this equation as

$\sum_{j=1}^{p}F_{j}\left(\omega_{Dj}\right)=1,\qquad
F_{j}\left(\omega_{Dj}\right)=\frac{5\rho_{j}T_{e}}{T_{j}}\
\frac{\left[1-i\left(\nu_{jn}/\omega_{Dj}\right)\right]k^{2}V_{Tj}^{2}}{3\omega_{Dj}^{2}-5k^{2}V_{Tj}^{2}\left[1-i\left(1+2\delta_{jn}/5\right)\nu_{jn}/\omega_{Dj}\right]}.$
(103)

To the first-order accuracy with respect to the small parameters
$i\nu_{sn}/\omega_{Ds}$ and $i\gamma/\omega_{Ds}$, expanding each
$F_{j}(\omega_{Dj}t)$ in Taylor series to the first-order (linear) terms, we
obtain

$F_{j}\left(\omega_{Dj}\right)\approx
F_{j0}(\omega_{Dj})+i\gamma\left.\frac{\partial
F_{j0}}{\partial\omega_{Dj}}\right|_{\omega_{Dj}=(\omega_{Dj})_{r}}+i\operatorname{Im}F_{j}(\omega_{Dj}),$

where $F_{j0}$ is the function $F_{j}(\omega_{Dj})$ with neglected terms
$\propto i\nu_{sn}/\omega_{Ds}$,
$F_{j0}(\omega_{Dj})\approx\operatorname{Re}F_{j}((\omega_{Dj})_{r})$, while
the argument of $i\operatorname{Im}F_{j}(\omega_{Dj})$ still includes full
$\omega_{Dj}$ with linear $i\nu_{sn}/\omega_{Ds}$ corrections. Assuming that
we know all roots $\omega_{Dj}\approx(\omega_{Dj})_{r}$ of the zeroth-order
equation $\sum_{j=1}^{p}\operatorname{Re}F_{j}(\omega_{Dj})=1$, for each of
these $n$ roots we have the equation

$i\gamma\sum_{j=1}^{p}\frac{\partial
F_{j0}}{\partial\omega_{Dj}}+i\sum_{j=1}^{p}\operatorname{Im}F_{j}\left(\omega_{Dj}\right)=0,$

yielding

$\gamma=-\
\left.\frac{\sum_{j=1}^{p}\operatorname{Im}F_{j}(\omega_{Dj})}{\sum_{j=1}^{p}\partial\operatorname{Re}F_{j}/\partial\omega_{Dj}}\right|_{\omega_{Dj}=(\omega_{Dj})_{r}},$
(104)

where $\operatorname{Im}F_{j}(\omega_{Dj})$ with
$\omega_{Dj}=(\omega_{Dj})_{r}$ contain only small linear terms
$\propto\nu_{sn}/\omega_{Ds}$.

According to Eq. (103), we have

$\operatorname{Re}F_{j}=F_{j0}(\omega_{Dj})\approx\frac{5\rho_{j}T_{e}k^{2}V_{Tj}^{2}}{T_{j}\left(3\omega_{Dj}^{2}-5k^{2}V_{Tj}^{2}\right)},$
(105)

yielding

$\frac{\partial F_{j0}}{\partial\omega_{Dj}}\approx-\
\frac{30\rho_{j}T_{e}k^{2}V_{Tj}^{2}\omega_{Dj}}{T_{j}\left(3\omega_{Dj}^{2}-5k^{2}V_{Tj}^{2}\right)^{2}}=-\
\frac{6\omega_{Dj}F_{j0}\left(\omega_{Dj}\right)}{3\omega_{Dj}^{2}-5k^{2}V_{Tj}^{2}}.$
(106)

Expanding the expression for $F_{j}\left(\omega_{Dj}\right)$ in Taylor series
to the linear term $\propto i\nu_{jn}/\omega_{Dj}$, we obtain

$\operatorname{Im}F_{j}\left(\omega_{Dj}\right)\approx-\
\frac{i\nu_{j}}{\omega_{Dj}}\
\frac{3\omega_{Dj}^{2}+2k^{2}V_{Tj}^{2}\delta_{jn}}{3\omega_{Dj}^{2}-5k^{2}V_{Tj}^{2}}\
F_{j0}\left(\omega_{Dj}\right),$ (107)

so that Eqs. (104)-(107) yield

$\displaystyle\gamma$ $\displaystyle\approx-\
\frac{\sum_{j=1}^{p}\frac{\nu_{jn}}{\omega_{Dj}}\frac{3\omega_{Dj}^{2}+2k^{2}V_{Tj}^{2}\delta_{jn}}{3\omega_{Dj}^{2}-5k^{2}V_{Tj}^{2}}\
F_{j0}\left(\omega_{Dj}\right)}{2\sum_{j=1}^{p}\frac{3\omega_{Dj}}{3\omega_{Dj}^{2}-5k^{2}V_{Tj}^{2}}\
F_{j0}(\omega_{Dj})}$ $\displaystyle=-\
\frac{\sum_{j=1}^{p}\frac{\nu_{jn}}{\omega_{Dj}}\
\frac{5\rho_{j}T_{e}k^{2}V_{Tj}^{2}\left(3\omega_{Dj}^{2}+2k^{2}V_{Tj}^{2}\delta_{jn}\right)}{T_{j}\left(3\omega_{Dj}^{2}-5k^{2}V_{Tj}^{2}\right)^{2}}}{2\sum_{j=1}^{p}\frac{15\rho_{j}T_{e}k^{2}V_{Tj}^{2}\omega_{Dj}}{T_{j}\left(3\omega_{Dj}^{2}-5k^{2}V_{Tj}^{2}\right)^{2}}}.$
(108)

In particular, in the SSI case ($p=1$, $j\rightarrow i$), we have

$\gamma\approx-\
\frac{\nu_{i}\left(3\omega_{Di}^{2}+2k^{2}V_{Ti}^{2}\delta_{in}\right)}{6\omega_{Di}^{2}}.$
(109)

These expressions clearly demonstrate that in the SWL the growth/damping rate
$\gamma$ is always negative, regardless of the driving electric field
amplitude. This means that in the large-$k$ limit all waves are absolutely
stable. Hence, somewhere in the intermediate range between the LWL and SWL,
there must be some optimal values of $\vec{k}$ where the instability growth
rate reaches one or several maxima and then goes down to the negative values
described by Eqs. (108) or (109). This leads to the conclusion that the
employed fluid model can be safely used for instabillity modeling with no need
for any additional damping mechanisms at short wavelengths to stabilize there
wave behavior. Though this analysis has neglected a few minor factors, such as
the charge separation, etc., the neglected factors are mostly wave-stabilizing
and could not change the main conclusion.

## Appendix B: LIST OF MAJOR NOTATIONS

$A_{s}$, is defined by Eq. (19), see also Eq. (27);

$B_{s}$, is defined by Eq. (19), see also Eq. (28);

$\vec{B}_{0}$ is the external magnetic field ($B_{0}=|\vec{B}_{0}|$);

$\hat{b}=$ $\vec{B}_{0}/B_{0}$ is the unit vector along $\vec{B}_{0}$;

$C_{s}$ is the isothermal ion-acoustic speed [see Eq. (11)];

$D(\omega,\vec{k})$ is the dispersion function in the LWL, see Eq. (34)
[$D_{0}(\omega_{r},\vec{k})$ is the dominant real part of $D(\omega,\vec{k})$,
Eq. (36)];

$\vec{E}_{0}$ is the external electric and magnetic field
($E_{0}=|\vec{E}_{0}|$);

$E_{\mathrm{Thr}}$ is the instability threshold field;

$E_{\mathrm{Thr}}^{\min}$ is the temperature-modified minimum FBI threshold
field [see Eq. (12)];

$N_{s}$ is defined by Eq. (21);

$F(\zeta_{e})$, see Eq. (47);

$G_{j}$ is the quantity defined in Eq. (43);

$\vec{K}_{s}=\delta\vec{V}_{s}/\left(\alpha_{s}\phi+\eta_{s}+\tau_{s}\right)$
is a temporary notation used in Sec. IV;

$\vec{k}$ is the wavevector ($k=|\vec{k}|$ is the wavenumber);

$M_{sn}=m_{s}m_{n}/(m_{s}+m_{n})$ is the effective mass of the two colliding
particles ($s$ and $n)$;

$m_{s}$ is the $s$-species particle mass;

$n_{s}$ is the $s$-species particle number density;

$p$ is the total number of the ion species;

$q_{s}$ is the $s$-species particle electric charge ($q_{e}=-e$);

$R$ is defined by Eq. (85);

$T_{s}$ is the $s$-species particle temperature (in energy units);

$\vec{U}_{j}\equiv\vec{V}_{e0}-\vec{V}_{j0}$ is the difference between the
undisturbed electron and ion drifts [see Eq. (7)];

$\vec{V}_{0}$ is the $\vec{E}_{0}\times\vec{B}_{0}$-drift velocity;

$\vec{V}_{s0}$ is the $s$-species mean fluid velocity;

$V_{Ts}=(T_{s0}/m_{s})^{1/2}$ is the mean thermal speed of the $s$-species
particles;

$\vec{V}_{\mathrm{ph}}=\omega/\vec{k}$ is the wave phase velocity

$\alpha_{s}\equiv T_{e0}q_{s}/(T_{s0}e)$ is a temporary parameter introduced
in Eq. (17);

$\alpha_{n}$ is the neutral-particle polarizability, Eq. (52);

$\gamma$ is the wave growth/damping rate;

