1 Introduction

The Zeeman effect is a phenomenon of energy level split which stems from interaction between spin and magnetic field [1]. This split may be applied to THz wave detection [2,3,4]. Researches on spintronic devices utilizing spin degree of freedom are being actively conducted [5]. Furthermore, in recent years, quantum computers employing electron spins as qubits have been developed [6,7,8]. Electron spin has become the subject of studies in engineering.

Electrons possess both particle and wave natures. Electrons are represented by probablistic particles in circuit simulator models of active quantum effect devices like single-electron transistors [9,10,11] while electrons have been depicted by quantum waves in the research of passive electron waveguides and filters [12,13,14]. The authors have asserted unification of the representation of electrons for integrated simulation of computing and communication systems built of both passive and active quantum devices [15,16,17]. Since the circuit simulator models of the active quantum devices are available today, electrons in the passive quantum devices should be modeled as probabilistic particles. Particle-represented models require less computation for providing samples of time and position of electrons to the simulator model of an active device than wave-represented models. This is because electrons as particles draw one-dimensional trajectories, while electron waves evolve in two and higher-dimensional space.

Theories on the transformation of quantum expression from waves to particles were established by Bohm [18] and Nelson [19]. However, the theories are restrictedly applicable to the transformations for spin-less quantum particles or in the case that spin effects may be neglected [20, 21]. The restriction should be removed since electron spin degree plays important roles in engineering as mentioned previously.

In this paper, taking electron spin into account, we propose a method of constructing a stochastic classical electron particle model. Using the model, we compute virtual electron trajectories and spin expectation. The wave nature of a non-relativistic electron with spin is governed by the Pauli equation. From the Pauli equation and its complex conjugate equation, a continuity equation for the probability density of electron location is derived. The continuity equation of probability density is the Fokker-Planck equation (FPE) [22]. Using the drift and diffusion coefficients of the FPE, we construct the generalized Langevin equation (LvE) as a mathematical particle model. This equation describes a probabilistic particle motion which follows the probability density satisfying the FPE. Virtual electron trajectories are computed by numerically integrating the LvE. Furthermore, we show how to compute the expectation of spin orientation of the electron on the trajectory.

The rest of the paper is organized as follows: Sect. 2 presents the probability density function and the probability current density derived from the Pauli equation. Section 3 builds the classical particle model. In addition, the section formulates expectation of spin orientation. The two sections accompany examples for an electron in symmetric gauge potential. In Sect. 4, we carry out numerical experiments to evaluate the proposed model of the electron in the potential. Finally, Sect. 5 presents a summary of this paper and future subjects.

2 Description of an electron as waves

2.1 Pauli equation

The wavelike behavior of relativistic electrons is governed by the Dirac equation. The Dirac equation for an electron in an electromagnetic field is given as follows:

$$\begin{aligned} i\hbar \dfrac{\partial }{\partial t} {\varvec{\Psi }}({\varvec{r}}, t)&= \hat{H}_{\mathrm {Dirac}} {\varvec{\Psi }}({\varvec{r}}, t) ,\\ \hat{H}_{\mathrm {Dirac}}&= -i\hbar c {\varvec{\alpha }} \cdot \left( \nabla + i \dfrac{e_0}{\hbar }{\varvec{A}} \right) + {\varvec{\beta }} m_e c^{2} - e_0 A_{0}, \nonumber \\ {\varvec{\alpha }}&= \begin{pmatrix} {\varvec{\alpha }}_x,&{\varvec{\alpha }}_y,&{\varvec{\alpha }}_z \end{pmatrix}, \nonumber \\ {\varvec{\alpha }}_x&= \begin{pmatrix} {\varvec{O}} &{} {\varvec{\sigma }}_x \\ {\varvec{\sigma }}_x &{} {\varvec{O}} \end{pmatrix},\quad {\varvec{\alpha }}_y = \begin{pmatrix} {\varvec{O}} &{} {\varvec{\sigma }}_y \\ {\varvec{\sigma }}_y &{} {\varvec{O}} \end{pmatrix},\nonumber \\ {\varvec{\alpha }}_z&= \begin{pmatrix} {\varvec{O}} &{} {\varvec{\sigma }}_z \\ {\varvec{\sigma }}_z &{} {\varvec{O}} \end{pmatrix},\quad {\varvec{\beta }} = \begin{pmatrix} {\varvec{O}} &{} {\varvec{I}} \\ {\varvec{I}} &{} {\varvec{O}} \end{pmatrix}, \nonumber \end{aligned}$$
(1)

where \({\varvec{r}} = (x, y, z)\) and t are independent variables of position and time, respectively. Notation i is the imaginary unit. Constant c is the speed of light, \(\hbar = h / (2 \pi )\), h is the Planck constant, \(m_e\) is the mass of the electron, and \(e_0\) is the elementary charge. Operator \(\nabla = (\nabla _x, \nabla _y, \nabla _z) = (\partial /\partial x, \partial /\partial y, \partial /\partial z)\) is the gradient operator. Functions \({\varvec{A}} = {\varvec{A}}({\varvec{r}}, t) = (A_x({\varvec{r}}, t), A_y({\varvec{r}}, t), A_z({\varvec{r}}, t))\) and \(A_0 = A_0 ({\varvec{r}}, t)\) are vector and scalar potential, respectively. Matrices \({\varvec{O}}\) and \({\varvec{I}}\) are \(2 \times 2\) zero and identity matrices, respectively and \({\varvec{\sigma }}_x\), \({\varvec{\sigma }}_y\), and \({\varvec{\sigma }}_z\) are the Pauli matrices given by

$$\begin{aligned} {\varvec{\sigma }}_x&= \begin{pmatrix} 0 &{} 1 \\ 1 &{} 0 \end{pmatrix}, {\varvec{\sigma }}_y = \begin{pmatrix} 0 &{} -i \\ i &{} 0 \end{pmatrix}, {\varvec{\sigma }}_z = \begin{pmatrix} 1 &{} 0 \\ 0 &{} -1 \end{pmatrix}. \end{aligned}$$
(2)

The solution \({\varvec{\Psi }}({\varvec{r}}, t)\) of Eq. (1) is a four-component spinor.

