Pauli equation

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.[1]

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:

Pauli equation (general)

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

Here ${\displaystyle {\boldsymbol {\sigma }}=(\sigma _{x},\sigma _{y},\sigma _{z})}$ are the Pauli operators collected into a vector for convenience, and ${\displaystyle \mathbf {p} =-i\hbar \nabla }$ is the momentum operator. 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):

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

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

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

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

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

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

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

Pauli equation (standard form)

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

Weak magnetic fields

Let us expand ${\textstyle (\mathbf {p} -q\mathbf {A} )^{2}}$ using the Landau gauge ${\textstyle \mathbf {A} ={\frac {1}{2}}\mathbf {B} \times \mathbf {r} }$, where ${\textstyle \mathbf {r} }$ is the position vector. We obtain

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

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)

${\displaystyle \left[{\frac {1}{2m}}\left[\left(|\mathbf {p} |^{2}-q(\mathbf {L} +2\mathbf {S} )\cdot \mathbf {B} \right)\right]+q\phi \right]|\psi \rangle =i\hbar {\frac {\partial }{\partial t}}|\psi \rangle }$

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 electron 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 by

${\displaystyle g_{J}={\frac {3}{2}}+{\frac {s(s+1)-\ell (\ell +1)}{2j(j+1)}}.}$

From Dirac equation

The Pauli equation is the non-relativistic limit of Dirac equation, the relativistic quantum equation of motion for particles spin-½.[3]

Derivation

Dirac equation can be written as:

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

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

Using the following ansatz:

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

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

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

In the non-relativistic limit, ${\displaystyle \partial _{t}\chi }$ 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):

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

See also

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

References

1. Pauli, Wolfgang (1927). "Zur Quantenmechanik des magnetischen Elektrons". Zeitschrift für Physik (in German). 43 (9–10): 601–623. doi:10.1007/BF01397326. ISSN 0044-3328.
2. Bransden, BH; Joachain, CJ (1983). Physics of Atoms and Molecules (1st ed.). Prentice Hall. p. 638–638. ISBN 0-582-44401-2.
3. Greiner, Walter (2012-12-06). Relativistic Quantum Mechanics: Wave Equations. Springer. ISBN 978-3-642-88082-7.

Books

• 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.