# Electron Wave Spin in Excited States

Ju Gao and Fang Shen<EMAIL_ADDRESS>University of Illinois, Department
of Electrical and Computer Engineering, Urbana, 61801, USA

###### Abstract

The wave spin of an electron can be fully characterized by the current density
calculated from the exact four-spinor solution of the Dirac equation. In the
excited states of the electron in a magnetic field-free quantum well, the
current density has a multiple vortex topology. The interaction of the current
with a magnetic potential produces a finer structure of anomalous Zeeman
splitting. When the magnetic potential is comparable to the size of the
individual vortices, fractional or zero spin effects can be observed.

###### pacs:

3.50, 32.80, 42.50

††preprint: APS/123-QED

##  I Electron wave spin described by current density

Spin is a fundamental property of an electron that represents the electron’s
internal angular momentum. However, what actually spins remains an open
question, because the particle spin interpretation would require the electron
to spin on its own axis at a speed greater than that of light. In
electromagnetism, the charge behavior is fully described by the Lorentz
covariant four-current, defined as $j=(c\rho,\pmb{j})$, where $\rho$ and
$\pmb{j}$ stand for the charge and current densities, respectively, since it
is responsible for both the generation and interaction of an electromagnetic
field. This leads to the question: could the spin also be described by the
current?

In a recent paper [1], we have shown that a stable circulating current density
exists for a Dirac electron in a quantum well without a magnetic field. The
circulating current density, denoted by $j$, forms a spinning vortex around
the center of the charge density, often referred to as an electron cloud.
Expressing the current density in terms of the charge density and a spinning
velocity distribution $v(\pmb{x})$ in $j(\pmb{x})=\rho(\pmb{x})v(\pmb{x})$, we
find that $v(\pmb{x})$ is limited to the speed of light everywhere in space.
In other words, the entire electron wave, or the electron cloud, spins.

Essentially, this is the wave spin interpretation that was first proposed by
Belinfante [2, 3] who argued that spin should be regarded as a circulating
flow of energy of the electron field. Ohanian [4] further elaborated the
connection between the circulating momentum density and current density with
the electron spin and the magnetic moment. Gao [1] showed that a confined
electron has a stable circulating current density, but the wave packet of a
free electron discussed by Ohanian is not stable and de-coherences quickly
because the wavepacket is not constructed by a pure state.

In this paper we continue the discussion of wave spin. To first show that spin
is an embedded property of the electron wave, we derive explicit expressions
for the momentum and current densities of an electron in an eigenstate
wavefunction $\Psi$ of the Dirac equation, where
$i\hbar\frac{\partial}{\partial t}\Psi=\mathcal{E}\Psi$ and $\mathcal{E}$ is
the eigen energy. These equations are as follows:

$\displaystyle\pmb{j}=\frac{ec^{2}}{\mathcal{E}}\left\\{\pmb{\nabla}\times\left(\Psi^{\dagger}\frac{\hbar}{2}\pmb{\Sigma}\Psi\right)+i\frac{\hbar}{2}\left[\left(\pmb{\nabla}\Psi^{\dagger}\right)\Psi-\Psi^{\dagger}\left(\pmb{\nabla}\Psi\right)\right]\right\\},$
$\displaystyle\pmb{G}=\left\\{\frac{1}{2}\pmb{\nabla}\times\left(\Psi^{\dagger}\frac{\hbar}{2}\pmb{\Sigma}\Psi\right)+i\frac{\hbar}{2}\left[\left(\pmb{\nabla}\Psi^{\dagger}\right)\Psi-\Psi^{\dagger}\left(\pmb{\nabla}\Psi\right)\right]\right\\}.$

These equations show that both the current density $\pmb{j}$ and the momentum
density $\pmb{G}$ contain the spin term with the spin operator
$\frac{\hbar}{2}\pmb{\Sigma}$ and the translation term with the momentum
operator $-i\hbar\pmb{\nabla}$. It is worth noting that these expressions are
derived from different origins. The momentum density is derived from a
symmetrical energy-momentum tensor known as the Belinfante-Rosenfeld tensor
[5]. This symmetrical tensor is constructed to serve as a source for the
gravitational field as required by the general relativity. The current
density, on the other hand, is derived from the definition
$\pmb{j}(\pmb{x})=ec\Psi^{\dagger}(\pmb{x})\pmb{\alpha}\Psi(\pmb{x})$ using
the Gordon decomposition, where $\pmb{\alpha}$ is the $\alpha-$matrix in the
Dirac equation. This definition ensures the conservation of charge for the
Dirac electron. The momentum and current densities represent the mechanical
and electrical nature of the wave spin, and have the same spin and translation
terms, except for the factor $\frac{1}{2}$ in the momentum term, which gives
the gyromagnetic ratio $g=2$ for the Dirac field.