$\delta A\propto\exp[i(\vec{k}\cdot\vec{r}-\omega t)]$ denotes a harmonic wave
perturbation of any scalar or vector quantity $A$ ($A_{0}$ is the undisturbed
value);

$\delta_{sn}$ is the mean collisional energy-loss fraction
($\delta_{sn}=\delta_{sn}^{\mathrm{elas}}=2m_{s}/(m_{s}+m_{n})$ for elastic
collisions);

$\epsilon_{0}$ is the permittivity of free space;

$\varepsilon$ is a small parameter, see Eq. (63);

$\zeta_{s}=\Omega_{s}/kV_{0}$ is a normalized quantity introduced in Sec. V.1
(there $\Omega_{s}\approx\Omega_{sr}$);

$\eta_{s}$ is a normalized perturbation of the $s$-species particle density,
$n_{s}$ [see Eq. (14)];

$\Theta_{j}=(\kappa_{j}/\kappa_{e})^{1/2}$ is a small parameter introduced in
Sec. (48);

$\theta$ is the angle (in radians) from $\vec{V}_{0}$ to $\vec{k}$ (the ‘flow’
angle);

$\kappa_{s}=\omega_{cs}/\nu_{sn}$ is the magnetization ratio of the
$s$-species particles;

$\lambda_{Ds}=[\epsilon_{0}T_{s0}/(e^{2}n_{s0})]^{1/2}$ is the ‘partial’ Debye
length of the $s$-species;

$\mu_{s}$ is a complex quantity introduced in Eq. (19);

$\nu_{sn}$ is the mean collision frequency of the $s$-species particles with
neutrals;

$\xi_{j}$, see Eq. (47);

$\rho_{j}=(q_{j}/e)(n_{j0}/n_{e0})$ is introduced in Eq. (22);

$\sigma_{sn}$ is the $s$-$n$ collisional cross-section;

$\tau_{s}$ is a normalized perturbation of the $s$-species particle
temperature, $T_{s}$ [see Eq. (14)];

$\Phi$ is the electrostatic potential;

$\phi$ is a normalized perturbation of the electrostatic potential $\Phi$ [see
Eq. (14)];

$\chi_{j}=\theta+\arctan\kappa_{j}$ is an angle (in radians), see also Eq.
(31);

$\psi_{j}$ is the quantity defined by Eq. (37);

$\Psi$ is the quantity defined by Eq. (42);

$\omega_{Ds}\equiv\omega-\vec{k}\cdot\vec{V}_{s0}$ is the Doppler-shifted
frequency in the frame of reference moving with the $s$-species mean flow,
$\vec{V}_{s0}$ [see Eq. (13)];

$\Omega_{s}$ is the gyrofrequency of the $s$-species particles;

$\omega=\omega_{r}+i\gamma$ is the wave frequency (both $\omega_{r}$ and
$\gamma$ are real);

Subscripts $\parallel$ and $\perp$ relate to the vector components parallel
and perpendicular to $\vec{B}_{0}$, respectively.

###### Acknowledgements.

We acknowledge the support of this work by NSF Grant No. 1903416, NASA grants
80NSSC20K1272, 80NSSC21K0737, 80NSSC21K1684, and contract NNG09FA40C.

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

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# Energy densities in quantum mechanics

V. Stepanyan1 and A.E. Allahverdyan1,2 1Yerevan State University, Yerevan,
Armenia
2Alikhanian National Laboratory, Yerevan, Armenia

###### Abstract

Quantum mechanics does not provide any ready recipe for defining energy
density in space, since the energy and coordinate do not commute. To find a
well-motivated energy density, we start from a possibly fundamental,
relativistic description for a spin-$\frac{1}{2}$ particle: Dirac’s equation.
Employing its energy-momentum tensor and going to the non-relativistic limit
we find a locally conserved non-relativistic energy density that is defined
via the Terletsky-Margenau-Hill quasiprobability (which is hence selected
among other options). It coincides with the weak value of energy, and also
with the hydrodynamic energy in the Madelung representation of quantum
dynamics, which includes the quantum potential. Moreover, we find a new form
of spin-related energy that is finite in the non-relativistic limit, emerges
from the rest energy, and is (separately) locally conserved, though it does
not contribute to the global energy budget. This form of energy has a
holographic character, i.e., its value for a given volume is expressed via the
surface of this volume. Our results apply to situations where local energy
representation is essential; e.g. we show that the energy transfer velocity of
a free Gaussian wave-packet (and also Airy wave-packet) is larger than its
group (i.e. coordinate-transfer) velocity.

##  I Introduction

Born’s rule of quantum mechanics provides for the time-depedendent
Schroedinger equation the locally conserved coordinate density and the
corresponding current Landau and Lifshitz (1958). The very understanding of
quantum mechanics is based on this picture. It will be useful to have its
energy analogue: a conserved energy density and the corresponding energy
current. They would describe how energy moves and distributes in space and
time. They would also describe the energy content of non-normalizable states.
Such states are employed in quantum mechanics Landau and Lifshitz (1958);
Berry and Balazs (1979), but the mean energy for them is not defined. The
major obstacle for defining operationally the energy density and current in
quantum mechanics is that the energy and coordinate do not commute Cohen
(1996). This may be solved via a quasiprobability for the energy and
coordinate, but it is unclear which one to select due to the variety of
quasiprobabilities and their results.

We start from a possibly fundamental description that underlies the non-
relativistic quantum mechanics, viz. Dirac’s equation for relativistic
covariant bispinor field Davydov (1991); Berestetskii  _et al._ (1982);
Thaller (2013). It has an energy-momentum tensor that contains among its
components a unique energy density Berestetskii  _et al._ (1982). As seen
below, the energy density is unique for two reasons. First, it follows from
the unique relativistically invariant Lagrangian. In contrast, non-
relativistic Lagrangian is not unique and different Lagrangians produce
different results for non-relativistic energy current. Second, the energy
density is immune to the known dilemma of symmetric versus anti-symmetric
energy-momentum tensor Landau and Lifshitz (1958).

Dirac’s equation is a cornerstone of quantum field theory, but the time-
dependent dynamics of the single-particle Dirac’s equation is still
controversial Davydov (1991); Thaller (2013); Berestetskii  _et al._ (1982).
It is unclear what the definition of the operational coordinate is, how to
deal with negative energies, etc. These issues disappear in the non-
relativistic limit. This limit employs the power of relativistic field theory
with its important predictions (spin, antiparticles, etc) Davydov (1991);
Berestetskii  _et al._ (1982); Thaller (2013), but avoids problems related to
the relativistic time-space description.

We show that in the non-relativistic limit Dirac’s energy density naturally
separates into the non-relativistic energy and the rest energy. The first of
them is based on the Terletsky-Margenau-Hill quasiprobability, and also
coincides with Madelung’s energy density that includes the quantum potential.
It correctly recovers the full (mean) energy of the quantum particle. In
energy eigenstates with wave-function $\phi_{n}({\bf r})$ and energy $E_{n}$,
this expression predicts energy density $E_{n}|\phi_{n}({\bf r})|^{2}$.

We applied this non-relativistic energy density for deducing the energy
transfer velocity of a wave-packet and showed that for Gaussian (and also
Airy) packets this velocity is generically larger than the coordinate transfer
velocity. Energy transfer velocity cannot be studied without defining the
energy density, but several conclusions on this velocity are independent from
details of this definition.

An interesting feature of the non-relativistic energy density is that for free
motion it does assume negative values. The overall energy is still positive,
but lower than the local rest energy. We found states, where this negativity
takes place for finite times only. We conjecture that normalized pure states
cannot provide non-negative energy density for all times and coordinates.

There is a fine-grained structure to the rest energy, since it can be
separated into the bulk contribution and an additional non-relativistic, spin-
dependent part that is locally conserved. This new form of energy has a
holographic character, since it is a divergence of a local vector; i.e. its
content in a volume $V$ is expressed via the surface integral over $\partial
V$. It nullifies for finite-motion stationary states (without magnetic field),
but is not zero already for stationary states that describe scattering. The
full space-integral of this new energy density is zero, i.e. it does not
contribute to the total energy balance. We shall illustrate this new form of
energy via examples.

This paper is organized as follows. Next section recalls the energy density
and current deduced from Dirac’s equation. Section III describes the non-
relativistic limit of the energy density. This section also makes connection
with the hydrodynamic approach and discusses the negativity of the energy
density, Section IV shows that there is an additional, holographic, locally-
conserving energy that comes from the rest-energy; its features are
exemplified in section VI. Section V shows how to define the energy transfer
velocity and discusses some of its features for Gaussian and Airy wave-
packets. We briefly summarize in the last section.

Appendices contain several relevant messages, as well as some derivations.
Appendix A recalls how expressions (4, 5) for the energy density and current
are derived from the unique relativistic invariant Langrangian and the energy-
momentum tensor. Appendix B studies the derivation of the energy density (12)
from a non-relativistic Lagrangian. This Lagrangian is not unique, and the two
main choices are presented in detail. Appendix C looks at 1d Gaussian wave-
packets, their energy density and energy transfer speed comparing it with the
coordinate transfer speed. Appendix D does the same for Airy wave-packets and
also explains the physical meaning of these packets via the energy density
(12). Eq. (64) in Appendix C provides an example of 1d normalizable state,
whose energy density $\rho(\bm{r},t)$ is negative for a finite ranges of
$\bm{r}$ and $t$ only. Appendix E calculates the holographic energy (20) for
Landau levels (free 2d electrons in a magnetic field) that are basic for the
modern solid state physics.

