1;1+2\lambda+1;-1),$
(158d)
$\mathcal{F}_{5}={{}_{2}F_{1}}(-\lambda+\nu-\mu,-\lambda+\nu+\mu;1-2\lambda;-1),$
(158e)
$\mathcal{F}_{6}={{}_{2}F_{1}}(-\lambda+\nu-\mu+1,-\lambda+\nu+\mu+1;1-2\lambda+1;-1)$
(158f)

and

$\mathcal{D}_{1}=2E_{1}-(qA_{2}-qA_{1})-\left(p-qA_{2}\right)-\left(p-qA_{1}\right),$
(159a)
$\mathcal{D}_{2}=i\frac{4}{\tau}\frac{(-\mu+\nu-\lambda)(-\mu+\nu+\lambda)}{1-2\mu},$
(159b)
$\mathcal{D}_{3}=2E_{2}+(qA_{2}-qA_{1})-\left(p-qA_{2}\right)-\left(p-qA_{1}\right),$
(159c)
$\mathcal{D}_{4}=i\frac{4}{\tau}\frac{(\lambda+\nu-\mu)(\lambda+\nu+\mu)}{1-2\lambda},$
(159d)
$\mathcal{D}_{5}=2E_{2}-(qA_{2}-qA_{1})+\left(p-qA_{2}\right)+\left(p-qA_{1}\right),$
(159e)
$\mathcal{D}_{6}=i\frac{4}{\tau}\frac{(-\lambda+\nu-\mu)(-\lambda+\nu+\mu)}{1-2\lambda}.$
(159f)

We can now apply the boundary conditions to the wave functions (157a) to
(157b), and (157c) to (157d) as

$\varphi_{\mathrm{e}}^{\mathrm{D}}(t=t_{0})=\varphi_{\mathrm{l}}^{\mathrm{D}}(t=t_{0})$
(160a) and
$\vartheta_{\mathrm{e}}^{\mathrm{D}}(t=t_{0})=\vartheta_{\mathrm{l}}^{\mathrm{D}}(t=t_{0}),$
(160b)

which yields

$\begin{split}\mathrm{e}^{\pi\frac{\tau}{2}E_{1}}\left[\left(2m+\mathcal{D}_{1}\right)\mathcal{F}_{1}-\mathcal{D}_{2}\mathcal{F}_{2}\right]&=\frac{C_{3}}{C_{2}}\mathrm{e}^{-\pi\frac{\tau}{2}E_{2}}\left[\left(2m+\mathcal{D}_{3}\right)\mathcal{F}_{3}+\mathcal{D}_{4}\mathcal{F}_{4}\right]\\\
&+\frac{C_{4}}{C_{2}}\mathrm{e}^{\pi\frac{\tau}{2}E_{2}}\left[\left(2m-\mathcal{D}_{5}\right)\mathcal{F}_{5}+\mathcal{D}_{6}\mathcal{F}_{6}\right]\end{split}$
(161a) and
$\begin{split}\mathrm{e}^{\pi\frac{\tau}{2}E_{1}}\left[\left(2m-\mathcal{D}_{1}\right)\mathcal{F}_{1}+\mathcal{D}_{2}\mathcal{F}_{2}\right]&=\frac{C_{3}}{C_{2}}\mathrm{e}^{-\pi\frac{\tau}{2}E_{2}}\left[\left(2m-\mathcal{D}_{3}\right)\mathcal{F}_{3}-\mathcal{D}_{4}\mathcal{F}_{4}\right]\\\
&+\frac{C_{4}}{C_{2}}\mathrm{e}^{\pi\frac{\tau}{2}E_{2}}\left[\left(2m+\mathcal{D}_{5}\right)\mathcal{F}_{5}-\mathcal{D}_{6}\mathcal{F}_{6}\right],\end{split}$
(161b)