We intend to direct our attention especially to the current density, because
the wave spin manifests itself by the current density through the interaction
with the electromagnetic field

$jA=\rho\phi+\pmb{j}\cdot\pmb{A}.$ (2)

The above equation accounts for the full electromagnetic interactions in both
classical and quantum electrodynamics. In this work, we will show that the
current-field interaction $jA$ not only recovers the conventional spin-field
interaction, but also reveals other geometric and topological properties that
are missing in the particle spin picture, in particular for electrons in
excited states.

##  II Wave spin in excited states

To investigate the wave spin in the excited states of a confined electron, we
first seek the exact solution of the Dirac equation

$i\hbar\frac{\partial}{\partial
t}\Psi(\pmb{r},t)=\left[c\pmb{\alpha}\cdot(-i\hbar\pmb{\nabla})+\gamma^{0}\text{mc}^{2}+U(\pmb{r})\right]\Psi(\pmb{r},t),$
(3)

in a two-dimensional quantum well

$U(\pmb{r})=U(x,y)=\begin{cases}0,-L_{x}<x<L_{x},-L_{y}<y<L_{y}\\\
\infty,\text{elsewhere.}\\\ \end{cases}$ (4)

The four-component spinor wavefunction is expressed by the separation of the
temporal and $z$-coordinate variables

$\Psi(\pmb{r},t)=Ne^{-i\mathcal{E}t/\hbar}e^{iP_{z}z/\hbar}\left(\begin{array}[]{c}\mu_{A}(x,y)\\\
\mu_{B}(x,y)\end{array}\right),$ (5)

where $\mathcal{E}$ is the energy, $P_{z}$ is the momentum along
$z$-direction, and $\mu_{A}(x,y)$ and $\mu_{B}(x,y)$ are two-component spinor
wavefunctions.

We now plug Eq. 5 into the Dirac equation Eq. 3 and set $P_{z}=0$ to obtain
the coupled equations

$\displaystyle\left(\mathcal{E}-mc^{2}\right)\mu_{A}(x,y)$ $\displaystyle=$
$\displaystyle-i\hbar c\left(\sigma_{x}\frac{\partial}{\partial
x}+\sigma_{y}\frac{\partial}{\partial y}\right)\mu_{B}(x,y);$
$\displaystyle\left(\mathcal{E}+mc^{2}\right)\mu_{B}(x,y)$ $\displaystyle=$
$\displaystyle-i\hbar c\left(\sigma_{x}\frac{\partial}{\partial
x}+\sigma_{y}\frac{\partial}{\partial y}\right)\mu_{A}(x,y),$

where $\sigma_{x}$ and $\sigma_{y}$ are the Pauli matrixes.

Eqs. II is combined to obtain a second-order differential equation for
$\mu_{A}(x,y)$

$\left(\mathcal{E}^{2}-m^{2}c^{4}\right)\mu_{A}(x,y)=-\hbar^{2}c^{2}\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right)\mu_{A}(x,y),$ (7)

whose eigen solution for the spin-up electron is found

$\mu_{A}(x,y)=\sin[k_{x}(x+L_{x})]\sin[k_{y}(y+L_{y})]\left(\begin{array}[]{c}1\\\
0\end{array}\right),$ (8)

where $k_{x}=\frac{\pi n_{x}}{2L_{x}};k_{y}=\frac{\pi n_{y}}{2L_{y}}$ are the
wave vectors and $n_{x}$, $n_{y}=1,2,3...$ are the quantum numbers of the
eigen states. The eigen energy

$\mathcal{E}=mc^{2}\sqrt{1+\eta^{2}}$ (9)

is quantized by $n_{x}$ and $n_{y}$ according to the expression of a
dimensionless geometric factor

$\eta=\sqrt{n_{x}^{2}\left(\frac{\lambda_{c}}{4L_{x}}\right)^{2}+n_{y}^{2}\left(\frac{\lambda_{c}}{4L_{y}}\right)^{2}}$
(10)

that measures the dimensions of the quantum well $(L_{x},L_{y})$ against the
Compton wavelength $\lambda_{c}=\frac{\hbar}{mc}$ at different states
$(n_{x},n_{y})$.