##  II Energy density and flux

Dirac’s equation governs bispinor $\psi$ for a relativistic spin-$\frac{1}{2}$
particle Berestetskii  _et al._ (1982); Davydov (1991); Thaller (2013):

$\displaystyle i\hbar\dot{\psi}={\cal H}\psi\equiv
mc^{2}\beta\psi+U(\bm{r},t)\psi-i\hbar c(\bm{\nabla}\bm{\alpha}\psi),$ (1)
$\displaystyle\bm{\alpha}=\begin{pmatrix}0&\bm{\sigma}\\\
\bm{\sigma}&0\end{pmatrix},~{}~{}\beta=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix},~{}~{}\partial_{t}{\psi}=\dot{\psi},~{}~{}\bm{\nabla}=\partial_{\bm{r}},$
(2)

where $U(\bm{r},t)$ is a potential energy,
$\bm{\alpha}=\\{\alpha_{i}\\}_{i=1}^{3}$ and $\beta$ are $4\times 4$ Dirac’s
matrices, $\bm{\sigma}=\\{\sigma_{i}\\}_{i=1}^{3}$ are Pauli’s matrices.
$U(\bm{r},t)$ is externally time-dependent indicating on processes of work-
exchange. The local energy conservation, energy density and flux read (resp.)
Berestetskii  _et al._ (1982); Greiner (1990):

$\displaystyle\dot{\varrho}(\bm{r},t)+\bm{\nabla}\bm{\mathcal{J}}(\bm{r},t)-\dot{U}(\bm{r},t)\psi^{\dagger}\psi=0,$
(3)
$\displaystyle\varrho=\frac{i\hbar}{2}\bigg{(}\psi^{\dagger}\dot{\psi}-\dot{\psi}^{\dagger}\psi\bigg{)}=$
(4) $\displaystyle=\frac{i\hbar
c}{2}\bigg{[}(\bm{\nabla}\psi^{\dagger}\bm{\alpha})\psi-\psi^{\dagger}(\bm{\nabla}\bm{\alpha}\psi)\bigg{]}+mc^{2}\psi^{\dagger}\beta\psi+U\psi^{\dagger}\psi,$
$\displaystyle\bm{\mathcal{J}}=\frac{i\hbar
c}{2}\bigg{(}\psi^{\dagger}\bm{\alpha}\dot{\psi}-\dot{\psi}^{\dagger}\bm{\alpha}\psi\bigg{)}=$
$\displaystyle=\frac{i\hbar
c^{2}}{2}\bigg{[}(\bm{\nabla}\psi^{\dagger}\bm{\alpha})\bm{\alpha}\psi-\psi^{\dagger}\bm{\alpha}(\bm{\nabla}\bm{\alpha}\psi)\bigg{]}+Uc\psi^{\dagger}\bm{\alpha}\psi,$
(5)

where (3) is the local conservation law with
$\dot{U}(\bm{r},t)\psi^{\dagger}\psi$ being the local source of work. Eqs.
(3–5) are derived via the unique, relativistically invariant Lagrangian and
the energy-momentum tensor; see Appendix A. We emphasize that the uniqueness
of (3–5) follows from the relativism, and it is absent in the non-relativistic
physics. In particular, it is absent in the non-relativistic Lagrangian
formalism; see Appendix B.

Eq. (1) can be employed for checking (3–5). $\varrho$ in (4) relates to the
$00$-component of the energy-momentum tensor of Dirac’s field Berestetskii
_et al._ (1982). For relativistic fields with non-zero spin, the definition of
the latter tensor is known to be non-unique: the canonic (Noether’s) tensor is
not symmetric, and there is symmetric (Belinfante-Rosenfeld) tensor that is
employed in gravitation and that agrees with the canonic one globally, but not
locally Berestetskii  _et al._ (1982). However, for Dirac’s field the two
tensors relate to each other via the symmetrization of indices; hence this
ambiguity of energy-momentum tensors does not affect $\varrho$ and
$\bm{\mathcal{J}}$ (up to a rotor field for the latter) Berestetskii  _et al._
(1982). Hence the energy-density for Dirac’s field is indeed well-defined in
contrast to e.g. the momentum density.

Eqs. (1, 4) imply expectedly for the mean energy:

$\displaystyle\int{\rm d}^{3}r\varrho(\bm{r},t)=\int{\rm
d}^{3}r\psi^{\dagger}(\bm{r},t){\cal H}\psi(\bm{r},t).$ (6)

The conservation of density $\psi^{\dagger}\psi$ reads from (1) Berestetskii
_et al._ (1982); Davydov (1991); Thaller (2013):

$\displaystyle\partial_{t}(\psi^{\dagger}\psi)+\bm{\nabla}[c\psi^{\dagger}\bm{\alpha}\psi]=0.$
(7)

Note that for $U=0$ the density $\psi^{\dagger}\psi$ is non-negative, in
constrast to the energy density $\varrho$ in (4) that can hold
$\varrho(\bm{r})<0$ for some $\bm{r}$; e.g. because $\beta$ in (2) has a
negative eigenvalue Davydov (1991). This is an aspect of the spin-statistics
theorem: half-integer spin wave-equations lead to non-negative density, but
not a non-negative energy density. For integer spins the situation is
opposite: the energy density is non-negative, while the density is not Davydov
(1991).

##  III The non-relativistic limit

The introduction of this limit in (1) starts with representing the bispinor
$\psi$ via two spinors $\varphi$ and $\chi$ and introducing a phase-factor
with the rest energy $mc^{2}$ Berestetskii  _et al._ (1982); Greiner (1990):

$\psi=\begin{bmatrix}\varphi\\\ \chi\end{bmatrix}e^{-\frac{i}{\hbar}mc^{2}t},$
(8)

Eqs. (1, 8) lead to differential equations for two spinors $\varphi$ and
$\chi$. Now take $\phi=\varphi+O(c^{-2})$ such that for $c\to\infty$ the
Schroedinger equation holds for $\phi$ with Hamiltonian $H$:

$\displaystyle i\hbar\dot{\phi}=H\phi,\quad
H\equiv-\frac{\hbar^{2}}{2m}\Delta+U.$ (9)

while the differential equation for $\chi$ is solved as

$\displaystyle\chi=-\frac{i\hbar}{2mc}\bm{\nabla}\bm{\sigma}\phi+\mathcal{O}(c^{-3}).$
(10)

We retain $\chi$, since (3) has terms $mc^{2}\chi^{\dagger}\chi$ that survive
in the non-relativistic limit. Using (10, 3, 9) we get

$\displaystyle\varrho=\rho+mc^{2}(\varphi^{\dagger}\varphi+\chi^{\dagger}\chi)+\mathcal{O}(c^{-2}),$
(11)
$\displaystyle\rho=-\frac{\hbar^{2}}{4m}(\,[\Delta\phi^{\dagger}]\phi+\phi^{\dagger}\Delta\phi)+U\phi^{\dagger}\phi.$
(12)

Since $\varphi^{\dagger}\varphi+\chi^{\dagger}\chi$ is the particle density
[cf. (8, 6)], the energy density $\varrho$ in (11) separates into the non-
relativistic energy density $\rho$ (12) and the rest energy. Now $\rho$ is
locally conserved as deduced from (9):

$\displaystyle\dot{\rho}+\bm{\nabla}\bm{J}-\dot{U}(\bm{r},t)\phi^{\dagger}\phi=0,$
(13) $\displaystyle{\rho}={\rm
tr}\Big{[}R\,\\{H,\,|\bm{r}\rangle\langle\bm{r}|\\}\Big{]},$ (14)
$\displaystyle{\bm{J}}=\frac{\hbar^{2}}{2m}\Re\Big{[}\phi^{\dagger}\bm{\nabla}\dot{\phi}-[\bm{\nabla}\phi^{\dagger}]\dot{\phi}\Big{]},$
(15)

where $\dot{U}(\bm{r},t)\phi^{\dagger}\phi$ in (13) is the work term,
$R=|\bm{\phi}\rangle\langle\bm{\phi}|$ is the density matrix, and
$\\{a,b\\}=\frac{1}{2}(ab+ba)$ is the anticommutator. Eq. (12) implies for the
mean non-relativistic energy [cf. (6)]:

$\displaystyle\int{\rm d}^{3}r\,\rho(\bm{r},t)=\int{\rm
d}^{3}r\,\phi^{\dagger}(\bm{r},t)H\phi(\bm{r},t),$ (16)

where $\int{\rm d}^{3}r\,\phi^{\dagger}(\bm{r},t)\phi(\bm{r},t)=1$. For
stationary states of (9) the spin and coordinate factorize,
$\phi_{s}=|s\rangle{\phi}(\bm{r})$, where
$H{\phi}(\bm{r})=E_{n}{\phi}(\bm{r})$. Now $\rho$ reduces to the particle’s
density

$\displaystyle\rho(\bm{r})=E_{n}|\phi_{n}({\bf r})|^{2}.$ (17)

Moreover, $\bm{J}=0$ whenever $\phi_{n}({\bf r})=\phi_{n}^{*}({\bf r})$, i.e.
the energy does not flow in stationary states with finite motion.

Eqs. (14, 15) show that $\rho$ and $\bm{J}$ relate to the Terletsy-Margenau-
Hill quasiprobability Kirkwood (1933); Terletsky (1937); Dirac (1945); Barut
(1957); Margenau and Hill (1961); Allahverdyan (2014); Lostaglio (2018); Hofer
(2017); Łobejko (2022); Francica (2022). $\rho$ is interpreted as the joint
probability of energy and coordinate and $\bm{J}$ as the joint probability of
energy and probability current. The usage of this quasiprobability for the
local energy was postulated in McLennan  _et al._ (1989). Here we derived this
postulate from Dirac’s equation. Eq. (14) is also found if assuming two non-
interacting particles in a state $\phi(\bm{r}_{1},\bm{r}_{2})$, we trace out
the second particle, and denote by $R$ the density matrix of the first
particle.