By applying a non-relativistic approximation to Eq. (1), we obtain the Pauli equation shown below [23, 24]:

$$\begin{aligned} \hat{H}_{\mathrm {Pauli}} {\varvec{\psi }}({\varvec{r}}, t)&= i \hbar \dfrac{\partial }{\partial t} {\varvec{\psi }}({\varvec{r}}, t), \end{aligned}$$
(3)
$$\begin{aligned} \hat{H}_{\mathrm {Pauli}}&= \dfrac{1}{2 m_e} \! \left( {\varvec{\sigma }} \cdot \hat{{\varvec{\pi }}} \right) ^{2} + e_0 A_{0}, \\ {\varvec{\sigma }}&= \begin{pmatrix} {\varvec{\sigma }}_x,&{\varvec{\sigma }}_y,&{\varvec{\sigma }}_z \end{pmatrix}, \nonumber \\ \hat{{\varvec{\pi }}}&= \begin{pmatrix} \hat{\pi }_x,&\hat{\pi }_y,&\hat{\pi }_z \end{pmatrix} = \hat{{\varvec{p}}} - e_0 {\varvec{A}}, \nonumber \end{aligned}$$
(4)

where \(\hat{{\varvec{p}}} = -i \hbar \nabla\) is the momentum operator. The solution \({\varvec{\psi }}({\varvec{r}}, t)\) of Eq. (3) is a 2-component spinor. In addition to possessing wave nature such as interference, this spinor along with the Pauli matrices gives expectation of spin orientation.

We introduce the following formulas [24]:

$$\begin{aligned} ( {\varvec{\sigma }} \cdot {\varvec{u}} ) ( {\varvec{\sigma }} \cdot {\varvec{v}} )&= {\varvec{u}} \cdot {\varvec{v}} + i {\varvec{\sigma }} \cdot ( {\varvec{u}} \times {\varvec{v}} ), \end{aligned}$$
(5)
$$\begin{aligned} {\varvec{\sigma }}_d {\varvec{\sigma }}_e&= \delta _{d, e} {\varvec{I}} + i \sum _f \epsilon _{d e f} {\varvec{\sigma }}_{f}, \end{aligned}$$
(6)

where \({\varvec{u}}\) and \({\varvec{v}}\) are vectors of 3 complex-valued or operator elements and \(\delta _{d, e}\) and \(\epsilon _{def}\), \(d, e, f \in \{x, y, z\}\), are Kronecker delta and the Levi-Civita symbol [24], respectively. By using Eqs. (5) and (6), Hamiltonian (4) is transformed into the following expression [24]:

$$\begin{aligned} \hat{H}_{\mathrm {Pauli}}&= \dfrac{1}{2m_e} \left( \hat{{\varvec{p}}} \! - \! e_0 {\varvec{A}} \right) ^{2} {\varvec{I}} + e_0 A_{0} {\varvec{I}} - \dfrac{e_0 \hbar }{2 m_{e}} ({\varvec{\sigma }} \! \cdot \! {\varvec{B}}) , \end{aligned}$$
(7)

where \({\varvec{B}}\) is a magnetic field associated with \({\varvec{A}}\) as \({\varvec{B}} = {\varvec{\nabla }} \times {\varvec{A}}\). In Eq. (7), the interaction between a magnetic field and the spin appears explicitly. The first and second terms on the right-hand side of Eq. (7) are the position-dependent terms. The third term expresses a spin-dependent term. Historically, the Pauli equation (7) was built empirically of these different terms.

In this paper, we disregard the spin-orbit interaction. Then, Hamiltonians (4) and (7) do not contain the interaction term.

2.2 Probability density function and probability current density

Spinors in Eqs. (1) and (3) are not substantial in themselves. The square of the magnitude of the spinor is the probability density function (PDF) \(\rho ({\varvec{r}}, t)\) in terms of location \({\varvec{r}}\) of an electron at time t. We denote current of \(\rho ({\varvec{r}}, t)\), probability current density (PCD), by \({\varvec{j}}({\varvec{r}}, t)\). It is expressed with \(\rho ({\varvec{r}}, t)\) and the velocity \({\varvec{v}}({\varvec{r}}, t)\) of the electron as

$$\begin{aligned} {\varvec{j}}({\varvec{r}}, t) = {\varvec{v}}({\varvec{r}}, t) \rho ({\varvec{r}}, t). \end{aligned}$$
(8)

The following continuity equation generally holds for the PDF:

$$\begin{aligned} \dfrac{\partial }{\partial t} \rho ({\varvec{r}}, t) + \nabla \cdot {\varvec{j}}({\varvec{r}}, t) = 0. \end{aligned}$$
(9)

The first and second components of the spinor of the Pauli equation describe electron states with spin orientation up and down, respectively. PDF \(\rho ({\varvec{r}}, t)\) is the sum of the two PDFs for the two spin orientations. Then, \(\rho ({\varvec{r}}, t)\) is given by

$$\begin{aligned} \rho ({\varvec{r}}, t) = {\varvec{\psi }}^{\dagger }({\varvec{r}}, t) {\varvec{\psi }}({\varvec{r}}, t), \end{aligned}$$
(10)

where \({\varvec{\psi }}^{\dagger }({\varvec{r}}, t)\) is the conjugate transpose of spinor \({\varvec{\psi }}({\varvec{r}}, t)\).