which resolve to

$\frac{C_{3}}{C_{2}}=\mathrm{e}^{\pi\frac{\tau}{2}(E_{1}+E_{2})}\frac{\left[\left(2m-\mathcal{D}_{1}\right)\mathcal{F}_{1}+\mathcal{D}_{2}\mathcal{F}_{2}\right]\left[\left(2m-\mathcal{D}_{5}\right)\mathcal{F}_{5}+\mathcal{D}_{6}\mathcal{F}_{6}\right]-\left[\left(2m+\mathcal{D}_{1}\right)\mathcal{F}_{1}-\mathcal{D}_{2}\mathcal{F}_{2}\right]\left[\left(2m+\mathcal{D}_{5}\right)\mathcal{F}_{5}-\mathcal{D}_{6}\mathcal{F}_{6}\right]}{\left[\left(2m-\mathcal{D}_{3}\right)\mathcal{F}_{3}-\mathcal{D}_{4}\mathcal{F}_{4}\right]\left[\left(2m-\mathcal{D}_{5}\right)\mathcal{F}_{5}+\mathcal{D}_{6}\mathcal{F}_{6}\right]-\left[\left(2m+\mathcal{D}_{3}\right)\mathcal{F}_{3}+\mathcal{D}_{4}\mathcal{F}_{4}\right]\left[\left(2m+\mathcal{D}_{5}\right)\mathcal{F}_{5}-\mathcal{D}_{6}\mathcal{F}_{6}\right]}$
(162a) and
$\frac{C_{4}}{C_{2}}=\mathrm{e}^{\pi\frac{\tau}{2}(E_{1}-E_{2})}\frac{\left[\left(2m+\mathcal{D}_{1}\right)\mathcal{F}_{1}-\mathcal{D}_{2}\mathcal{F}_{2}\right]\left[\left(2m-\mathcal{D}_{3}\right)\mathcal{F}_{3}-\mathcal{D}_{4}\mathcal{F}_{4}\right]-\left[\left(2m-\mathcal{D}_{1}\right)\mathcal{F}_{1}+\mathcal{D}_{2}\mathcal{F}_{2}\right]\left[\left(2m+\mathcal{D}_{3}\right)\mathcal{F}_{3}+\mathcal{D}_{4}\mathcal{F}_{4}\right]}{\left[\left(2m-\mathcal{D}_{3}\right)\mathcal{F}_{3}-\mathcal{D}_{4}\mathcal{F}_{4}\right]\left[\left(2m-\mathcal{D}_{5}\right)\mathcal{F}_{5}+\mathcal{D}_{6}\mathcal{F}_{6}\right]-\left[\left(2m+\mathcal{D}_{3}\right)\mathcal{F}_{3}+\mathcal{D}_{4}\mathcal{F}_{4}\right]\left[\left(2m+\mathcal{D}_{5}\right)\mathcal{F}_{5}-\mathcal{D}_{6}\mathcal{F}_{6}\right]}.$
(162b)

###  7.7 Asymptotic Spinor Wavefunctions and Scattering Probabilities

We shall finally calculate here the sought after scattering probabilities. For
this purpose, we first recall the asymptotic Dirac spinors in Eqs. (147) and
(152):

$\psi_{\mathrm{e}}^{\mathrm{D}}(t\rightarrow-\infty,z)=G_{\mathrm{i}}\begin{pmatrix}1\\\
\frac{E_{1}-m}{p-qA_{1}}\end{pmatrix}\mathrm{e}^{-iE_{1}(t-t_{0})}e^{ipz}$
(163a) and
$\begin{split}\psi_{\mathrm{l}}^{\mathrm{D}}(t\rightarrow+\infty,z)=G_{\mathrm{f}}\begin{pmatrix}1\\\
\frac{E_{2}-m}{p-qA_{2}}\end{pmatrix}\mathrm{e}^{-iE_{2}(t-t_{0})}e^{ipz}+G_{\mathrm{b}}\begin{pmatrix}1\\\
\frac{-E_{2}-m}{p-qA_{2}}\end{pmatrix}\mathrm{e}^{iE_{2}(t-t_{0})}e^{ipz},\end{split}$
(163b) where
$G_{\mathrm{i}}=\frac{1}{\sqrt{2}}\frac{C_{2}}{m}\mathrm{e}^{\pi\frac{\tau}{2}E_{1}}\left(m+E_{1}-(p-qA_{1})\right),$
(163c)
$G_{\mathrm{f}}=\frac{1}{\sqrt{2}}\frac{C_{3}}{m}\mathrm{e}^{-\pi\frac{\tau}{2}E_{2}}\left(m+E_{2}-(p-qA_{2})\right),$
(163d) and
$G_{\mathrm{b}}=\frac{1}{\sqrt{2}}\frac{C_{4}}{m}\mathrm{e}^{\pi\frac{\tau}{2}E_{2}}\left(m-E_{2}-(p-qA_{2})\right).$
(163e)