Appendix B explains that (12, 15) can be derived via a Lagrangian for (9). The
choice of the non-relativistic Lagrangian is not unique in contrast to the
relativistic situation. The non-relativistic Lagrangian that produces (12, 15)
is unusual, since contains second-order space-derivatives.

###  III.1 Relations with the hydrodynamic approach

Eq. (12) coincides with the prediction of the quantum hydrodynamic approach
Madelung ; Takabayasi (1952). Define
$\phi_{s}(\bm{r},t)=|\phi_{s}(\bm{r},t)|e^{i\Gamma_{s}(\bm{r},t)/\hbar}$,
where $s=1,2$ refers to the spinor components. Now the quantum dynamics
reduces to a classical hydrodynamics (in Euler’s picture), for two fluids
($s=1,2$), where $\bm{v}_{s}(\bm{r},t)=\bm{\nabla}\Gamma_{s}(\bm{r},t)$ is the
local velocity, while $|\phi_{s}(\bm{r},t)|$ is the local density Takabayasi
(1952). The energy density of this hydrodynamical system,

${\sum}_{s=1}^{2}|\phi_{s}(\bm{r},t)|\bigg{[}\frac{m\bm{v}_{s}^{2}(\bm{r},t)}{2}+U(\bm{r},t)-\frac{\hbar^{2}}{2m}\frac{\Delta{|\phi_{s}|}}{|\phi_{s}|}\bigg{]},$
(18)

coincides with $\rho(\bm{r},t)$ in (12, 14). In (18) we have kinetic energy,
potential energy, and Bohm’s (quantum) potential Takabayasi (1952) that comes
from the kinetic energy in (12). Eq. (14) is more general than (18) since it
applies to mixed states. Note that (14) contains the weak value of the kinetic
energy Cohen (1996); Aharonov  _et al._ (1993); Popławski and Del Grosso
(2021).

###  III.2 Negativity of non-relativistic energy density for free motion

For a free motion ($U=0$) $\rho(\bm{r},t)$ in (12) can be negative for certain
$\bm{r}$ and $t$ despite of the positive mean energy (16). I.e. the total
energy is lower than the rest-mass energy; cf. (11). This contrasts to the
fully relativistic situation, where the whole energy can be negative Thaller
(2013). This negativity is a quantum effect. Its origin can be traced back to
the Terletsky-Margenau-Hill quasiprobability that also needs to be sometimes
negative due to non-commutativity of coordinate and momentum Allahverdyan
(2014). Now $\rho=\frac{\hbar|\bm{k}|^{2}}{2m}$ is positive for a plane-wave
$\phi(\bm{r},t)\propto e^{-i\frac{\hbar|\bm{k}|}{2m}t+i\bm{k}\bm{r}}$ with
wave-vector $k$. However, a plane-wave is not normalizable. For normalizable
states we were able to find a Gaussian wave-packet (pure state), where
$\phi(\bm{r},t)<0$ only for a finite range of $t$; see Appendix C. We
conjecture that there are no normalizable states for which $\phi(\bm{r},t)\geq
0$ for all $\bm{r}$ and all $t$ ($t<0$ and $t>0$). Further research should
clarify this conjecture and perhaps relate to quantum inequalities in field
theory Fewster (2012).

##  IV Rest-mass energy and holographic energy

Returning to (11), note that the particle density
$\varphi^{\dagger}\varphi+\chi^{\dagger}\chi$ is also conserved locally; cf.
(7). Moreover, using
$\sigma_{i}\sigma_{j}=\delta_{ij}+i\epsilon_{ijk}\sigma_{k}$ in (10), we note
that in the non-relativistic limit the rest energy (12) can be written as a
sum of two quantities that are conserved separately:

$\displaystyle
mc^{2}(\varphi^{\dagger}\varphi+\chi^{\dagger}\chi)=mc^{2}n_{0}+\rho_{s},$
(19)
$\displaystyle\rho_{s}=-\frac{i\hbar^{2}}{4m}{\bm{\nabla}}\phi^{\dagger}\cdot\bm{\sigma}\times\bm{\nabla}\phi,$
(20) $\displaystyle
n_{0}=\varphi^{\dagger}\varphi+\frac{\hbar^{2}}{4m^{2}c^{2}}\bm{\nabla}\phi^{\dagger}\cdot\bm{\nabla}\phi\geq
0.$ (21)

$\rho_{s}$ is a ${\cal O}(c^{-2})$ part of the particle’s density that leads
to a non-relativistic, and locally conserved energy density. The local
conservation of $\rho_{s}$ can be deduced from (9). Alternatively, we note
that $\rho_{s}$ is a divergence of $\bm{\Upsilon}(\bm{r},t)$:

$\displaystyle\rho_{s}=\bm{\nabla}\bm{\Upsilon},~{}~{}~{}~{}\dot{\rho}_{s}+\bm{\nabla}J_{s}=0,~{}~{}~{}~{}\bm{J}_{s}=-\dot{\bm{\Upsilon}},$
(22)
$\displaystyle\bm{\Upsilon}=\frac{\hbar}{4m}\Re[\phi^{\dagger}\bm{\sigma}\times\bm{P}\phi]=\frac{\hbar}{4m}\Re\,{\rm
tr}\Big{[}|\bm{r}\rangle\langle\bm{r}|\bm{\sigma}\times\bm{P}R\Big{]},$ (23)

where $R$ is the density matrix [cf. (14)], and $\bm{P}=-i\hbar\bm{\nabla}$ is
the momentum operator. Note that the current $\bm{J}_{s}$ is expressed via the
time-derivative of $\bm{\Upsilon}$. Eq. (22) means that $\rho_{s}$ does not
have a global content [cf. (20)]:

$\displaystyle\int{\rm d}r^{3}\rho_{s}(\bm{r},t)=0~{}~{}{\rm
for}~{}~{}\int{\rm d}r^{3}\phi^{\dagger}(\bm{r},t)\phi(\bm{r},t)=1.$ (24)

Eq. (22) also shows that the content $\int_{V}{\rm d}^{3}r\rho_{s}$ of
$\rho_{s}$ in a finite volume $V$ is expressed as the integral of
$\bm{\Upsilon}$ over the boundary $\partial V$; i.e. $\rho_{s}$ is a
holographic quantity. Now $\bm{\Upsilon}$ is essentially spin-dependent and
hence quantum, as witnessed by the $\hbar$ factor in (23). For a finite-motion
stationary state (without magnetic field) we get $\rho_{s}=0$, since spin and
coordinate factorize, $\phi_{s}(\bm{r})=|s\rangle\phi(\bm{r})$, and
$\phi(\bm{r})=\phi^{*}(\bm{r})$; see (20).

$\rho_{s}$ in (20) is invariant with respect to space-inversion
$\bm{r}\to-\bm{r}$, since $\bm{\Upsilon}$ in (23) is vector product of vector
and pseudo-vector (hence $\Upsilon$ is vector), while the non-relativistic
spinor under $\bm{r}\to-\bm{r}$ is just multiplied by a phase factor
Berestetskii  _et al._ (1982) that disappears from (20). In the context of
(23) recall that the quantity $\bm{\sigma}\cdot\bm{P}$ is a pseudo-vector and
refers to particle’s helicity Berestetskii  _et al._ (1982).

$\rho_{s}$ is also invariant with respect to time-inversion, as any energy
should be Davydov (1991); Landau and Lifshitz (1958). Recall that the time-
inversion of a spinor $\phi$ is defined as Davydov (1991); Landau and Lifshitz
(1958)

$\displaystyle\phi^{[T]}=\hat{\pi}\phi^{*},\qquad\hat{\pi}=\begin{pmatrix}0&-1\\\
1&0\end{pmatrix},$ (25)

where $\phi^{*}$ means complex conjugation. Eq. (25) means that besides the
complex conjugation, which is related to time-inverting of a wave-function,
the spinor should be subject to an additional unitary transformation
$\hat{\pi}$, which reflects the fact that the eigenvalues of the spin itself
change sign under time-inversion Landau and Lifshitz (1958). We now employ
$\bm{\sigma}^{*}=\hat{\pi}\bm{\sigma}\hat{\pi}$ in (20), and deduce the time-
invariance of $\rho_{s}$.

Thus, we have two non-relativistic forms of energy, $\rho$ and $\rho_{s}$ that
are conserved locally and expressed via the Schroedinger wave-function $\phi$
from (9). Note from (7, 12, 22, 23) that
$mc^{2}(n_{0}-\phi^{\dagger}\phi)={\cal O}(1)$ is also locally conserved, and
has the non-relativistic order of magnitude. But it is not expressed only via
$\phi$, i.e. it demands solving a quasi-relativistic equation for $\varphi$;
see (8). Hence only (12) and (20) define new observables for non-relativisitic
quantum mechanics. We focus on $\rho$ and $\rho_{s}$, which we regard as
different forms of energy.

##  V Energy transfer velocity

The following important question cannot be addressed without the notion of
quantum mechanical energy density, though (as we shall see) the answer to an
extent does not not depend on the details of the definition. What is the
energy transfer velocity for a freely propagating ($U(\bm{r},t)=0$) quantum
wave-packet and how it differs from velocity of the coordinate transfer? This
question is relevant for all quantum fields dealing with excitation transfer,
but it does not seem to be addressed so far. It has a remote analogue in
optics, where the energy transfer velocity (group velocity) is compared with
the unphysical phase-velocity Brillouin (2013); Milonni (2004). In our
situations both velocities are physical, and the group velocity refers to the
coordinate transfer.