According to the preceding researches [24, 25], PCD \({\varvec{j}}({\varvec{r}}, t)\) is derived in the following way: The time-derivative function of \(\rho ({\varvec{r}}, t)\) is given by

$$\begin{aligned} \dfrac{\partial \rho ({\varvec{r}}, t)}{\partial t} = {\varvec{\psi }}^{\dagger }({\varvec{r}}, t)\dfrac{\partial {\varvec{\psi }}({\varvec{r}}, t)}{\partial t} + \dfrac{\partial {\varvec{\psi }}^{\dagger }({\varvec{r}}, t)}{\partial t} {\varvec{\psi }}({\varvec{r}}, t). \end{aligned}$$
(11)

Substituting Eq. (3) and its conjugate transpose

$$\begin{aligned} -i \hbar \dfrac{\partial }{\partial t} {\varvec{\psi }}^{\dagger }({\varvec{r}}, t)&= \dfrac{1}{2 m_e} \left( ( {\varvec{\sigma }} \cdot \hat{{\varvec{\pi }}} )^2 {\varvec{\psi }}({\varvec{r}}, t) \right) ^{\dagger } \nonumber \\&+ e_0 A_0 {\varvec{\psi }}^{\dagger }({\varvec{r}}, t) \end{aligned}$$
(12)

into Eq. (11) and dividing it by \(i \hbar\), we get

$$\begin{aligned} \dfrac{\partial \rho ({\varvec{r}}, t)}{\partial t}&= \dfrac{1}{i 2 m_e \hbar } \left( {\varvec{\psi }}^{\dagger } ( {\varvec{\sigma \cdot \hat{{\varvec{\pi }}}}} )^{2} {\varvec{\psi }} - \left( ( {\varvec{\sigma }} \cdot \hat{{\varvec{\pi }}} )^{2} {\varvec{\psi }} \right) ^{\dagger } {\varvec{\psi }} \right) \nonumber \\&= \dfrac{1}{i 2 m_e \hbar } \times \nonumber \\&\quad \sum _{d \in \{x, y, z\}} \sum _{e\in \{x, y, z\}} ( {\varvec{\psi }}^{\dagger } {\varvec{\sigma }}_d {\varvec{\sigma }}_e (\hat{\pi }_d \hat{\pi }_e {\varvec{\psi }}) \nonumber \\&\qquad \qquad \qquad - (\hat{\pi }_d \hat{\pi }_e {\varvec{\psi }} )^{\dagger } {\varvec{\sigma }}_e {\varvec{\sigma }}_d {\varvec{\psi }} ) \nonumber \\&= -\dfrac{1}{2m_e} \times \nonumber \\&\quad \sum _{d \in \{x, y, z\}} \sum _{e\in \{x, y, z\}} \nabla _d ( {\varvec{\psi }}^{\dagger } {\varvec{\sigma }}_d {\varvec{\sigma }}_e (\hat{\pi }_e {\varvec{\psi }}) \nonumber \\&\qquad \qquad \qquad + (\hat{\pi }_e {\varvec{\psi }})^{\dagger } {\varvec{\sigma }}_e {\varvec{\sigma }}_d {\varvec{\psi }} ) \nonumber \\&\quad + \dfrac{1}{i 2 m_e \hbar } \times \nonumber \\&\quad \sum _{d \in \{x, y, z\}} \sum _{e\in \{x, y, z\}} ( \hat{\pi }_e {\varvec{\psi }} )^{\dagger } {\varvec{\sigma }}_e {\varvec{\sigma }}_d (\hat{\pi }_d {\varvec{\psi }}) \nonumber \\&\qquad \qquad \qquad - (\hat{\pi }_d {\varvec{\psi }}){\dagger } {\varvec{\sigma }}_d {\varvec{\sigma }}_e (\hat{\pi }_e {\varvec{\psi }}). \end{aligned}$$
(13)

In equating last two expressions in Eq. (13), we applied an operation rule [24] given by

$$\begin{aligned} \phi ^{\dagger } (\hat{\pi }_d \varphi ) - (\hat{\pi }_d \phi )^{\dagger } \varphi = \dfrac{\hbar }{i} \nabla _d( \phi ^\dagger \varphi ) \end{aligned}$$
(14)

where \(\phi\) and \(\varphi\) are 2-components spinors. The second summation in the last expression of Eq. (13) is 0. Comparing Eq. (9) and (13), we find that PCD \({\varvec{j}}({\varvec{r}}, t)\) is given by

$$\begin{aligned} {\varvec{j}}({\varvec{r}}, t) \!&= \! \dfrac{1}{2 m_e} \times \nonumber \\&\sum _{d\in \{x, y, z\}} \! \left( \! (\hat{\pi }_d {\varvec{\psi }})^{\dagger } {\varvec{\sigma }}_d {\varvec{\sigma }} {\varvec{\psi }} + {\varvec{\psi }}^{\dagger } {\varvec{\sigma }} {\varvec{\sigma }}_d ( \hat{\pi }_{d} {\varvec{\psi }} ) \! \right) . \end{aligned}$$
(15)