The wavefunction $\mu_{B}(x,y)$ is subsequently derived via Eq. II and the
complete four-component spinor wavefunction is then obtained

$\Psi(\pmb{r},t)=Ne^{-i\mathcal{E}t/\hbar}\left(\begin{array}[]{c}\sin[k_{x}(x+L_{x})]\sin[k_{y}(y+L_{y})]\\\
0\\\ 0\\\
-i\frac{\eta_{x}}{1+\sqrt{\eta^{2}+1}}\cos[k_{x}(x+L_{x})]\sin[k_{y}(y+L_{y})]+\frac{\eta_{y}}{1+\sqrt{\eta^{2}+1}}\sin[k_{x}(x+L_{x})]\cos[k_{y}(y+L_{y})]\\\
\end{array}\right).$ (11)

where $\eta_{x}=\frac{\hbar k_{x}}{mc},\eta_{y}=\frac{\hbar k_{y}}{mc}$ are
dimensionless factors along $x$ and $y$ directions. The normalization factor
of the wavefunction is found

$N=\sqrt{\frac{1+\sqrt{1+\eta^{2}}}{\sqrt{1+\eta^{2}}}}.$ (12)

The wavefunction in Eq. 11 is used to calculate the stable charge density
$\rho(x,y)=e\Psi^{\dagger}(x)\Psi(x)$ inside the quantum well

$\displaystyle\rho(x,y)=eN^{2}\sin^{2}[k_{x}(x+L_{x})]\sin^{2}[k_{y}(y+L_{y})]$
$\displaystyle+eN^{2}\frac{\eta_{x}^{2}}{(1+\sqrt{1+\eta^{2}})^{2}}\cos^{2}[k_{x}(x+L_{x})]\sin^{2}[k_{y}(y+L_{y})]$
$\displaystyle+eN^{2}\frac{\eta_{y}^{2}}{(1+\sqrt{1+\eta^{2}})^{2}}\sin^{2}[k_{x}(x+L_{x})]\cos^{2}[k_{y}(y+L_{y})].$

and the corresponding current density

$\displaystyle j_{x}$ $\displaystyle=$ $\displaystyle
ec\frac{2\eta_{y}}{\sqrt{1+\eta^{2}}}\sin^{2}[k_{x}(x+L_{x})]\sin[2k_{y}(y+L_{y})],$
$\displaystyle j_{y}$ $\displaystyle=$ $\displaystyle-
ec\frac{2\eta_{x}}{\sqrt{1+\eta^{2}}}\sin^{2}[k_{y}(y+L_{y})]\sin[2k_{x}(x+L_{x})].$

Both the charge density and the current density exhibit properties of a
standing wave, characterized by the quantum numbers $(n_{x},n_{y})$. As an
example, we choose an excited state $(n_{x}=2,n_{y}=2)$ of a quantum well
$(L_{x}=10~{}\text{nm},L_{y}=10~{}\text{nm})$. Fig. 1 is the density plot of
the charge expected for the behavior of an electron cloud. Fig. 2 is the
density plot of the current density of the same electron, showing multiple
vortices around the peaks of the electron cloud. The wave spin picture is
further illustrated by the vector plot of the current in Fig. 3, which shows
the circulation of each vortex in the same direction. It is clear that the
wave spin in the excited state is distributed among multiple vortices that are
holographic to each other. This topology suggests that each wave spin vortex
represents a part or fraction of the total spin that can be studied and
observed by interaction with an external field, as we will elaborate later.

In Fig. 3 it can also be seen that the current flows continuously along the
edge of the quantum well. A similar behavior of the edge current is observed
and studied in the quantum Hall effect [6] when a strong magnetic field is
applied to topological insulator materials. The intrinsic edge current shown
here suggests that topological spin effects can be observed even without the
presence of any magnetic field, internally and externally.

Figure 1: Charge distribution for the excited state $n_{x}=2$ and $n_{y}=2$
for a Dirac electron in a quantum well of $L_{x}=10~{}\text{nm}$ and
$L_{y}=10~{}\text{nm}$. The z-axis represents the charge density of the
relative unit. Figure 2: Density plot of the current density circulates for
the excited state $n_{x}=2$ and $n_{y}=2$ for a Dirac electron in a quantum
well of $L_{x}=10~{}\text{nm}$ and $L_{y}=10~{}\text{nm}$. Figure 3: Vector
plot of the current density for the excited state $(n_{x}=2,n_{y}=2)$ for a
Dirac electron in a quantum well of $L_{x}=10~{}\text{nm}$ and
$L_{y}=10~{}\text{nm}$.