We start with a wave-packet with 1d Schroedinger wave-function $\phi(x,t)$,
energy density (12) and energy current (15). The spin degree of freedom is
irrelevant provided that its wave-function factorizes
$\phi_{s}(x,t)=|s\rangle\phi(x,t)$; see (12, 15). Looking at the local
conservation

$\displaystyle\partial_{t}(\phi^{*}\phi)+\partial_{x}j=0,~{}j(x,t)=\frac{i\hbar}{2m}(\partial_{x}\phi^{*}\phi-\phi^{*}\partial_{x}\phi),$
(26)

of density, we define the coordinate transfer velocity as

$\displaystyle v_{\rm cor}(t)=\int{\rm d}x\,j(x,t)=\frac{1}{m}\int{\rm
d}x\,\phi(x,t)P\phi(x,t)$ (27) $\displaystyle=({1}/{m})\langle
P\rangle(t),~{}~{}~{}P=({i}/{\hbar})\partial_{x}.$ (28)

$v_{\rm cor}(t)$ relates to the mean momentum, which is also the mean group
velocity Brillouin (2013); Milonni (2004); cf. (18). Likewise, a sensible
(though by no means exclusive) definition of the energy transfer velocity will
be [cf. (12, 15, 16)]:

$\displaystyle v_{\rm en}(t)=\frac{\int{\rm d}x\,J(x,t)}{\int{\rm
d}x\,\rho(x,t)}=\frac{\langle P^{3}\rangle}{m\langle P^{2}\rangle},$ (29)

where we used (16) and employed $\int{\rm d}x\,J(x,t)=\frac{1}{2m^{2}}\langle
P^{3}\rangle$ from (15). Note that this intuitive relation should hold for any
sensible definition of the non-relativistic energy density, and not only (12,
15). (Eq. (16) has the same general status.) Altogether, (29) should hold for
any definition of energy density and its current.

Let us now assume a paradigmatic situation that $\phi(x,t)$ is a Gaussian
wave-packet. It is also Gaussian in the momentum representation, i.e. we can
employ in (29) the formula $\langle P^{3}\rangle=3\langle P\rangle\langle
P^{2}\rangle-2\langle P\rangle^{3}$ and obtain:

$\displaystyle v_{\rm en}-v_{\rm cor}=\frac{2\langle P\rangle\,[\,\langle
P^{2}\rangle-\langle P\rangle^{2}\,]}{m\langle P^{2}\rangle}\geq 0,$ (30)

i.e. energy is transferred at a larger velocity. Now $v_{\rm en}(t)\to v_{\rm
cor}(t)$ for nearly plane-waves, where $\langle P^{2}\rangle\approx\langle
P\rangle^{2}$. The message of (30) does not depend on the employed definitions
(28, 29), at least for Gaussian states. Appendix C shows the same message
$v_{\rm en}(t)\geq v_{\rm cor}(t)$ for another definition, where analogues of
$v_{\rm en}(t)$ and $v_{\rm cor}(t)$ are defined via the velocity of peaks
(most probable values) for $\rho(x,t)$ and $\phi^{*}(x,t)\phi(x,t)$,
respectively. The same conclusion holds for Airy wave-packets Berry and Balazs
(1979); Siviloglou  _et al._ (2007) that move without dispersion (i.e. as a
whole) and with an acceleration; see Appendix D. Once these packets are not
normalizable, we need to employ for the energy transfer velocity defined via
the most probable values. Moreover, Appendix D also shows that the energy
density (12) does explain the physical meaning of Airy wave-packets.

##  VI Stationary states with non-zero holographic energy (20)

Recall that $\rho_{s}=0$ whenever the stationary $\phi(\bm{r})$ is real.
Likewise, $\rho_{s}=0$ if $\phi(\bm{r})$ is a plane wave. Hence we focus on
stationary scattering states that do describe an infinite motion, and we need
to look for interference effects there that go beyond a single plane wave. The
spin is crucial for the existence of (20), but it is sufficient to assume the
simplest factorized situation $\phi_{s}(\bm{r})=|s\rangle\phi(\bm{r})$. Thus,
(20) will contain the mean magnetization $\bm{\mu}\equiv\langle
s|\bm{\sigma}|s\rangle$. Now for $\phi(\bm{r})$ we assume the assume the
following stationary solution $\phi(\bm{r},t)=e^{-itE/\hbar}\phi(\bm{r})$ of
(9):

$\displaystyle\phi(\bm{r})=e^{ikz}+\frac{fe^{ikr}}{r},~{}~{}r=|\bm{r}|,~{}~{}E=\frac{\hbar^{2}k^{2}}{2m},$
(31)

where $\bm{r}=(x,y,z)$, $E>0$ is the energy, and $k$ is the wave-vector. Eq.
(31) describes the incident wave $e^{ikz}$ that is scattered on a potential
$U(\bm{r})\propto\delta(\bm{r})$ centered at $\bm{r}=0$ Landau and Lifshitz
(1958); Davydov (1991). This produced the scattered expanding spherical wave
in (31), where $f$ is a constant scattering amplitude. We get from (20, 31):

$\displaystyle\rho_{s}=\frac{\hbar^{2}k^{2}(\mu_{x}y-\mu_{y}x)f}{2mr^{2}}\Big{[}\sin(kz-
kr)-\frac{\cos(kz-kr)}{kr}\Big{]},$ $\displaystyle\bm{\mu}\equiv\langle
s|\bm{\sigma}|s\rangle.$ (32)

Note that the interference between the two waves in (31) is essential for
$\rho_{s}\not=0$ in (32). We see in (32) that ${\cal R}^{2}\oint{\rm
d}\Omega\,\rho_{s}=0$, where $\int{\rm d}\Omega$ is the surface integral over
the sphere with radius ${\cal R}$. However, if we integrate over a part of
this sphere, ${\cal R}^{2}\int{\rm d}\Omega\,\rho_{s}$ can scale as ${\cal
O}({\cal R})$. In that sense $\rho_{s}$ concentrates on the spherical surface.

For a non-zero magnetic field $\bm{B}\not=0$ the stationary wave-function
$\phi$ need not be real. Hence $\rho_{s}\not=0$ in (20) is possible for
normalizable stationary states. An important example of this type is provided
by Landau levels for a weakly-confined 2d electron gas under constant and
homogeneous $\bm{B}$; see Tong (2016) for review. This system is basic e.g.
for the quantum Hall effects and related macroscopic quantum states Tong
(2016). Appendix E studies this system, shows that $\rho_{s}\not=0$ due to
$\bm{B}\not=0$, and that $\rho_{s}$ concentrates at the surface of the system.

##  VII Summary

We show that the energy density and current in quantum mechanics can be
defined consistently with the fundamental relativistic physics and that this
definition coincides with that provided by Terletsky-Margenau-Hill coordinate-
momentum quasiprobability. We applied this result to deducing the energy
transfer velocity for Gaussian and Airy wave-packets and showing that it
generically exceeds the coordinate transfer (i.e. group) velocity. Further
interesting questions stay open; e.g. the energy motion in tunneling or the
extension of our results to open, discrete systems, as was recently done for
the probability current Hovhannisyan and Imparato (2019). We also uncovered a
new form of energy with a non-relativistic magnitude which is essentially
spin-dependent. It is holographic, i.e. it does not contribute to the global
energy budget of finite-motion states, but its local contribution is sizable.

###### Acknowledgements.

This work was supported by SCS of Armenia, grants 20TTAT-QTa003, 21AG-1C038
and 22AA-1C023. We thank K. Hovhannisyan for discussions.

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##  Appendix A Derivation of Dirac’s energy density and current from the
relativistic Lagrangian and the energy-momentum tensor

The relativistic Lagrangian of spin-$1/2$ particle reads ($\hbar=c=1$)
Berestetskii  _et al._ (1982):

$\displaystyle{\cal
L}=\frac{i}{2}\left(\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-\partial_{\mu}\bar{\psi}\gamma^{\mu}\psi\right)-m\bar{\psi}\psi,$
(33)

where $m$ is the mass, $\psi$ is the four-component column bispinor,
$\mu=0,...,4$, $A_{\mu}B^{\mu}=A^{0}B^{0}-\bm{A}\cdot\bm{B}$,
$\partial_{\mu}=(\partial_{t},\vec{\nabla})$, and $\gamma^{\mu}$ are gamma
matrices [see (2)]:

$\displaystyle\gamma^{0}=\beta,\qquad\gamma^{k}=\beta\alpha_{k},\qquad
i,j,k=1,2,3,$ (34)
$\displaystyle\alpha_{i}\alpha_{j}+\alpha_{j}\alpha_{i}=2\delta_{ij},\quad\alpha_{k}\beta+\beta\alpha_{k}=0,\quad\beta^{2}=1.$
(35)

Recall that (33) is a 4-scalar Berestetskii  _et al._ (1982). Within Dirac’s
representation $\vec{\alpha}$ and $\beta$ are given by (2).