The x component of Eq. (15) is

$$\begin{aligned} j_x({\varvec{r}}, t) \!&= \! \dfrac{1}{2 m_e} \times \nonumber \\&\sum _{d \in \{x, y, z\}} \! (\hat{\pi }_d {\varvec{\psi }})^{\dagger } {\varvec{\sigma }}_d {\varvec{\sigma }}_x {\varvec{\psi }} + {\varvec{\psi }}^{\dagger } {\varvec{\sigma }}_x {\varvec{\sigma }}_d (\hat{\pi }_d {\varvec{\psi }}). \end{aligned}$$
(16)

Equation (16) is transformed by using Eqs. (6) and (14) to the following equation:

$$\begin{aligned} j_x({\varvec{r}}, t)&= \dfrac{1}{2 m_e} \left( (\hat{\pi }_x {\varvec{\psi }})^{\dagger } {\varvec{\psi }} + {\varvec{\psi }}^{\dagger } (\hat{\pi }_x {\varvec{\psi }}) \right) \nonumber \\&\quad + \dfrac{i}{2 m_e} \sum _{d \in \{x, y, z\}} \sum _{e\in \{x, y, z\}} \nonumber \\&\quad \quad \epsilon _{x d e} \left( -( \hat{\pi }_d {\varvec{\psi }} )^{\dagger } {\varvec{\sigma }}_e {\varvec{\psi }} + {\varvec{\psi }}^{\dagger } {\varvec{\sigma }}_e ( \hat{\pi }_d {\varvec{\psi }} ) \right) \nonumber \\&= \dfrac{1}{2 m_e} \left[ \left\{ \left( -i \hbar \dfrac{\partial }{\partial x} - e_0 A_x \right) {\varvec{\psi }} \right\} ^{\dagger } {\varvec{\psi }} \right. \nonumber \\&\qquad \qquad \left. + {\varvec{\psi }}^{\dagger } \left\{ \left( -i \hbar \dfrac{\partial }{\partial x} - e_0 A_x \right) {\varvec{\psi }} \right\} \right] \nonumber \\&\quad + \dfrac{i}{2 m_e} \sum _{d \in \{x, y, z\}} \sum _{e\in \{x, y, z\}} \epsilon _{x d e} \dfrac{\hbar }{i} \nabla _d ( {\varvec{\psi }}^{\dagger } {\varvec{\sigma }}_{e} {\varvec{\psi }} ) \nonumber \\&= \dfrac{i \hbar }{2 m_e} \left( \dfrac{\partial {\varvec{\psi }}^{\dagger }}{\partial x} {\varvec{\psi }} - {\varvec{\psi }}^{\dagger } \dfrac{\partial {\varvec{\psi }}}{\partial x} \right) - \dfrac{e_0}{m_e} A_x \rho \nonumber \\&\quad + \dfrac{\hbar }{2 m_e} \left( \dfrac{\partial }{\partial y} ( {\varvec{\psi }}^{\dagger } {\varvec{\sigma }}_z {\varvec{\psi }} ) - \dfrac{\partial }{\partial z} ( {\varvec{\psi }}^{\dagger } {\varvec{\sigma }}_y {\varvec{\psi }} ) \right) . \end{aligned}$$
(17)

The y and z components are obtained in the same way. Then, \({\varvec{j}}({\varvec{r}}, t)\) is formulated by the following expressions:

$$\begin{aligned} {\varvec{j}}({\varvec{r}}, t)&= {\varvec{j}}_{\mathrm {orbital}}({\varvec{r}}, t) + {\varvec{j}}_{\mathrm {spin}}({\varvec{r}}, t) , \end{aligned}$$
(18)
$$\begin{aligned} {\varvec{j}}_{\mathrm {orbital}}({\varvec{r}}, t)&= i \dfrac{\hbar }{2 m_e} \left[ \left\{ \nabla {\varvec{\psi }}({\varvec{r}}, t)\right\} ^{\dagger } {\varvec{\psi }}({\varvec{r}}, t) \right. \nonumber \\&\quad \left. - {\varvec{\psi }}^{\dagger }({\varvec{r}}, t) \left\{ \nabla {\varvec{\psi }}({\varvec{r}}, t)\right\} \right] - \dfrac{e_0}{m_e} {\varvec{A}} \rho \end{aligned}$$
(19)
$$\begin{aligned} {\varvec{j}}_{\mathrm {spin}}({\varvec{r}}, t)&= \dfrac{\hbar }{2 m_e} \left\{ \nabla \times ( {\varvec{\psi }}^{\dagger } {\varvec{\sigma }} {\varvec{\psi }} ) \right\} . \end{aligned}$$
(20)

As the Pauli equation is derived from the non-relativistic approximation of the Dirac equation, the PCD for the Pauli spinor is derived also by applying the non-relativistic approximation to the PCD for the Dirac equation [26]. Preceding researches [25, 27,28,29,30,31] presented various ways of deriving the PCD, for example, a way starting from the expectation of spin orientation toward the density current.