Figure 4:  The upper figure shows the magnetic potential (blue) of Eq. 19
overlapping with the upper left current vortex (red) at
$a=-L_{x}/2,b=L_{y}/2$, producing a quarter spin-field coupling. The lower
figure shows that the magnetic potential (blue) of Eq. 19 lies in the middle
of the quantum well at $a=0,b=0$, creating zero spin-field coupling.

##  III Topological current-field interaction

The interaction of a particle spin with an external field is expressed by
$\frac{e}{m}\frac{\hbar}{2}\pmb{\Sigma}\cdot\pmb{B}$, which excludes all
geometrical and topological effects since $\pmb{\Sigma}$ depicts a
dimensionless point. However, since the wave spin is encoded in the current
density, its topological property should be transferred to the interaction

$\pmb{j}\cdot\pmb{A}=\Psi^{\dagger}(\pmb{r},t)ec\pmb{\alpha}\cdot\pmb{A}(\pmb{r})\Psi(\pmb{r},t),$
(15)

where the vector potential $\pmb{A}(\pmb{r})$ may itself be a vortex field

$\pmb{A}(x,y)=\frac{B}{2}(-y,x,0)$ (16)

to represent a uniform magnetic field along the $z$ direction by the
definition $\pmb{\nabla}\times\pmb{A}(x,y)=(0,0,B)$. In this case, this
interaction shall be characterized as the vortex-vortex interaction. When the
vector potential $\pmb{A}(\pmb{r})$ is larger in size than the electron wave,
which is usually the case, the interaction of Eq. 15 can be evaluated by
integration over the entire quantum well, yielding the following

$\displaystyle\mathcal{E}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2L_{x}}\frac{1}{2L_{y}}\int_{-L_{y}}^{L_{y}}\int_{-L_{x}}^{L_{x}}\pmb{j}\cdot\pmb{A}(x,y)dxdy$
(17) $\displaystyle=$ $\displaystyle\frac{e\hbar
B}{2m\sqrt{1+\eta^{2}}}=\frac{\mu_{B}B}{\sqrt{1+\eta^{2}}},$

where $\mathcal{E}^{(1)}$ is the first-order energy shift for a weak field and
$\mu_{B}=\frac{e\hbar}{2m}$ is the Bohr magneton. Thus, the difference between
the spin-up and spin-down energy shifts is

$\Delta\mathcal{E}=2\mathcal{E}^{(1)}=\frac{2\mu_{B}B}{\sqrt{1+\eta^{2}}},$
(18)

which recovers the expression for the anomalous Zeeman splitting $2\mu_{B}B$
modified by additional geometric factors and quantum numbers via Eq. 10. This
indicates a finer structure of the anomalous Zeeman effect. Since in the wave
spin picture the spin state and the spatial state are coupled in the spinor
expression, each excited state corresponds to a unique spin state of a unique
topology to interact with an external field.

To show the topological footprints of the vortex-vortex interaction described
by Eq. 15, we postulate a vector potential smaller in size than the electron
wave,

$\tilde{\pmb{A}}(\pmb{r})=\begin{cases}\frac{B}{2}(-y+b,x-a,0),\begin{array}[]{c}-L_{x}/2<x-a<L_{x}/2,\\\
-L_{y}/2<y-b<L_{y}/2\\\ \end{array}\\\ \\\ 0,\text{ elsewhere.}\end{cases}$
(19)

to represent a field enclosed in a square that is only a quarter of the
quantum well with center $(a,b)$. The vector potential of Eq. 19 produces the
same uniform magnetic field within the enclosed region. Fig. 4 shows the
vector plot of such a field superimposed on the electron current distribution.
We now perform the same integration over the quantum well as in Eq. 17. We
find that the current-field interaction depends on the relative position of
the vortices. The energy shift for $(n_{x}=2,n_{y}=2)$ is now

