Pauli equation

Equation

For a particle of mass ${\displaystyle m}$ and electric charge ${\displaystyle q}$, in an electromagnetic field described by the magnetic vector potential ${\displaystyle \mathbf{A}}$ and the electric scalar potential ${\displaystyle \varphi }$, the Pauli equation reads:

Pauli equation(general)

${\displaystyle \left[\frac{1}{2m}(\mathit{\sigma }\cdot (\hat{\mathbf{p}}-q\mathbf{A}){)}^2+q\varphi \right]|\psi ⟩=i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}|\psi ⟩}$

Here ${\displaystyle \mathit{\sigma }=({\sigma }_x,{\sigma }_y,{\sigma }_z)}$ are the Pauli operators collected into a vector for convenience, and ${\displaystyle \hat{\mathbf{p}}=-i\hslash \mathrm{\nabla }}$ is the momentum operator in position representation. The state of the system, ${\displaystyle |\psi ⟩}$ (written in Dirac notation), can be considered as a two-component spinor wavefunction, or a column vector (after choice of basis):

${\displaystyle |\psi ⟩={\psi }_{+}|\uparrow ⟩+{\psi }_{-}|\downarrow ⟩\stackrel{\cdot }{=}\left[\begin{array}{c}{\psi }_{+}\\ {\psi }_{-}\end{array}\right]}$.

The Hamiltonian operator is a 2 × 2 matrix because of the Pauli operators.

${\displaystyle \hat{H}=\frac{1}{2m}{\left[\mathit{\sigma }\cdot (\hat{\mathbf{p}}-q\mathbf{A})\right]}^2+q\varphi }$

Substitution into the Schrödinger equation gives the Pauli equation. This Hamiltonian is similar to the classical Hamiltonian for a charged particle interacting with an electromagnetic field. See Lorentz force for details of this classical case. The kinetic energy term for a free particle in the absence of an electromagnetic field is just ${\displaystyle \frac{{\mathbf{p}}^2}{2m}}$ where ${\displaystyle \mathbf{p}}$ is the kinetic momentum, while in the presence of an electromagnetic field it involves the minimal coupling ${\displaystyle \mathbf{\Pi }=\mathbf{p}-q\mathbf{A}}$, where now ${\displaystyle \mathbf{\Pi }}$ is the kinetic momentum and ${\displaystyle \mathbf{p}}$ is the canonical momentum.

The Pauli operators can be removed from the kinetic energy term using the Pauli vector identity:

${\displaystyle (\mathit{\sigma }\cdot \mathbf{a})(\mathit{\sigma }\cdot \mathbf{b})=\mathbf{a}\cdot \mathbf{b}+i\mathit{\sigma }\cdot \left(\mathbf{a}\times \mathbf{b}\right)}$

Note that unlike a vector, the differential operator ${\displaystyle \hat{\mathbf{p}}-q\mathbf{A}=-i\hslash \mathrm{\nabla }-q\mathbf{A}}$ has non-zero cross product with itself. This can be seen by considering the cross product applied to a scalar function ${\displaystyle \psi }$:

${\displaystyle \begin{array}{cc}\left[\left(\hat{\mathbf{p}}-q\mathbf{A}\right)\times \left(\hat{\mathbf{p}}-q\mathbf{A}\right)\right]\psi & =-q\left[\hat{\mathbf{p}}\times \left(\mathbf{A}\psi \right)+\mathbf{A}\times \left(\hat{\mathbf{p}}\psi \right)\right]\\ & =iq\hslash \left[\mathrm{\nabla }\times \left(\mathbf{A}\psi \right)+\mathbf{A}\times \left(\mathrm{\nabla }\psi \right)\right]\\ & =iq\hslash \left[\psi \left(\mathrm{\nabla }\times \mathbf{A}\right)-\mathbf{A}\times \left(\mathrm{\nabla }\psi \right)+\mathbf{A}\times \left(\mathrm{\nabla }\psi \right)\right]=iq\hslash \mathbf{B}\psi \end{array}}$

where ${\displaystyle \mathbf{B}=\mathrm{\nabla }\times \mathbf{A}}$ is the magnetic field.

For the full Pauli equation, one then obtains

Pauli equation(standard form)

${\displaystyle \hat{H}|\psi ⟩=\left[\frac{1}{2m}\left[{\left(\hat{\mathbf{p}}-q\mathbf{A}\right)}^2-q\hslash \mathit{\sigma }\cdot \mathbf{B}\right]+q\varphi \right]|\psi ⟩=i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}|\psi ⟩}$

for which only a few analytic results are known, e.g., in the context of Landau quantization with homogenous magnetic fields or for an idealized, Coulomb-like, inhomogeneous magnetic field.

Weak magnetic fields

For the case of where the magnetic field is constant and homogenous, one may expand $(\hat{\mathbf{p}}-q\mathbf{A}{)}^2$ using the symmetric gauge $\hat{\mathbf{A}}=\frac{1}{2}\mathbf{B}\times \hat{\mathbf{r}}$, where $\mathbf{r}$ is the position operator and A is now an operator. We obtain

${\displaystyle (\hat{\mathbf{p}}-q\hat{\mathbf{A}}{)}^2=|\hat{\mathbf{p}}{|}^2-q(\hat{\mathbf{r}}\times \hat{\mathbf{p}})\cdot \mathbf{B}+\frac{1}{4}{q}^2\left(|\mathbf{B}{|}^2|\hat{\mathbf{r}}{|}^2-|\mathbf{B}\cdot \hat{\mathbf{r}}{|}^2\right)\approx {\hat{\mathbf{p}}}^2-q\hat{\mathbf{L}}\cdot \mathbf{B},}$

where $\hat{\mathbf{L}}$ is the particle angular momentum operator and we neglected terms in the magnetic field squared $B^2$. Therefore, we obtain

Pauli equation(weak magnetic fields)

${\displaystyle \left[\frac{1}{2m}\left[|\hat{\mathbf{p}}{|}^2-q(\hat{\mathbf{L}}+2\hat{\mathbf{S}})\cdot \mathbf{B}\right]+q\varphi \right]|\psi ⟩=i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}|\psi ⟩}$

where $\mathbf{S}=\hslash \mathit{\sigma }/2$ is the spin of the particle. The factor 2 in front of the spin is known as the Dirac g-factor. The term in $\mathbf{B}$, is of the form $-\mathit{\mu }\cdot \mathbf{B}$ which is the usual interaction between a magnetic moment $\mathit{\mu }$ and a magnetic field, like in the Zeeman effect.

For an electron of charge $-e$ in an isotropic constant magnetic field, one can further reduce the equation using the total angular momentum $\mathbf{J}=\mathbf{L}+\mathbf{S}$ and Wigner-Eckart theorem. Thus we find

${\displaystyle \left[\frac{|\mathbf{p}{|}^2}{2m}+{\mu }_{\mathrm{B}}{g}_J{m}_j|\mathbf{B}|-e\varphi \right]|\psi ⟩=i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}|\psi ⟩}$

where ${\mu }_{\mathrm{B}}=\frac{e\hslash }{2m}$ is the Bohr magneton and $m_j$ is the magnetic quantum number related to $\mathbf{J}$. The term $g_J$ is known as the Landé g-factor, and is given here by

${\displaystyle g_J=\frac{3}{2}+\frac{\frac{3}{4}-\ell (\ell +1)}{2j(j+1)},}$

where ${\displaystyle \ell }$ is the orbital quantum number related to ${\displaystyle L^2}$ and ${\displaystyle j}$ is the total orbital quantum number related to ${\displaystyle J^2}$.