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 ${\displaystyle \psi }$ :

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

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]

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

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

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

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

${\displaystyle \left[{\frac {|\mathbf {p} |^{2}}{2m}}+\mu _{\rm {B}}g_{J}m_{j}|\mathbf {B} |-e\phi \right]|\psi \rangle =i\hbar {\frac {\partial }{\partial t}}|\psi \rangle }$ 

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

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

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

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-1/2 particles.^[4]

The 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 ${\displaystyle mc^{2}}$ , leading to the Lévy-Leblond equation.^[5] 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): $${\displaystyle i\hbar \,\partial _{t}\,\psi =\left[{\frac {({\boldsymbol {\sigma }}\cdot {\boldsymbol {\Pi }})^{2}}{2\,m}}+q\,\phi \right]\psi .}$$ 

The rigorous derivation of the Pauli equation follows from Dirac equation in an external field and performing a Foldy–Wouthuysen transformation^[4] considering terms up to order ${\displaystyle {\mathcal {O}}(1/mc)}$ . Similarly, higher order corrections to the Pauli equation can be determined giving rise to spin-orbit and Darwin interaction terms, when expanding up to order ${\displaystyle {\mathcal {O}}(1/(mc)^{2})}$  instead.^[6]

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 ${\displaystyle \gamma ^{\mu }}$ .^[7][8] 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

• Schwabl, Franz (2004). Quantenmechanik I. Springer. ISBN 978-3540431060.
• Schwabl, Franz (2005). Quantenmechanik für Fortgeschrittene. Springer. ISBN 978-3540259046.
• Claude Cohen-Tannoudji; Bernard Diu; Frank Laloe (2006). Quantum Mechanics 2. Wiley, J. ISBN 978-0471569527.