$\displaystyle\mathcal{E}^{(1)}_{a,b}$ $\displaystyle=$
$\displaystyle\frac{1}{2L_{x}}\frac{1}{2L_{y}}\int_{-L_{y}}^{L_{y}}\int_{-L_{x}}^{L_{x}}\pmb{j}\cdot\tilde{\pmb{A}}(x,y)dxdy$
(20) $\displaystyle=$
$\displaystyle\begin{cases}\frac{1}{4}\frac{\mu_{B}B}{\sqrt{1+\eta^{2}}},a=\pm
L_{x}/2,b=\pm L_{y}/2\\\ ~{}~{}~{}~{}0,~{}~{}~{}~{}~{}a=0,b=0,\\\ \end{cases}$

which is only $\frac{1}{4}$ of the spin and magnetic field interaction in Eq.
17 when the field overlaps with one of the current vortices. The fractional
spin effect is due to the partial participation of the wave spin represented
by the current density. A special situation arises when the field lies at the
center of the quantum well. Then a zero interaction is observed, because equal
fractions of the current flowing in different directions are exactly cancelled
out.

The above results are certainly not to be expected from the particle spin
picture and should be considered as properties of the wave spin alone. It is
conceivable that we can image the entire current density profile by scanning
the field against the current distribution. It is now possible to generate
vortex optical fields [7] that can be focused on a smaller region than the
quantum well. Then the interaction of wave spin and orbital angular momentum
can be studied via the vortex-vortex interaction on a controllable scale. All
of these discussions and developments mean that we can study and manipulate
partial spins to gain knowledge and control over the entire wave spin due to
the holographic nature of the multi-vortex current density.

##  IV Discussions and Conclusions

In summary:

  1. 1.

We argue that spin is not an abstract two-valued property of the electron
particle but a property of the electron wave that can be fully described by
its momentum and current densities.

  2. 2.

We show that in the excited state, the current density of an electron forms
multiple vortices in a magnetic field-free quantum well. The topology of these
vortices depends on the quantum numbers of the states and is holographic in
nature.

  3. 3.

We investigate the anomalous Zeeman effect of the wave spin and show that the
anomalous Zeeman splitting contains finer structures than those of the
conventional particle spin picture.

  4. 4.

We show that the geometrical and topological properties of the wave spin can
be observed and studied through its interaction with an electromagnetic field.
By studying the interaction with a field smaller in size than the wave spin
itself, we show that fractional and even zero spin effects can be observed.

Apparently, the discussion of wave spin could have implications in many areas,
and each area requires in-depth study. Here we offer some preliminary
discussions.

  1. 1.

In the field of quantum technology, the electron spin has been proposed as a
candidate for use as a quantum bit (qubit), which is a superposition of the
spin-up and spin-down states $\alpha\left(\begin{array}[]{c}1\\\
0\end{array}\right)+\beta\left(\begin{array}[]{c}0\\\ 1\end{array}\right)$,
where $\alpha$ and $\beta$ are complex numbers. In the wave spin picture, the
spin states $\left(\begin{array}[]{c}1\\\ 0\end{array}\right)$ and
$\left(\begin{array}[]{c}0\\\ 1\end{array}\right)$ are replaced by the spinor
wave functions, as in Eq. 11, where the spatial wave function serves as the
spinor component. Therefore, we have argued that the spin cannot be completely
isolated from the physical environment in which it resides. Any change of the
boundary conditions alters the wave functions, and hence the spin states. We
further argue that the spin cannot be completely isolated from the Hilbert
space of the electron either. Any transition to or from other spatial states
alters the original spin state, leading to decoherence of the prepared qubit
and loss of the quantum information. This means that additional protection and
correction mechanisms need to be implemented to protect and preserve the spin
qubit.

  2. 2.

The multi-vortex wave spin topology and its partial interaction with
electromagnetic fields show that the spin-field interaction is not
dimensionless, suggesting novel schemes for parallel information processing
using spin. Each vortex of the current is a holographic part of the entire
spin and can interact simultaneously with multiple electromagnetic fields,
potentially enabling parallel computing. Building such parallel computers
shall benefit from the advancement of both spintronics and optics.

  3. 3.

It is conceivable that the holographic spin interactions could already exist
in nature. It is suggested that the electron spin could play a role in bio-
homochirality [8], which makes the wave spin perspective interesting for this
biological topic [9] and the general spin effects in molecules. The wave spin
could interact holographically with all atoms within the molecule via the
shared electron cloud, leaving a coherent spin footprint on the entire
molecular structure.

##  V Acknowledgement

The authors would like to thank Jora L. Gao for her careful review of the
manuscript. The authors would also like to thank Jane Y. Gao and W. Wang for
stimulating discussions on the role of spin in biology.

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