# Pauli equation

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In quantum mechanics, the Pauli equation or Schrödinger–Pauli equation is the formulation of the Schrödinger equation for spin-½ particles, which takes into account the interaction of the particle's spin with an external electromagnetic field. It is the non-relativistic limit of the Dirac equation and can be used where particles are moving at speeds much less than the speed of light, so that relativistic effects can be neglected. It was formulated by Wolfgang Pauli in 1927. 

## Equation

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

Pauli equation(general)

$\left[{\frac {1}{2m}}({\boldsymbol {\sigma }}\cdot (\mathbf {p} -q\mathbf {A} ))^{2}+q\phi \right]|\psi \rangle =i\hbar {\frac {\partial }{\partial t}}|\psi \rangle$ Here ${\boldsymbol {\sigma }}=(\sigma _{x},\sigma _{y},\sigma _{z})$ are the Pauli operators collected into a vector for convenience, and $\mathbf {p} =-i\hbar \nabla$ is the momentum operator. The state of the system, $|\psi \rangle$ (written in Dirac notation), can be considered as a two-component spinor wavefunction, or a column vector (after choice of basis):

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

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

${\hat {H}}={\frac {1}{2m}}\left[{\boldsymbol {\sigma }}\cdot (\mathbf {p} -q\mathbf {A} )\right]^{2}+q\phi$ 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 ${\frac {\mathbf {p} ^{2}}{2m}}$ where $\mathbf {p}$ is the kinetic momentum, while in the presence of an electromagnetic field it involves the minimal coupling $\mathbf {\Pi } =\mathbf {p} -q\mathbf {A}$ , where now $\mathbf {\Pi }$ is the kinetic momentum and $\mathbf {p}$ is the canonical momentum.

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

$({\boldsymbol {\sigma }}\cdot \mathbf {a} )({\boldsymbol {\sigma }}\cdot \mathbf {b} )=\mathbf {a} \cdot \mathbf {b} +i{\boldsymbol {\sigma }}\cdot \left(\mathbf {a} \times \mathbf {b} \right)$ Note that unlike a vector, the differential operator $\mathbf {p} -q\mathbf {A} =-i\hbar \nabla -q\mathbf {A}$ has non-zero cross product with itself. This can be seen by considering the cross product applied to a scalar function $\psi$ :

$\left[\left(\mathbf {p} -q\mathbf {A} \right)\times \left(\mathbf {p} -q\mathbf {A} \right)\right]\psi =-q\left[\mathbf {p} \times \left(\mathbf {A} \psi \right)+\mathbf {A} \times \left(\mathbf {p} \psi \right)\right]=iq\hbar \left[\nabla \times \left(\mathbf {A} \psi \right)+\mathbf {A} \times \left(\nabla \psi \right)\right]=iq\hbar \left[\psi \left(\nabla \times \mathbf {A} \right)-\mathbf {A} \times \left(\nabla \psi \right)+\mathbf {A} \times \left(\nabla \psi \right)\right]=iq\hbar \mathbf {B} \psi$ where $\mathbf {B} =\nabla \times \mathbf {A}$ is the magnetic field.

For the full Pauli equation, one then obtains 

Pauli equation(standard form)

${\hat {H}}|\psi \rangle =\left[{\frac {1}{2m}}\left[\left(\mathbf {p} -q\mathbf {A} \right)^{2}-q\hbar {\boldsymbol {\sigma }}\cdot \mathbf {B} \right]+q\phi \right]|\psi \rangle =i\hbar {\frac {\partial }{\partial t}}|\psi \rangle$ ### Weak magnetic fields

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

$(\mathbf {p} -q\mathbf {A} )^{2}=|\mathbf {p} |^{2}-q(\mathbf {r} \times \mathbf {p} )\cdot \mathbf {B} +{\frac {1}{4}}q^{2}\left(|\mathbf {B} |^{2}|\mathbf {r} |^{2}-|\mathbf {B} \cdot \mathbf {r} |^{2}\right)\approx \mathbf {p} ^{2}-q\mathbf {L} \cdot \mathbf {B} \,,$ where ${\textstyle \mathbf {L} }$ is the particle angular momentum and we neglected terms in the magnetic field squared ${\textstyle B^{2}}$ . Therefore we obtain

Pauli equation(weak magnetic fields)