The later-forward and later-backward scattering coefficients may then directly
be obtained from Eqs. (163c), (163d) and (163e) according to their
definitions, as

$f=\frac{G_{\mathrm{f}}}{G_{\mathrm{i}}}=\frac{\frac{1}{\sqrt{2}}\frac{C_{3}}{m}\mathrm{e}^{-\pi\frac{\tau}{2}E_{2}}\left(m+E_{2}-(p-qA_{2})\right)}{\frac{1}{\sqrt{2}}\frac{C_{2}}{m}\mathrm{e}^{\pi\frac{\tau}{2}E_{1}}\left(m+E_{1}-(p-qA_{1})\right)}=\frac{C_{3}}{C_{2}}\mathrm{e}^{-\pi\frac{\tau}{2}(E_{1}+E_{2})}\frac{\left(m+E_{2}-(p-qA_{2})\right)}{\left(m+E_{1}-(p-qA_{1})\right)}$
(164a) and
$b=\frac{G_{\mathrm{b}}}{G_{\mathrm{i}}}=\frac{\frac{1}{\sqrt{2}}\frac{C_{4}}{m}\mathrm{e}^{\pi\frac{\tau}{2}E_{2}}\left(m-E_{2}-(p-qA_{2})\right)}{\frac{1}{\sqrt{2}}\frac{C_{2}}{m}\mathrm{e}^{\pi\frac{\tau}{2}E_{1}}\left(m+E_{1}-(p-qA_{1})\right)}=\frac{C_{4}}{C_{2}}\mathrm{e}^{-\pi\frac{\tau}{2}(E_{1}-E_{2})}\frac{\left(m-E_{2}-(p-qA_{2})\right)}{\left(m+E_{1}-(p-qA_{1})\right)},$
(164b)

which leads, using the approach in Sec. 3.2.4, to the probabilities

$F=\frac{f^{2}}{b^{2}+f^{2}}$ (165a) and $B=\frac{b^{2}}{b^{2}+f^{2}},$ (165b)

where

$\frac{C_{3}}{C_{2}}=\mathrm{e}^{\pi\frac{\tau}{2}(E_{1}+E_{2})}\frac{\left[\left(2m-\mathcal{D}_{1}\right)\mathcal{F}_{1}+\mathcal{D}_{2}\mathcal{F}_{2}\right]\left[\left(2m-\mathcal{D}_{5}\right)\mathcal{F}_{5}+\mathcal{D}_{6}\mathcal{F}_{6}\right]-\left[\left(2m+\mathcal{D}_{1}\right)\mathcal{F}_{1}-\mathcal{D}_{2}\mathcal{F}_{2}\right]\left[\left(2m+\mathcal{D}_{5}\right)\mathcal{F}_{5}-\mathcal{D}_{6}\mathcal{F}_{6}\right]}{\left[\left(2m-\mathcal{D}_{3}\right)\mathcal{F}_{3}-\mathcal{D}_{4}\mathcal{F}_{4}\right]\left[\left(2m-\mathcal{D}_{5}\right)\mathcal{F}_{5}+\mathcal{D}_{6}\mathcal{F}_{6}\right]-\left[\left(2m+\mathcal{D}_{3}\right)\mathcal{F}_{3}+\mathcal{D}_{4}\mathcal{F}_{4}\right]\left[\left(2m+\mathcal{D}_{5}\right)\mathcal{F}_{5}-\mathcal{D}_{6}\mathcal{F}_{6}\right]}$
(166a) and
$\frac{C_{4}}{C_{2}}=\mathrm{e}^{\pi\frac{\tau}{2}(E_{1}-E_{2})}\frac{\left[\left(2m+\mathcal{D}_{1}\right)\mathcal{F}_{1}-\mathcal{D}_{2}\mathcal{F}_{2}\right]\left[\left(2m-\mathcal{D}_{3}\right)\mathcal{F}_{3}-\mathcal{D}_{4}\mathcal{F}_{4}\right]-\left[\left(2m-\mathcal{D}_{1}\right)\mathcal{F}_{1}+\mathcal{D}_{2}\mathcal{F}_{2}\right]\left[\left(2m+\mathcal{D}_{3}\right)\mathcal{F}_{3}+\mathcal{D}_{4}\mathcal{F}_{4}\right]}{\left[\left(2m-\mathcal{D}_{3}\right)\mathcal{F}_{3}-\mathcal{D}_{4}\mathcal{F}_{4}\right]\left[\left(2m-\mathcal{D}_{5}\right)\mathcal{F}_{5}+\mathcal{D}_{6}\mathcal{F}_{6}\right]-\left[\left(2m+\mathcal{D}_{3}\right)\mathcal{F}_{3}+\mathcal{D}_{4}\mathcal{F}_{4}\right]\left[\left(2m+\mathcal{D}_{5}\right)\mathcal{F}_{5}-\mathcal{D}_{6}\mathcal{F}_{6}\right]},$
(166b)