2.3 Probability density function and probability current density

As an example, we derive a PDF \(\rho ({\varvec{r}}, t)\) and a PCD \({\varvec{j}}({\varvec{r}}, t)\) in a symmetric gauge potential

$$\begin{aligned} {\varvec{A}} = \left( A_r, A_\theta , A_z \right) = \left( 0, \dfrac{B_0}{2} r, 0 \right) . \end{aligned}$$
(21)

In Eq. (21) and hereafter, the cylindrical coordinate system \({\varvec{r}} = (r, \theta , z)\) may be used. The gradient operator in the cylindrical coordinate system is \(\nabla = (\partial / \partial r, (1/r)\partial / \partial \theta , \partial / \partial z)\). Then, the magnetic field is given by

$$\begin{aligned} {\varvec{B}} = \nabla \times {\varvec{A}} = \begin{pmatrix} 0, 0, B_{0} \end{pmatrix}. \end{aligned}$$
(22)

The scalar potential is assumed to be \(A_{0} = 0\).

Eigen functions of the Pauli equation (3) are denoted by

$$\begin{aligned} {\varvec{\psi }}_{n, m_z}({\varvec{r}}, t; k_z) = \begin{pmatrix} \psi _{n, m_z, \uparrow }({\varvec{r}}, t; k_z) \\ \psi _{n, m_z,\downarrow }({\varvec{r}}, t; k_z) \end{pmatrix}, \end{aligned}$$
(23)

where n is the principal quantum number, \(m_z\) is the angular quantum number, and \(k_z\) is the wave number in the z direction. The eigen functions and eigen energies are given by the following equations [32]:

$$\begin{aligned} \psi _{n, m_z, \uparrow / \downarrow }({\varvec{r}}, t; k_z)&= \dfrac{1}{\sqrt{2}} R_{n, m_z}(\xi (r)) \exp { \left( i m_z \theta \right) } \nonumber \\&\quad \! \times \! \mathrm {e}^{ i k_z z } \exp { \left( -i \dfrac{\varepsilon _{n, m_z, \uparrow / \downarrow }}{\hbar } t\right) }, \\ \xi (r)&= \frac{e_0 B_0}{2 \hbar }r^2 \nonumber \\ R_{n, m_z}(x)&= \sqrt{\dfrac{m_e \omega _c}{\pi \hbar }} \sqrt{\dfrac{n!}{(n + \left|m_z\right|)!}} L_{n}^{\left|m_z\right|}(x) \nonumber \\&\quad \times \exp \left( -\dfrac{x}{2} \right) x^{\frac{\left|m_z\right|}{2}}, \nonumber \\ \varepsilon _{n, m_z, \uparrow / \downarrow }(k_z)&= \left( 2n + \left|m_z\right| + 1 \right) \hbar \omega _C + m_z \hbar \omega _C \nonumber \\&\quad + \dfrac{\hbar ^2 k_{z}^{2}}{2 m_e} \pm \omega _C B_0, \nonumber \\ \omega _C&= \dfrac{e_0 B_0}{2 m_e} \nonumber \end{aligned}$$
(24)

where \(L_{n}^{\left|m_z\right|}(x)\) is the associated Laguerre polynomial.

When spinors are given by Eq. (24), PDF \(\rho ({\varvec{r}}, t)\) and PCD \({\varvec{j}}({\varvec{r}}, t)\) described by Eqs. (10) and (18) become

$$\begin{aligned} \rho ({\varvec{r}}, t)&= \left\{ R_{n, m_z}(\xi (r)) \right\} ^{2}, \end{aligned}$$
(25)
$$\begin{aligned} j_r({\varvec{r}}, t)&= 0, \end{aligned}$$
(26)
$$\begin{aligned} j_{\theta }({\varvec{r}}, t)&= -\dfrac{\hbar m_z}{m_e} \dfrac{\rho ({\varvec{r}}, t)}{r} - \dfrac{e_0 B_0}{2 m_e} r \rho ({\varvec{r}}, t), \end{aligned}$$
(27)
$$\begin{aligned} j_{z}({\varvec{r}}, t)&= -\dfrac{\hbar }{m_e} k_z \rho ({\varvec{r}}, t) \nonumber \\&\quad + \dfrac{\hbar }{2 m_e} \left[ \dfrac{\rho ({\varvec{r}}, t)}{r} - \rho ({\varvec{r}}, t) + \dfrac{\partial }{\partial r}\rho ({\varvec{r}}, t) \right] \nonumber \\&\quad \times \sin { \left( \dfrac{e_0 B_0}{m_e}t - \theta \right) }. \end{aligned}$$
(28)

3 Constructing a classical particle model

3.1 Particle model

We describe how to construct a classical particle model of an electron.

Evolution of PDF \(\rho _{FP}({\varvec{r}}, t)\) for a probabilistic classical particle affected by white Gaussian fluctuation is described by the following FPE [22]:

$$\begin{aligned} \dfrac{\partial \rho _{FP}({\varvec{r}}, t)}{\partial t}\! =\! - \nabla \! \cdot \! \left( {\varvec{b}}({\varvec{r}}, t) \rho _{FP}({\varvec{r}}, t) \right) + \nu \nabla ^2 \rho _{FP}({\varvec{r}}, t). \end{aligned}$$
(29)

According to reference [19], quantum fluctuation of the motion of an electron is \(\hbar / m_e\) in variance. Then, we set the diffusion coefficient in Eq. (29) as

$$\begin{aligned} \nu = \dfrac{\hbar }{2 m_e}. \end{aligned}$$
(30)