The energy-momentum tensor reads from (33)

$\displaystyle T^{\mu}_{\quad\nu}=\frac{\partial{\cal
L}}{\partial[\partial_{\mu}\psi]}\partial_{\nu}\psi+\partial_{\nu}\bar{\psi}\frac{\partial{\cal
L}}{\partial[\partial_{\mu}\bar{\psi}]}-\delta^{\mu}_{\nu}{\cal L}$
$\displaystyle=\frac{i}{2}\left(\bar{\psi}\gamma^{\mu}\partial_{\nu}\psi-\partial_{\nu}\bar{\psi}\gamma^{\mu}\psi\right).$
(36)

Note that $T^{\mu}_{\quad\nu}\not=T^{\nu}_{\quad\mu}$. It holds four
conservation laws:

$\displaystyle\partial_{\mu}T^{\mu}_{\quad\nu}=0.$ (37)

We shall look at the energy conservation. The energy density and energy
current reads from (A):

$\displaystyle T^{0}_{\quad
0}=-\frac{i}{2}\left({\psi}^{\dagger}{\alpha_{k}}{\nabla_{k}}\psi-{\nabla_{k}}{\psi}^{\dagger}{\alpha_{k}}\psi\right)+m\psi^{\dagger}\beta\psi,$
(38) $\displaystyle T^{i}_{\quad
0}=-\frac{i}{2}\left({\psi}^{\dagger}{\alpha}_{i}\alpha_{j}\partial_{j}\psi-\partial_{j}{\psi}^{\dagger}\alpha_{j}{\alpha}_{i}\psi\right)$
(39)

where $i=1,2,3$. Using in (39):

$\displaystyle{\psi}^{\dagger}{\alpha}_{i}\alpha_{j}\partial_{j}\psi-\partial_{j}{\psi}^{\dagger}\alpha_{j}{\alpha}_{i}\psi={\psi}^{\dagger}\partial_{i}\psi$
$\displaystyle-\partial_{i}{\psi}^{\dagger}\psi+\sum_{j\not=i}\left({\psi}^{\dagger}{\alpha}_{i}\alpha_{j}\partial_{j}\psi-\partial_{j}{\psi}^{\dagger}\alpha_{j}{\alpha}_{i}\psi\right),$
(40)

and noting from (35)

$\displaystyle{\alpha}_{i}\alpha_{j}=i\epsilon_{ijk}\tau_{k},\qquad{\tau_{i}}=\begin{pmatrix}{\sigma_{i}}&0\\\
0&{\sigma_{i}}\end{pmatrix},$ (41)

we see that the 3-vector of energy current in (39) is a sum of two
contributions:

$\displaystyle T^{i}_{\quad 0}=\bm{{\cal
J}}+\frac{1}{2}{\bm{\nabla}}\times(\psi^{\dagger}{\bm{\tau}}\psi),$ (42)
$\displaystyle\bm{{\cal
J}}=-\frac{i}{2}(\psi^{\dagger}{\bm{\nabla}}\psi-{\bm{\nabla}}\psi^{\dagger}\psi),$
(43)

where $\times$ means vector product. The contribution
$\frac{1}{2}{\bm{\nabla}}\times(\psi^{\dagger}{\bm{\tau}}\psi)$ in (42) is
interesting, but we shall not focus on it, since it is a rotor. Finally, we
can write the energy conservation in the dimensional form, also incuding there
the potential energy $U$; see (4, 5).

##  Appendix B Lagrangian perspective of the non-relativistic energy density
and current

The energy density (12) for the Schroedinger wave-function $\phi$ [cf. (9)]
can be found via the following Lagrangian density

$\mathcal{L}=\frac{\hbar^{2}}{4m}\bigg{(}\phi^{\dagger}\Delta\phi+\Delta\phi^{\dagger}\phi\bigg{)}-U\phi^{\dagger}\phi+\frac{i\hbar}{2}(\phi^{\dagger}\dot{\phi}-\dot{\phi}^{\dagger}\phi).$
(44)

For simplicity we assume that the potential $U(\bm{r})$ is time-independent.
Now $\mathcal{L}$ contains only second-order space-derivatives. Equations of
motion generated by $\mathcal{L}$,

$\displaystyle\frac{{\rm d}}{{\rm
d}t}\frac{\partial\mathcal{L}}{\partial\dot{q}}=\frac{\partial\mathcal{L}}{\partial
q}+\partial^{2}_{kk}\frac{\partial\mathcal{L}}{\partial[\partial^{2}_{kk}{q}]},\qquad
q=\phi,\phi^{\dagger},$ (45)

coincide with Schrodinger’s equation (9). Using (45) we verify that the local
energy conservation law generated due to $\partial_{t}\mathcal{L}=0$,

$\displaystyle\frac{{\rm d}}{{\rm
d}t}\Big{[}\frac{\partial\mathcal{L}}{\partial\dot{\phi}}\dot{\phi}+\dot{\phi}^{\dagger}\frac{\partial\mathcal{L}}{\partial\dot{\phi}^{\dagger}}-\mathcal{L}\Big{]}$
$\displaystyle+\partial_{k}\Big{[}\frac{\partial\mathcal{L}}{\partial[\partial^{2}_{kk}{\phi}]}\partial_{k}\dot{\phi}-\partial_{k}\Big{(}\frac{\partial\mathcal{L}}{\partial[\partial^{2}_{kk}{\phi}]}\Big{)}\dot{\phi}$
$\displaystyle+\Big{(}\partial_{k}\dot{\phi}^{\dagger}\Big{)}\frac{\partial\mathcal{L}}{\partial[\partial^{2}_{kk}{\phi^{\dagger}}]}-\dot{\phi}^{\dagger}\partial_{k}\frac{\partial\mathcal{L}}{\partial[\partial^{2}_{kk}{\phi^{\dagger}}]}\Big{]}=0,$
(46)

coincides with (12, 15). More precisely for current generated from (46) we get
[cf. (15)]

$\displaystyle{\bm{J}}=\frac{\hbar^{2}}{2m}\Re\Big{[}\phi^{\dagger}\bm{\nabla}\dot{\phi}-[\bm{\nabla}\phi^{\dagger}]\dot{\phi}\Big{]},$
(47)

where one can employ the Schroedinger equation (9). Note that the non-
relativistic limit taken in (43) produces a different current:

$\displaystyle\bm{J}_{\rm D}=\frac{i\hbar
U}{2m}(\bm{\nabla}\phi^{\dagger}\phi-\phi^{\dagger}\bm{\nabla}\phi)+\frac{i\hbar^{3}}{8m^{2}}(\phi^{\dagger}\bm{\nabla}\Delta\phi$
$\displaystyle-[\partial_{i}\phi^{\dagger}]\bm{\nabla}\partial_{i}\phi+\bm{\nabla}\partial_{i}\phi^{\dagger}[\partial_{i}\phi]-[\bm{\nabla}\Delta\phi^{\dagger}]\phi).$
(48)

However, the difference between (47) and (B) amounts to a rotor as verified
directly. I.e. this is an allowed difference. Recall that one rotor was
already neglected in (43) compared with (42).

Note that the choice of the non-relativistic Lagrangian density (44) is not
unique. The usual, first-order space-derivative containing density is Greiner
(1990)

$\Lambda=-\frac{\hbar^{2}}{2m}{\bm{\nabla}}\phi^{\dagger}{\bm{\nabla}}\phi-U\phi^{\dagger}\phi+\frac{i\hbar}{2}(\phi^{\dagger}\dot{\phi}-\dot{\phi}^{\dagger}\phi).$
(49)

Eq. (49) generates the same (9):

$\displaystyle\frac{{\rm d}}{{\rm
d}t}\frac{\partial\mathcal{L}}{\partial\dot{q}}=\frac{\partial\mathcal{L}}{\partial
q}-\partial_{k}\frac{\partial\mathcal{L}}{\partial[\partial_{k}{q}]},\qquad
q=\phi,\phi^{\dagger}.$ (50)

The difference between (49) and (44) is that the former contains only first
space-derivatives, while the latter only second-order space-derivatives. The
very possibility of such a difference is due to non-relativism, where the
space and time enters differently. Recall in this context that the
relativistic Dirac’s Lagrangian (33) which we write again as

$\begin{split}\mathcal{L}&=\frac{ihc}{2}(\psi^{\dagger}\bm{\alpha}\bm{\nabla}\psi-\bm{\nabla}\psi^{\dagger}\bm{\alpha}\psi)+\frac{i\hbar}{2}(\psi^{\dagger}\dot{\psi}-\dot{\psi}^{\dagger}\psi)\\\
&-mc^{2}\psi^{\dagger}\beta\psi-U\psi^{\dagger}\psi,\end{split}$ (51)

does not contain such a non-uniqueness, since the time and space enters into
(51) equivalently, i.e. via the first derivatives over space and time
Berestetskii  _et al._ (1982).

Returning to $\mathcal{L}$ and $\Lambda$, we note that

$\displaystyle\int{\rm d}^{3}x\,\mathcal{L}=\int{\rm d}^{3}x\,\Lambda.$ (52)

Now $\int{\rm d}^{3}x\,\Lambda$ equals the Dirac-Frenkel Lagrangian Frenkel
_et al._ (1934). We do not suggest defining the stationary action principe via
$\mathcal{L}$ or $\Lambda$. Such a principle is problematic (and needs
additional considerations) for the first order (in time) Schroedinger equation
Van Leeuwen (1998); Vignale (2008). For us $\mathcal{L}$ or $\Lambda$ are
tools for studying the symmetries, including time-translation invariance and
the ensuing local energy conservation.