where

$\mathcal{F}_{1}={{}_{2}F_{1}}(-\mu+\nu-\lambda,-\mu+\nu+\lambda;1-2\mu;-1),$
(167a)
$\mathcal{F}_{2}={{}_{2}F_{1}}(-\mu+\nu-\lambda+1,-\mu+\nu+\lambda+1;1-2\mu+1;-1),$
(167b)
$\mathcal{F}_{3}={{}_{2}F_{1}}(\lambda+\nu-\mu,\lambda+\nu+\mu;1+2\lambda;-1),$
(167c)
$\mathcal{F}_{4}={{}_{2}F_{1}}(\lambda+\nu-\mu+1,\lambda+\nu+\mu+1;1+2\lambda+1;-1),$
(167d)
$\mathcal{F}_{5}={{}_{2}F_{1}}(-\lambda+\nu-\mu,-\lambda+\nu+\mu;1-2\lambda;-1),$
(167e)
$\mathcal{F}_{6}={{}_{2}F_{1}}(-\lambda+\nu-\mu+1,-\lambda+\nu+\mu+1;1-2\lambda+1;-1)$
(167f)

and

$\mathcal{D}_{1}=2E_{1}-(qA_{2}-qA_{1})-\left(p-qA_{2}\right)-\left(p-qA_{1}\right),$
(168a)
$\mathcal{D}_{2}=i\frac{4}{\tau}\frac{(-\mu+\nu-\lambda)(-\mu+\nu+\lambda)}{1-2\mu},$
(168b)
$\mathcal{D}_{3}=2E_{2}+(qA_{2}-qA_{1})-\left(p-qA_{2}\right)-\left(p-qA_{1}\right),$
(168c)
$\mathcal{D}_{4}=i\frac{4}{\tau}\frac{(\lambda+\nu-\mu)(\lambda+\nu+\mu)}{1-2\lambda},$
(168d)
$\mathcal{D}_{5}=2E_{2}-(qA_{2}-qA_{1})+\left(p-qA_{2}\right)+\left(p-qA_{1}\right),$
(168e)
$\mathcal{D}_{6}=i\frac{4}{\tau}\frac{(-\lambda+\nu-\mu)(-\lambda+\nu+\mu)}{1-2\lambda},$
(168f)

with

$\mu=i\sqrt{\alpha_{1}}=i\frac{\tau}{2}E_{1},$ (169a)
$\nu=\frac{1-\sqrt{1-4(\alpha_{0}+\alpha_{1}+\alpha_{2})}}{2}=i\frac{\tau}{2}(qA_{2}-qA_{1}),$
(169b) $\lambda=i\sqrt{\alpha_{2}}=i\frac{\tau}{2}E_{2}.$ (169c)

In this analysis, $E_{1}$ and $E_{2}$ represent the incident electron energy
$E_{\mathrm{i}}$ and the later-forward energy $E_{\mathrm{f}}$, respectively,
and the later-backward energy is defined as $E_{\mathrm{b}}=-E_{\mathrm{f}}$.