Equation (29) is expressed differently by using the differential operator \(\nabla\) and PCD \({\varvec{j}}_{FP}({\varvec{r}}, t)\) for the classical particle as follows [22]:

$$\begin{aligned} \dfrac{\partial \rho _{FP}({\varvec{r}}, t)}{\partial t}&+ \nabla \cdot {\varvec{j}}_{FP}({\varvec{r}}, t) = 0, \end{aligned}$$
(31)
$$\begin{aligned} {\varvec{j}}_{FP}({\varvec{r}}, t)&= {\varvec{b}}({\varvec{r}}, t) \rho _{FP}({\varvec{r}}, t) - \dfrac{\hbar }{2 m_{\mathrm {e}}} \nabla \rho _{FP}({\varvec{r}}, t). \end{aligned}$$
(32)

Comparing Eqs. (9) and (31), we find that Eq. (9) is the FPE for a quantum electron. Then, PCD \({\varvec{j}}({\varvec{r}}, t)\) for the electron can be expressed by Eq. (32) with \(\rho _{FP}({\varvec{r}}, t)\) and \({\varvec{j}}_{FP}({\varvec{r}}, t)\) replaced by \(\rho ({\varvec{r}}, t)\) and \({\varvec{j}}({\varvec{r}}, t)\), respectively. We then obtain the drift term of the FPE for the electron as

$$\begin{aligned} {\varvec{b}}({\varvec{r}}, t)&= \dfrac{\hbar }{2 m_{\mathrm {e}}} \dfrac{\nabla \rho ({\varvec{r}}, t)}{\rho ({\varvec{r}}, t)} + \dfrac{{\varvec{j}}({\varvec{r}}, t)}{\rho ({\varvec{r}}, t)} \nonumber \\&= \dfrac{\hbar }{2 m_{\mathrm {e}}} \nabla \ln {\rho ({\varvec{r}}, t)} + \dfrac{{\varvec{j}}({\varvec{r}}, t)}{\rho ({\varvec{r}}, t)}. \end{aligned}$$
(33)

A classical particle model of the electron should follow \(\rho ({\varvec{r}}, t)\) statistically. Therefore, as mentioned in Sect. 1, the equation of motion for the model should be described by the following LvE possessing drift term (33) and a markov process \({\varvec{\Gamma }}(t)\):

$$\begin{aligned} d{\varvec{r}}(t) = {\varvec{b}}({\varvec{r}}, t)dt + d{\varvec{\Gamma }}(t), \end{aligned}$$
(34)
$$\begin{aligned} d{\varvec{\Gamma }}(t) = {\varvec{\Gamma }}(t + dt) - {\varvec{\Gamma }}(t). \end{aligned}$$
(35)

The components of \(d{\varvec{\Gamma }}(t)\) are white Gaussian fluctuation. They should be zero in average and \(2 \nu\) in variance.

3.2 Electron spin of the particle model

The spin of an electron gives it angular momentum. This spin angular momentum is determined in average with an operator

$$\begin{aligned} \hat{{\varvec{S}}} = \dfrac{\hbar }{2} {\varvec{\sigma }} \end{aligned}$$
(36)

on the Pauli spinor by the following integral:

$$\begin{aligned} \left\langle {\varvec{S}} \right\rangle _{{\varvec{r}}}\!(t) = \int {\varvec{\psi }}^{\dagger }({\varvec{r}}, t) \hat{{\varvec{S}}} {\varvec{\psi }}({\varvec{r}}, t) d{\varvec{r}}. \end{aligned}$$
(37)

This spin angular momentum is regarded as an expectation obtained by averaging spatially with the probability distribution derived from the spinor. The classical particle model of an electron follows the same distribution. However, locations on a sample trajectory of the electron model are determined as a function of time unlike obscure location of an electron determined by the spinor. Then, it is natural that the spin angular momentum of the electron model which locates at point \(\tilde{{\varvec{r}}}\) is given by

$$\begin{aligned} \tilde{{\varvec{S}}} (\tilde{{\varvec{r}}}, t) = \dfrac{{\varvec{\psi }}^{\dagger }(\tilde{{\varvec{r}}}, t) \hat{{\varvec{S}}} {\varvec{\psi }}(\tilde{{\varvec{r}}}, t)}{{\varvec{\psi }}^{\dagger }(\tilde{{\varvec{r}}}, t){\varvec{\psi }}(\tilde{{\varvec{r}}}, t)}. \end{aligned}$$
(38)

This spin angular momentum \(\tilde{S}(\tilde{{\varvec{r}}}, t)\) is regarded as an ensemble average for the electrons on all trajectories passing through point \(\tilde{{\varvec{r}}}\), as illustrated in Fig. 1.

As \(\left\langle {\varvec{S}} \right\rangle _{{\varvec{r}}}\!(t)\) defined by Eq. (37) is constant in magnitude, \(\tilde{{\varvec{S}}} (\tilde{{\varvec{r}}}, t)\) is normalized as in Eq. (38). The orientation of electron spin mentioned in previous and succeeding sections means the direction of these spin angular momenta \(\left\langle {\varvec{S}} \right\rangle _{{\varvec{r}}}\!(t)\) and \(\tilde{{\varvec{S}}} (\tilde{{\varvec{r}}}, t)\).