$\left[{\frac {1}{2m}}\left[\left(|\mathbf {p} |^{2}-q(\mathbf {L} +2\mathbf {S} )\cdot \mathbf {B} \right)\right]+q\phi \right]|\psi \rangle =i\hbar {\frac {\partial }{\partial t}}|\psi \rangle$ where ${\textstyle \mathbf {S} =\hbar {\boldsymbol {\sigma }}/2}$ is the spin of the particle. The factor 2 in front of the spin is known as the Dirac g-factor. The term in ${\textstyle \mathbf {B} }$ , is of the form ${\textstyle -{\boldsymbol {\mu }}\cdot \mathbf {B} }$ which is the usual interaction between a magnetic moment ${\textstyle {\boldsymbol {\mu }}}$ and a magnetic field, like in the Zeeman effect.

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

$\left[{\frac {|\mathbf {p} |^{2}}{2m}}+\mu _{\rm {B}}g_{J}m_{j}|\mathbf {B} |-e\phi \right]|\psi \rangle =i\hbar {\frac {\partial }{\partial t}}|\psi \rangle$ where ${\textstyle \mu _{\rm {B}}={\frac {e\hbar }{2m}}}$ is the Bohr magneton and ${\textstyle m_{j}}$ is the magnetic quantum number related to ${\textstyle \mathbf {J} }$ . The term ${\textstyle g_{J}}$ is known as the Landé g-factor, and is given here by

$g_{J}={\frac {3}{2}}+{\frac {{\frac {3}{4}}-\ell (\ell +1)}{2j(j+1)}},$ [lower-alpha 1]

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

## From Dirac equation

The Pauli equation is the non-relativistic limit of Dirac equation, the relativistic quantum equation of motion for particles spin-½. 

### Derivation

Dirac equation can be written as:

$\mathrm {i} \,\hbar \,\partial _{t}\,\left({\begin{array}{c}\psi _{1}\\\psi _{2}\end{array}}\right)=c\,\left({\begin{array}{c}{\boldsymbol {\sigma }}\cdot {\boldsymbol {\Pi }}\,\psi _{2}\\{\boldsymbol {\sigma }}\cdot {\boldsymbol {\Pi }}\,\psi _{1}\end{array}}\right)+q\,\phi \,\left({\begin{array}{c}\psi _{1}\\\psi _{2}\end{array}}\right)+mc^{2}\,\left({\begin{array}{c}\psi _{1}\\-\psi _{2}\end{array}}\right)$ ,

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

Using the following ansatz:

$\left({\begin{array}{c}\psi _{1}\\\psi _{2}\end{array}}\right)=\mathrm {e} ^{-\displaystyle i{\frac {mc^{2}t}{\hbar }}}\left({\begin{array}{c}\psi \\\chi \end{array}}\right)$ ,

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

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

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

Thus

$\chi \approx {\frac {{\boldsymbol {\sigma }}\cdot {\boldsymbol {\Pi }}\,\psi }{2\,mc}}\,.$ Inserted in the upper component of Dirac equation, we find Pauli equation (general form):

$\mathrm {i} \,\hbar \,\partial _{t}\,\psi =\left[{\frac {({\boldsymbol {\sigma }}\cdot {\boldsymbol {\Pi }})^{2}}{2\,m}}+q\,\phi \right]\psi .$ ### From a Foldy-Wouthuysen transformation

One can also rigorously derive Pauli equation, starting from Dirac equation in an external field and performing a Foldy-Wouthuysen transformation. 

## Pauli coupling

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

$p_{\mu }\to p_{\mu }-qA_{\mu }+a\sigma _{\mu \nu }F^{\mu \nu }$ where $p_{\mu }$ is the four-momentum operator, $A_{\mu }$ if the electromagnetic four-potential, $a$ is the anomalous magnetic dipole moment, $F^{\mu \nu }=\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }$ is 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 $\gamma ^{\mu }$ .   In the context, of non-relativistic quantum mechanics, instead of working with Schrödinger equation, Pauli coupling is equivalent to use Pauli equation (or to postulate Zeeman energy) for an arbitrary g-factor.

## Footnotes

1. The formula used here is for a particle with spin ½, with a g-factor ${\textstyle g_{S}=2}$ and orbital g-factor ${\textstyle g_{L}=1}$ .

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