###  7.8 Dimensional Analysis for Time Constant $\tau$

We have used natural units ($c=\hbar=1$) for convenience throughout the
calculations. However, to compare the theoretical results with experimental
results, it is necessary to denormalize some of the obtained formulas.
Specifically, we will need here to denormalize time in order to compare the
transition times in the smooth potential with the de Broglie period of the
electron.

This denormalization can be performed as follows. In _standard_ units, which
we shall distinguish from our natural units by tilde accents, the temporal
part of the wavefunction takes the form [1, 2, 3, 4, 5, 6]

$\psi\propto\mathrm{e}^{-i\tilde{E}\tilde{t}/\hbar},$ (170)

with $\tilde{E}$ and $\tilde{t}$ measured in joules and seconds, respectively.
For simplicity, we express the energy, $\tilde{E}$, in terms of multiple of
the electron rest mass energy, i.e., $\tilde{E}=\kappa\left(mc^{2}\right)$,
which transforms Eq. (170) into

$\psi\propto\mathrm{e}^{-i\kappa mc^{2}\tilde{t}/\hbar}.$ (171)

Moreover, it makes sense to also express the _natural_ -unit energy, $E$, in
terms of that multiple ($\kappa$) of the electron mass, i.e., $E=\kappa m$,
which alters Eq. (171) to

$\psi\propto\mathrm{e}^{-iEc^{2}\tilde{t}/\hbar}.$ (172)

Comparing this expression with its _natural_ -unit counterpart,

$\psi\propto\mathrm{e}^{-iEt},$ (173)

finally yields

$t=\frac{c^{2}\tilde{t}}{\hbar},$ (174)

whose dimension is $1/m$ since the dimension of $E$ is $m$.

Thus, the natural-unit dimensionless parameter $\tau$ in Eq. (109)
denormalizes as

$\tilde{\tau}=\tau\frac{\hbar}{mc^{2}},$ (175)

measured in seconds, which we call $\eta$ in the paper.

##  8 Non-relativistic Regime

###  8.1 Non-relativistic Limit of the Dirac Equation

In order to determine the _non-relativistic limit_ ,

$v\ll{c},$ (176)

of the Dirac equation, we shall first recast that equation in its (non-
covariant) Schrödinger form because the covariant form, introduced in Sec. 2
and applied in the report to address the core of the problem, does not provide
explicit access to the dynamic energy, which needs to be compared with the
rest energy to determine that limit.

Let us start by recalling the covariant form of the Dirac equation, given by
Eq. (17):

$\left[\gamma^{\mu}(i\partial_{\mu}-qA_{\mu})-m\right]\psi=0.$ (177)

Separating in this equation the temporal and spatial operators according to
Einstein summation convention yields

$\left[\gamma^{0}(i\partial_{0}-qA_{0})+\gamma^{i}(i\partial_{i}-qA_{i})-m\right]\psi=0,$
(178)

or, factoring out $\gamma^{0}$,

$\gamma^{0}\left[i\partial_{0}+\left(\gamma^{0}\right)^{-1}\gamma^{i}i\partial_{i}-qA_{0}-\left(\gamma^{0}\right)^{-1}\gamma^{i}qA_{i}-\left(\gamma^{0}\right)^{-1}m\right]\psi=0.$
(179)

Dropping $\gamma^{0}$ and subsequently using Eq. (12) with the fact that,
according to Eq. (10f), $\left(\gamma^{0}\right)^{-1}=\gamma^{0}$, simplifies
that equation to

$\left[i\partial_{0}+\alpha^{i}i\partial_{i}-qA_{0}-\alpha^{i}qA_{i}-\gamma^{0}m\right]\psi=0,$
(180)

or, isolating the temporal derivative term,

$i\partial_{0}\psi=\left[\alpha^{i}\left(-i\partial_{i}+qA_{i}\right)+\gamma^{0}m+qA_{0}\right]\psi.$
(181)

We may now write the last equation in vector form using the definitions