Fig. 1
figure 1

Illustration of spin angular momenta \(\left\langle {\varvec{S}} \right\rangle _{{\varvec{r}}}\!(t)\) and \(\tilde{{\varvec{S}}} (\tilde{{\varvec{r}}}, t)\)

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3.3 Classical particle model of an electron in a symmetric gauge potential

The drift term (33) of the LvE or the equation of motion for the classical particle model of an electron in a symmetric gauge is obtained by substituting \(\rho (r, \theta , z, t)\) and \({\varvec{j}}(r, \theta , z, t)\) given by Eqs. (25)–(28) into Eq. (33) as follows:

$$\begin{aligned} b_r({\varvec{r}}, t)&= \dfrac{\hbar }{2 m_e} \dfrac{1}{\rho ({\varvec{r}}, t)} \dfrac{\partial }{\partial r} \rho ({\varvec{r}}, t) + \dfrac{j_r({\varvec{r}}, t)}{\rho ({\varvec{r}}, t)} \nonumber \\&= \dfrac{\hbar }{2 m_e} \dfrac{\partial }{\partial r}\left( \ln { \rho ({\varvec{r}}, t) } \right) , \end{aligned}$$
(39)
$$\begin{aligned} b_{\theta }({\varvec{r}}, t)&= \dfrac{\hbar }{2 m_e} \dfrac{1}{\rho ({\varvec{r}}, t)} \dfrac{1}{r} \dfrac{\partial }{\partial \theta } \rho ({\varvec{r}}, t) + \dfrac{j_{\theta }({\varvec{r}}, t)}{\rho ({\varvec{r}}, t)} \nonumber \\&= -\dfrac{\hbar }{m_e} m_z \dfrac{1}{r} -\dfrac{e_0 B_0}{2 m_e} r, \end{aligned}$$
(40)
$$\begin{aligned} b_z({\varvec{r}}, t)&= \dfrac{\hbar }{2 m_e} \dfrac{1}{\rho ({\varvec{r}}, t)} \dfrac{\partial }{\partial z} \rho ({\varvec{r}}, t) + \dfrac{j_z({\varvec{r}}, t)}{\rho ({\varvec{r}}, t)} \nonumber \\&=\! \dfrac{\hbar }{2 m_e} \left\{ \dfrac{1}{r}\! -\! 1 + \dfrac{\partial }{\partial r}\left( \ln {\rho ({\varvec{r}}, t)} \right) \right\} \nonumber \\&\quad \times \sin { \left( \dfrac{e_0 B_0}{m_e}t\! -\! \theta \right) }\! -\! \dfrac{\hbar }{m_e} k_z. \end{aligned}$$
(41)

Sample trajectories of the electron model is computed by integrating numerically the LvE with the above drift term. Expectation of the spin orientation at any point on the trajectories is computed with Eq. (38).

4 Numerical experiments

4.1 Units

We employ the Hartree atomic units for convenience in computation. That is, we perform numerical experiments with the converted Planck constant \(\hbar\), the mass of the electron \(m_e\), and the electron charge \(e_0\) normalized to 1. As a result of the normalization, the units are converted as shown in Table 1.

Table 1 Unit Conversion
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4.2 Model and its conditions

A mathematical model of an electron in symmetric gauge potential (21) was established in Sect. 3.3. In this section, numerical experiments for the model will be conducted on the following three conditions: (1) The electron is confined in \(r-\theta\) or \(x-y\) plane with \(z = 0\), which can be done substantially with three layers of different semiconductor materials. Then, we set \(e^{i k_z z} = 1\) and \(b_z(r, t) = 0\). (2) The magnitude of magnetic field is set to \(B_0 = 1.0\). (3) The electron spin is assumed not to be relaxed since the Pauli equation in Sect. 2.1 describes energy-conserved quantum systems. It is expected that the electron takes cyclotron motion on the \(r - \theta\) plane and its spin orientation is conserved in z-direction.

4.3 Results

4.3.1 Probability distribution, probability current, and drift term

Figure 2 shows the PDF \(\rho ({\varvec{r}}) = \left|\psi _{n, m_z}\right|^2\) for \((n, m_z) = (1, 2)\) and (2, 3), derivative \(d\rho (r)/dr\) of marginal PDF \(\rho (r)\) of r, and argument component of PCD \(j_{\theta }(r)\) given by Eq. (27). We see that \(\rho (r, \theta )\) is rotationally symmetric. This is because the PDF given by Eq. (25) is independent of \(\theta\). We realize also that \(j_{\theta }(r)\) switches its polarity at the bottom of marginal PDF \(\rho (r)\) where \(d\rho (r)/dr = 0\), that is, PCD \(j_{\theta }(r)\) is clockwise and counterclockwise on adjacent lobes of \(\rho (r)\).

Figure 3 presents radius and argument components of the drift term, \(b_r(r)\) and \(b_\theta (r)\). The radius component given by Eq. (39) is zero where \(\rho (r)\) takes extrema and \(d\rho (r)/dr = 0\) since \(j_r(r, \theta ) = 0\) at the extremal points. The argument component given by Eq. (40) also switches its polarity at the bottom of marginal PDF \(\rho (r)\) since \(d\rho (r, \theta ) / d\theta = 0\) and then \(b_\theta (r) = j_\theta (r)/\rho (r)\). It is inferred that the electron moves almost circularly and clockwise or counterclockwise along circles where \(\rho (r)\) is locally maximum.