Eq. (49) leads to a different local energy density:

$\displaystyle\frac{{\rm d}}{{\rm
d}t}\Big{[}\frac{\partial\Lambda}{\partial\dot{\phi}}\dot{\phi}+\dot{\phi}^{\dagger}\frac{\partial\Lambda}{\partial\dot{\phi}^{\dagger}}-\Lambda\Big{]}$
$\displaystyle+\partial_{k}\Big{[}\dot{\phi}^{\dagger}\frac{\partial\Lambda}{\partial[\partial_{k}{\phi}^{\dagger}]}+\Big{(}\frac{\partial\Lambda}{\partial[\partial_{k}{\phi}]}\Big{)}\dot{\phi}\Big{]}=0,$
(53)
$\displaystyle\widetilde{\rho}=\frac{\hbar^{2}}{2m}{\bm{\nabla}}\phi^{\dagger}{\bm{\nabla}}\phi+U\phi^{\dagger}\phi$
(54)
$\displaystyle=\frac{1}{2m}\langle\bm{r}|\bm{P}R\bm{P}|\bm{r}\rangle+\langle\bm{r}|RU|\bm{r}\rangle,\quad\bm{P}\equiv-i\hbar\bm{\nabla}.$
(55)

There are three major differences between $\rho$ [given by (12)] and
$\widetilde{\rho}$. First, $\widetilde{\rho}(\bm{r})\geq 0$ for the free
motion with $U=0$, while ${\rho}(\bm{r})<0$ for certain values of $\bm{r}$.
This feature of $\widetilde{\rho}$ is welcome.

Second is that $\widetilde{\rho}$ does not hold (17), i.e. even in a state
with a definite energy it is possible to have energy without particle:
$\widetilde{\rho}(\bm{x})>0$, but $\phi^{\dagger}(\bm{x})\phi(\bm{x})=0$. This
feature of $\widetilde{\rho}$ is not welcome.

The third difference between $\rho$ and $\widetilde{\rho}$ is seen from
comparing (14) with (55). Eq. (14) contains the kinetic energy operator, while
(55) is composed from two momentum operators; i.e. (14) refers to a sort of
operator realism in contrast to (55). We emphasize again that (14) emerges
from a deeper theory, in contrast to (55); hence we do support (14).

##  Appendix C Gaussian wavefunctions

The Schroedinger equation for a single-particle wavefunction evolving under
the free propagation Hamiltonian $\hat{H}=\frac{1}{2m}\hat{p}^{2}$ in 1d is
easily solved in the momentum representation:

$\displaystyle i\hbar\partial_{t}\phi(p,t)=\frac{p^{2}}{2m}\phi(p,t).$ (56)

Eq. (56) leads to

$\phi(p,t)=e^{-itp^{2}/(2m\hbar)}\chi(p),$ (57)

where $\chi(p)$ is a normalized function, $\int{\rm d}p|\chi(p)|^{2}=1$. For
Gaussian wavefunctions we take $\chi(p)$ to be

$\displaystyle\chi(p)=\frac{\sqrt{a}}{\pi^{\frac{1}{4}}}\exp\left[-\frac{a^{2}}{2}(p-\frac{b}{a})^{2}\right],~{}~{}\int{\rm
d}p\,|\chi(p)|^{2}=1,~{}$ (58)

with constants $a>0$ and $b$ defined via the mean and variance of the
momentum:

$\langle p\rangle\equiv\int{\rm d}p\,p|\chi(p)|^{2}=\frac{b}{a},\qquad\langle
p^{2}\rangle-\langle p\rangle^{2}=\frac{1}{2a^{2}}.$ (59)

Eq. (58) defines a general pure Gaussian state. There can be two more real
parameters $\alpha$ and $\beta$ in (58) in the sense that it can be multiplied
by $e^{i\alpha p+i\beta p^{2}}$. However, $\alpha$ and $\beta$ shift
(respectively) the initial coordinate and initial time that for (58) are both
taken to be zero.

Figure 1: Energy transfer velocities $v_{\rm en}$ and $v_{\rm mp}$, and the
coordinate transfer velocity $v_{\rm cor}$ for the Gaussian wave-packet (60);
see (69, 70, 71). Parameters assume the following values: $a=2^{-1/2}$,
$m=2^{-1}$, $b=2^{-3/2}$, $\hbar=1$. Figure 2: For the Airy wave-function (72)
we show its energy density (77) as a function of $\xi=\xi_{t}$. Parameters are
given as: $\beta=\hbar=m=1$, $t=\sqrt{6}$. We see that the energy density has
a pronounced (first from the right) local maximum near the wave-front at
$\xi=0$. However, there is no a finite most probable value of energy, since it
grows indefinitely for $\xi\to-\infty$. Figure 3: The difference
$v_{r}(t)=v_{\rm en}(t)-v_{\rm cor}(t)$ between the energy transfer velocity
and coordinate transfer velocity [see (78, 80)] for the Airy wave-function
(72). Parameters read: $\beta=\hbar=m=1$.

Recalling the $\delta$-function normalized momentum eigenfunctions, $\langle
x|p\rangle=\frac{1}{\sqrt{2\hbar\pi}}\,e^{ipx/\hbar}$, we obtain from Eq. (58)
the wavefunction in the coordinate representation:

$\displaystyle\phi(x,t)=\frac{1}{\sqrt{2\pi\hbar}}\int{\rm
d}p\,e^{ipx/\hbar}\phi(p,t)$
$\displaystyle=\frac{1}{\pi^{\frac{1}{4}}\sqrt{a(1+i\tau)}}\exp\left[-\frac{b^{2}}{2}-\frac{(\xi-
ib)^{2}}{2(1+i\tau)}\right]$ (60)
$\displaystyle=\frac{1}{\pi^{\frac{1}{4}}\sqrt{a(1+i\tau)}}\exp\left[-\frac{(\xi-b\tau)^{2}}{2(1+\tau^{2})}+\frac{i}{2}\,\,\frac{2b\xi+\tau(\xi^{2}-b^{2})}{1+\tau^{2}}\right],$
(61)

where

$\displaystyle\xi\equiv x/(a\hbar),\qquad\tau\equiv t/(ma^{2}\hbar),$ (62)

are the dimensionless coordinate and time, respectively.

For the energy density, particle current and energy current we get,
respectively,

$\displaystyle\rho=-\frac{1}{2ma^{2}}\Re[\phi\frac{{\rm d}^{2}\phi^{*}}{{\rm
d}\xi^{2}}]$ (63)
$\displaystyle=\frac{|\phi|^{2}}{2ma^{2}}\,\frac{\left(\tau^{2}-1\right)\left(\xi^{2}-b^{2}\right)+4b\tau\xi+\tau^{2}+1}{\left(\tau^{2}+1\right)^{2}},$
(64) $\displaystyle|\phi|^{2}v=\frac{1}{ma}\Im[\phi^{*}\frac{{\rm d}\phi}{{\rm
d}\xi}],$ (65) $\displaystyle v=\frac{1}{ma}\,\frac{b+\xi\tau}{1+\tau^{2}},$
(66) $\displaystyle J=-\frac{1}{4m^{2}a^{3}}\Im[\phi^{*}\frac{{\rm
d}^{3}\phi}{{\rm d}\xi^{3}}+\frac{{\rm d}^{2}\phi^{*}}{{\rm
d}\xi^{2}}\frac{{\rm d}\phi}{{\rm d}\xi}]$ (67)
$\displaystyle=\frac{|\phi|^{2}}{4m^{2}a^{3}(1+\tau^{2})^{3}}\Big{[}2\tau(\tau^{2}-1)\xi^{3}+2b(5\tau^{2}-1)\xi^{2}$
$\displaystyle+2\tau(3(1+\tau^{2})+b^{2}(\tau^{2}-5))\xi$
$\displaystyle+2b(2+(1-\tau^{2})(b^{2}+\tau^{2})\Big{]}.$ (68)

Note from (64) that for $b=0$, $\rho(\xi,\tau)>0$ for all $\xi$ and
$|\tau|>1$; i.e. $\rho(\xi,\tau)<0$ only for a finite interval of $\tau$. Thus
(64) provides an example, when the energy density $\rho(\bm{r},t)$ is negative
only for a limited range of $\bm{r}$ and $t$. In the main text we conjectured
that such a limited negativity of $\rho(\bm{r},t)$ is unavoidable for any
normalizable state of the free quantum motion.

Eqs. (65, 66) show that the coordinate transfer velocity can be defined either
via the mean current $\int{\rm d}\xi|\phi|^{2}v$ or via $v$ calculated at the
maximally probable value $\xi=b\tau$ of the Gaussian wave-packet. In both
cases the coordinate transfer velocity reads [cf. (59)]

$\displaystyle v_{\rm cor}=\frac{b}{ma}=\frac{\langle p\rangle}{m}.$ (69)

The definition (29, 30) of the energy transfer velocity $v_{\rm en}$ is worked
out directly from (59):

$\displaystyle v_{\rm en}-v_{\rm cor}=\frac{b}{ma}\frac{1}{1+2b^{2}}\geq 0.$
(70)

It is seen that $v_{\rm en}-v_{\rm cor}\to 0$ both for low-speed and high-
speed situations, i.e. for $b\to 0$ and $b\to\infty$.

Another pertinent definition of energy transfer velocity, $v_{\rm mp}(t)$, is
given by looking at the maximum of the energy density (64) over $\xi$ for
fixed $\tau$ (i.e. over $x$ for fixed $t$). Denote this maximum as $\xi_{\rm
mp}(\tau)$ and then define:

$\displaystyle v_{\rm mp}=\frac{1}{ma}\,\frac{{\rm d}\xi_{\rm mp}}{{\rm
d}\tau}.$ (71)

Fig. 1 compares with each other $v_{\rm mp}(t)$ (which is time-dependent),
$v_{\rm en}$, and $v_{\rm cor}$. As expected, $v_{\rm en}>v_{\rm cor}$ and
$v_{\rm mp}(t)>v_{\rm cor}$, but the relation between $v_{\rm en}$ and $v_{\rm
mp}(t)$ is more intricate.

