Pauli 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 ${\displaystyle \phi }$, the Pauli equation reads:

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

${\displaystyle |\psi \rangle =\psi _{+}|{\mathord {\uparrow }}\rangle +\psi _{-}|{\mathord {\downarrow }}\rangle \,{\stackrel {\cdot }{=}}\,{\begin{bmatrix}\psi _{+}\\\psi _{-}\end{bmatrix}}}$.

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

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

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:

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

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

${\displaystyle \left[\left(\mathbf {\hat {p}} -q\mathbf {A} \right)\times \left(\mathbf {\hat {p}} -q\mathbf {A} \right)\right]\psi =-q\left[\mathbf {\hat {p}} \times \left(\mathbf {A} \psi \right)+\mathbf {A} \times \left(\mathbf {\hat {p}} \psi \right)\right]=iq\hbar \left[\nabla \times \left(\mathbf {A} \psi \right)+\mathbf {A} \times \left(\nabla \psi \right)\right]=iq\hbar \left[\psi \left(\nabla \times \mathbf {A} \right)-\mathbf {A} \times \left(\nabla \psi \right)+\mathbf {A} \times \left(\nabla \psi \right)\right]=iq\hbar \mathbf {B} \psi }$

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

For the full Pauli equation, one then obtains[2]

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.[3]

The Pauli equation can be inferred from the non-relativistic limit of the Dirac equation, which is the relativistic quantum equation of motion for spin-½ particles.[4]

Derivation

Dirac equation can be written as:

$${\displaystyle i\hbar \,\partial _{t}{\begin{pmatrix}\psi _{1}\\\psi _{2}\end{pmatrix}}=c\,{\begin{pmatrix}{\boldsymbol {\sigma }}\cdot {\boldsymbol {\Pi }}\,\psi _{2}\\{\boldsymbol {\sigma }}\cdot {\boldsymbol {\Pi }}\,\psi _{1}\end{pmatrix}}+q\,\phi \,{\begin{pmatrix}\psi _{1}\\\psi _{2}\end{pmatrix}}+mc^{2}\,{\begin{pmatrix}\psi _{1}\\-\psi _{2}\end{pmatrix}},}$$

where ${\textstyle \partial _{t}={\frac {\partial }{\partial t}}}$ and ${\displaystyle \psi _{1},\psi _{2}}$ are two-component spinor, forming a bispinor.

Using the following ansatz:

$${\displaystyle {\begin{pmatrix}\psi _{1}\\\psi _{2}\end{pmatrix}}=e^{-i{\tfrac {mc^{2}t}{\hbar }}}{\begin{pmatrix}\psi \\\chi \end{pmatrix}},}$$

with two new spinors ${\displaystyle \psi ,\chi }$, the equation becomes

$${\displaystyle i\hbar \partial _{t}{\begin{pmatrix}\psi \\\chi \end{pmatrix}}=c\,{\begin{pmatrix}{\boldsymbol {\sigma }}\cdot {\boldsymbol {\Pi }}\,\chi \\{\boldsymbol {\sigma }}\cdot {\boldsymbol {\Pi }}\,\psi \end{pmatrix}}+q\,\phi \,{\begin{pmatrix}\psi \\\chi \end{pmatrix}}+{\begin{pmatrix}0\\-2\,mc^{2}\,\chi \end{pmatrix}}.}$$

In the non-relativistic limit, ${\displaystyle \partial _{t}\chi }$ and the kinetic and electrostatic energies are small with respect to the rest energy ${\displaystyle mc^{2}}$.

Thus

$${\displaystyle \chi \approx {\frac {{\boldsymbol {\sigma }}\cdot {\boldsymbol {\Pi }}\,\psi }{2\,mc}}\,.}$$

Inserted in the upper component of Dirac equation, we find Pauli equation (general form):

$${\displaystyle i\hbar \,\partial _{t}\,\psi =\left[{\frac {({\boldsymbol {\sigma }}\cdot {\boldsymbol {\Pi }})^{2}}{2\,m}}+q\,\phi \right]\psi .}$$

Pauli's equation is derived by requiring minimal coupling, which provides a g-factor g=2. Most elementary particles have anomalous g-factors, different from 2. In the domain of relativistic quantum field theory, one defines a non-minimal coupling, sometimes called Pauli coupling, in order to add an anomalous factor

${\displaystyle \gamma ^{\mu }p_{\mu }\to \gamma ^{\mu }p_{\mu }-q\gamma ^{\mu }A_{\mu }+a\sigma _{\mu \nu }F^{\mu \nu }}$

where ${\displaystyle p_{\mu }}$ is the four-momentum operator, ${\displaystyle A_{\mu }}$ is the electromagnetic four-potential, ${\displaystyle a}$ is proportional to the anomalous magnetic dipole moment, ${\displaystyle F^{\mu \nu }=\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }}$ is the electromagnetic tensor, and ${\textstyle \sigma _{\mu \nu }={\frac {i}{2}}[\gamma _{\mu },\gamma _{\nu }]}$ are the Lorentzian spin matrices and the commutator of the gamma matrices ${\displaystyle \gamma ^{\mu }}$.[6][7] In the context of non-relativistic quantum mechanics, instead of working with the Schrödinger equation, Pauli coupling is equivalent to using the Pauli equation (or postulating Zeeman energy) for an arbitrary g-factor.

• Semiclassical physics
• Atomic, molecular, and optical physics
• Group contraction
• Gordon decomposition