4.3.2 Trajectory of an electron

Figure 4 shows samples of electron trajectories obtained by numerical integration of Eq. (34). The initial points of the trajectory samples are set to \((r_0, \theta _0) = (1.0, \frac{\pi }{6}), (4.0, \frac{\pi }{6})\) in Fig. 4a and \((r_0, \theta _0) = (1.0, \frac{\pi }{6}), (3.0, \frac{\pi }{6}), (5.0, \frac{\pi }{6})\) in Fig. 4b. Figure 4 shows that the behavior of the proposed electron model is roughly circular motion along higher points of the PDF as we surmised. Figure 5 presents the \(\theta\)-component of the electron trajectory samples. We see that the direction of the rotation of the electron model changes depending on the radius. The period of the rotation also depends on the radius. From Figs. 4 and 5, the proposed model seems to be following the PDF and the PCD derived from the Pauli spinor.

We computed the expectations of spin orientation \(\tilde{{\varvec{S}}}(\tilde{{\varvec{r}}}, t)\) given by Eq. (38) at points on the trajectories of the electron model and plotted them in Fig. 6. The spin rotation in this figure and the circular motion in Fig. 5 do not synchronize since \(\tilde{{\varvec{S}}}(\tilde{{\varvec{r}}}, t)\) is a function of not only location \({\varvec{r}}\) but also time t.

Fig. 2
figure 2

Probability density function and probability current density of electrons

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Fig. 3
figure 3

Drift Term of the model

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Fig. 4
figure 4

Electron trajectories and contour map of probability density function

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Fig. 5
figure 5

Time-varing argument \(\theta (t)\) of the location of electron particle model. a Upper: \(r \approx 1.0\), lower:\(r \approx 4.0\), b upper:\(r \approx 1.0\), middle:\(r \approx 3.0\), lower:\(r \approx 5.0\)

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Fig. 6
figure 6

Time variation of orientation of the electron spin. a Upper: \(r \approx 1.0\), lower:\(r \approx 4.0\), b upper:\(r \approx 1.0\), middle:\(r \approx 3.0\), lower:\(r \approx 5.0\)

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Fig. 7
figure 7

Marginal probability density functions computed with spinors and histograms obtained from electron trajectory samples

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4.4 Discussion

Figure 7 shows the marginal PDFs \(\rho (r)\) and \(\rho _p(r)\) given by Eq. (25) and computed from the electron trajectory samples, respectively. We compute \(\rho _{p}(r)\) as follows:

$$\begin{aligned} \rho _p (r)&= \dfrac{N_i (r)}{N_{total}}, \\ N_{total}&: \text{ The } \text{ number } \text{ of } \text{ sampled } \text{ locations }, \nonumber \\ N_i (r)&: \text{ The } \text{ number } \text{ of } \text{ sampled } \text{ locations } \text{ whose } \nonumber \\&\quad r \text{-coordinate } \text{ is } r \in \delta r_{i}, \nonumber \\ \delta r_{i}&= \left[ \dfrac{i}{N_{divide}} r_{max}, \dfrac{i + 1}{N_{divide}}r_{max} \right) , \nonumber \\ 0&\le i < N_{divide}, \nonumber \\ N_{divide}&: \text{ The } \text{ number } \text{ of } \text{ intervals } \delta r_{i}. \nonumber \end{aligned}$$
(42)

Marginal PDFs \(\rho _{p}(r)\) in Fig. 7 are computed under the following setting: \(N_{total} = 3000, N_{divide} = 50\), and \(r_{max} = 7.0\).

In order to evaluate closeness between \(\rho _p(r)\) and \(\rho (r)\), the following error was computed:

$$\begin{aligned} \text{ Error } = \int _{0}^{r_{max}} \left| \rho (r) - \rho _{p}(r) \right| dr. \end{aligned}$$
(43)

The errors for \((n, m_z) = (1, 2)\) and (2, 3) were 0.041 and 0.039, respectively. From these results, we confirm that proposed classical particle model and the spinor have almost the same statistical properties.

Spatially averaged spin orientation (37) is computed by substituting Eq. (24) into Eq. (37). As \(\hbar\), \(m_e\), and \(e_0\) are normalized and \(B_0 = 1\), we have

$$\begin{aligned} \langle {\varvec{S}} \rangle _{{\varvec{r}}}\! (t)&= A \begin{pmatrix} \cos { (t - t_0) },&\sin { (t - t_0) } \end{pmatrix} \\ A&= \dfrac{1}{2} \int R_{n, m_z}(\xi (r)) dr. \nonumber \end{aligned}$$
(44)

The orientation given by Eq. (44) is shown in Fig. 6 together with \(\tilde{{\varvec{S}}}({\varvec{r}}, t)\) when \(t_0 = 0\). It is interesting that the rotation periods of \(\tilde{{\varvec{S}}}({\varvec{r}}, t)\) are equal to that of \(\langle {\varvec{S}} \rangle _{{\varvec{r}}}\!(t)\) independently of whether the electron model takes inner, middle, or outer orbit.

5 Conclusions

We established the LvE to describe an electron as a stochastic classical particle model. Unlike preceding studies [15,16,17,18,19,20,21], the drift term of the LvE is constructed of Pauli spinor that determines spin state. Therefore, in addition to virtual electron trajectories, expectation of spin orientation was computed along the trajectories, as shown in Sect. 4.3. The statistical property of the trajectories and the period of the spin rotation are confirmed to be consistent with theoretical ones derived from the spinors.

Real quantum systems are more complex in structure and consist of more electrons than the example in this paper. Then, we rely on numerical computation to obtain wave functions of real systems. One of our future works is developing stable, accurate, and simple numerical computation required to construct stochastic classical particle models of real systems. Another future work is constructing a model with probabilistically dichotomic spin orientations {up, down} representing measured spin orientations.