##  Appendix D Airy wave packet

The free Airy wave-packet is given by a non-normalizable wave-function that
has two interesting properties Berry and Balazs (1979). First, it moves
without changing its form, i.e. it moves without dispersion. Second, it moves
with a constant acceleration, although it is not subject to any force. These
features have to do with its non-normalizable character, but (as expected)
both will survive at least for some times for suitably normalizable analogs of
the free Airy wave-packet that were observed experimentally Siviloglou  _et
al._ (2007).

The Airy wave function Berry and Balazs (1979) solves the free Schroedinger
equation (9):

$\displaystyle\phi(x,t)={\rm
Ai}\bigg{(}\frac{\beta}{\hbar^{2/3}}\,\xi_{t}\bigg{)}\exp\bigg{[}\frac{i}{\hbar}t\,\frac{\beta^{3}}{2m}\bigg{(}\xi_{t}+\frac{\beta^{3}t^{2}}{12m^{2}}\bigg{)}\bigg{]},~{}$
(72) $\displaystyle\xi_{t}\equiv x-\frac{\beta^{3}t^{2}}{4m^{2}},$ (73)

where $\beta>0$ is a constant, and ${\rm Ai}(x)$ is the Airy function. It is a
particular solution of the linear differential equation with two conditions:

$\displaystyle\frac{{\rm d}^{2}}{{\rm d}x^{2}}{\rm Ai}\equiv{\rm
Ai}^{\prime\prime}[x]=x{\rm Ai}[x],~{}~{}~{}{\rm Ai}[|x|\to\infty]\to 0.$ (74)

For $x\gg 1$, ${\rm Ai}[x]$ goes to zero quickly, while its decay is very slow
(and hence ${\rm Ai}[x]$ is not normalizable) for $x\ll-1$:

$\displaystyle{\rm
Ai}[x\ll-1]\simeq\frac{1}{\sqrt{\pi}\,(-x)^{1/4}}\sin\Big{[}\frac{\pi}{4}+\frac{2}{3}(-x)^{3/2}\Big{]},$
(75) $\displaystyle{\rm Ai}[x\gg
1]\simeq\frac{1}{\sqrt{\pi}\,x^{1/4}}\exp\Big{[}-\frac{2}{3}x^{3/2}\Big{]},$
(76)

Note that (74) can be visualized as the Newton equation of motion for a
particle subject to a time-dependent harmonic potential $-\frac{tX^{2}}{2}$,
where $t\to x$ and $X\to{\rm Ai}$ for (74). This potential changes from
confining (at $t<0$) to deconfining for $t>0$. In this physical sense the
oscillatory behavior (75) is more expected than (76).

Thus the density $|\phi(x,t)|^{2}$ of the wave-packet (72) has an oscillatory
tail extending to $\xi_{t}\to-\infty$ and a sharp wave-front around
$\xi_{t}\simeq 0$. The global maximum of $|\phi(x,t)|^{2}$ is located around
this wave-front that moves with acceleration according to (73).

The energy density of the Airy wave-packet (72) is deduced from (12, 74):

$\rho(x,t)=\frac{\beta^{3}}{2m}\bigg{[}\frac{\beta^{3}t^{2}}{4m^{2}}-\xi_{t}\bigg{]}{\rm
Ai}^{2}\bigg{(}\frac{\beta}{\hbar^{2/3}}\xi_{t}\bigg{)}.$ (77)

It is clear from (77, 75) that the energy density grows (with oscillations)
for $x\to-\infty$; see Fig. 2. This is the main qualitative difference between
$\rho(x,t)$ and $|\phi(x,t)|^{2}$. Such an infinite energy reservoir is the
physical reason for the acceleration behavior of the Airy wave-packet.

We turn to the coordinate group velocity of (72). We can define this as the
velocity $v_{\rm cor}(t)$ of the most probable value of $|\phi(x,t)|^{2}$,
which is located near the wave-front. Since $|\phi(x,t)|^{2}$ depends only on
$\xi_{t}$, we find

$v_{\rm cor}(t)=-\dot{\xi}_{t}=\frac{\beta^{3}}{2m^{2}}t,$ (78)

which again confirms that (72) moves with a constant acceleration. Since
$|\phi(x,t)|^{2}$ for the Airy wave-packet moves as a whole without
dispersion, $v_{\rm cor}(t)$ can be identified via the velocity of any of its
points. Note the following relation between (78) and (77):

$\partial_{t}{\rho}(y,t)|_{y=\xi_{t}}=|\phi(x,t)|^{2}\frac{{\rm d}}{{\rm
d}t}\frac{mv_{\rm cor}^{2}(t)}{2}.$ (79)

The energy transfer velocity for the Airy wave-packet can be defined as the
velocity of the local maximum of $\rho(x,t)$ [given by (77)]. One first
maximizes $\rho(x,t)$ as a function of $x$ for a given $t$, denotes this by
$x_{\rm en}(t)$ and then defines the energy transfer velocity as:

$\displaystyle v_{\rm en}(t)=\dot{x}_{\rm en}(t).$ (80)

Comparing (80) with (78) we see that $v_{\rm en}(t)\geq v_{\rm cor}(t)$ with
$v_{\rm en}(t)\to v_{\rm cor}(t)$ for long times; see Fig. 3.

##  Appendix E Spin-dependent energy density for Landau levels

So far we discussed the case with zero magnetic field. As usual, the magnetic
field elongates space-derivatives subtracting the vector potential times
charge, $e\bm{A}$ from the canonical momentum operator
$\bm{P}=-i\hbar\bm{\nabla}$ Cohen-Tannoudji  _et al._ (1977); Landau and
Lifshitz (1958). We confirmed this standard rule via direct calculation. Thus
the spin-dependent energy for the magnetic field reads from (23):

$\displaystyle\bm{\Upsilon}=\frac{\hbar}{4m}\Re\bigg{[}\phi^{\dagger}\bm{\sigma}\times(\bm{P}+|e|\bm{A})\phi\bigg{]},$
(81) $\displaystyle\rho_{s}=\bm{\nabla}\bm{\Upsilon},$ (82)

where we focus on electron: $e=-|e|$. We emphasize that (81) and (82) are
formally gauge-invariant.

We now turn to calculating (82) for electrons in Landau levels, one of the
most important example of magnetic field induced quantum states that is basic
for quantum Hall effects, strong correlations, topological states etc; see
Tong (2016) for further references and a review.

We assume a constant, uniform magnetic field $\bm{B}=[0,0,B]$ acts on
otherwise free electron with charge $-|e|$ and mass $m$. The gauge is fixed as

$\displaystyle\bm{A}=[-By,0,0].$ (83)

Various gauge choices and their differences are discussed in Van Enk (2020).
The stationary Schroedinger-Pauli equation reads

$\displaystyle
E\phi=-\frac{\hbar^{2}}{2m}\bigg{[}\bigg{(}\partial_{x}-i\frac{m}{\hbar}\omega_{B}y\bigg{)}^{2}+\partial_{y}^{2}+\partial_{z}^{2}\bigg{]}\phi+\frac{\hbar}{2}\omega_{B}\sigma_{z}\phi,$
$\displaystyle\omega_{B}={|e|B}/{m},$ (84)

where $\omega_{B}$ is the cyclotron frequency, and $E$ is the energy. To solve
(84) we take Landau and Lifshitz (1958)

$\phi_{n}(\bm{r})\propto
e^{ik_{x}x+ik_{z}z}\tilde{\phi}_{n}(y)[s_{1},s_{2}]^{T}.$ (85)

Eq. (85) is an eigenstate of the canonic momentum $\bm{P}=-i\hbar\bm{\nabla}$
along $x$ and $z$-axes. Recall that $\bm{P}$ is not gauge-invariant and hence
is not observable. Eq. (85) also assumes that the spin state is factorized and
amounts to the eigenstate of $\sigma_{z}$. Eventually, $\phi_{n}(y)$ is the
solution of the 1d harmonic oscillator problem:

$\displaystyle\frac{{\rm d}^{2}}{{\rm
d}y^{2}}\tilde{\phi}_{n}(y)+\frac{1}{\hbar^{2}}\bigg{(}2mE_{n}-2m\hbar
s\omega_{B}-\hbar^{2}k_{z}^{2}$
$\displaystyle-m^{2}\omega_{B}^{2}(y-y_{0})^{2}\bigg{)}\tilde{\phi}_{n}(y)=0,$
(86) $\displaystyle
s=\frac{1}{2}[s_{1},s_{2}]\sigma_{z}[s_{1},s_{2}]^{T}=\pm\frac{1}{2},$ (87)
$\displaystyle y_{0}=\frac{\hbar k_{x}}{m\omega_{B}}$ (88)

where $y_{0}$ is the center of the harmonic oscillator, and where
$[s_{1},s_{2}]^{T}$ is the eigenvector of $\sigma_{z}$. Energy levels
generated by (86) read (Landau levels):

$E_{n}=\bigg{(}n+\frac{1}{2}+s\bigg{)}\hbar\omega_{B}+\frac{\hbar^{2}k_{z}^{2}}{2m}.$
(89)

In these states $\bm{\Upsilon}=[0,\Upsilon,0]$ in (82), where

$\displaystyle\Upsilon=s\frac{\hbar\omega_{B}}{2}\tilde{\phi}^{2}_{n}(y)\bigg{[}y_{0}-y\bigg{]},$
(90)
$\displaystyle\rho_{s}=s\frac{\hbar\omega_{B}}{2}\bigg{[}2(y_{0}-y)\tilde{\phi}_{n}(y)\tilde{\phi}_{n}^{\prime}(y)-\tilde{\phi}_{n}^{2}(y)\bigg{]}.$
(91)

It is seen that $\rho_{s}$ is finite and nullifies together with magnetic
field.