Pauli equation

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

{{Equation box 1

|title=Pauli equation (general)

|indent =:

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

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

:$|\backslash psi\backslash rangle\; =\; \backslash psi\_+\; |\backslash mathord\backslash uparrow\backslash rangle\; +\; \backslash psi\_-|\backslash mathord\backslash downarrow\backslash rangle\; \backslash ,\backslash stackrel\{\backslash cdot\}\{=\}\backslash ,\; \backslash begin\{bmatrix\}$

\psi_+ \\

\psi_-

\end{bmatrix}.

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

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

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 $\backslash frac\{\backslash mathbf\{p\}^2\}\{2m\}$ where $\backslash mathbf\{p\}$ is the kinetic momentum, while in the presence of an electromagnetic field it involves the minimal coupling $\backslash mathbf\{\backslash Pi\}\; =\; \backslash mathbf\{p\}\; -\; q\backslash mathbf\{A\}$, where now $\backslash mathbf\{\backslash Pi\}$ is the kinetic momentum and $\backslash mathbf\{p\}$ is the canonical momentum.

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

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

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

:$$

\begin{align}

\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]\\

&= i q \hbar \left[\nabla \times \left(\mathbf{A}\psi\right) + \mathbf{A} \times \left(\nabla\psi\right)\right]\\

&= i q \hbar \left[\psi\left(\nabla \times \mathbf{A}\right) - \mathbf{A} \times \left(\nabla\psi\right) + \mathbf{A} \times \left(\nabla\psi\right)\right] = i q \hbar \mathbf{B} \psi

\end{align}

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

For the full Pauli equation, one then obtains{{Cite book|title=Physics of Atoms and Molecules|author=Bransden, BH|author2=Joachain, CJ|year=1983|publisher=Prentice Hall|edition=1st|pages=638|isbn=0-582-44401-2}}

{{Equation box 1

|title=Pauli equation (standard form)

|indent =:

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

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|background colour = #ECFCF4}}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.{{Cite journal |last1=Sidler |first1=Dominik |last2=Rokaj |first2=Vasil |last3=Ruggenthaler |first3=Michael |last4=Rubio |first4=Angel |date=2022-10-26 |title=Class of distorted Landau levels and Hall phases in a two-dimensional electron gas subject to an inhomogeneous magnetic field |url=https://link.aps.org/doi/10.1103/PhysRevResearch.4.043059 |journal=Physical Review Research |language=en |volume=4 |issue=4 |page=043059 |doi=10.1103/PhysRevResearch.4.043059 |bibcode=2022PhRvR...4d3059S |s2cid=253175195 |issn=2643-1564|hdl=10810/58724 |hdl-access=free }}

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

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

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

{{Equation box 1

|title=Pauli equation (weak magnetic fields)

|indent =:

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

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where $\backslash mathbf\{S\}=\backslash hbar\backslash boldsymbol\{\backslash sigma\}/2$ is the spin of the particle. The factor 2 in front of the spin is known as the Dirac g-factor. The term in $\backslash mathbf\{B\}$, is of the form $-\backslash boldsymbol\{\backslash mu\}\backslash cdot\backslash mathbf\{B\}$ which is the usual interaction between a magnetic moment $\backslash boldsymbol\{\backslash mu\}$ 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 $\backslash mathbf\{J\}=\backslash mathbf\{L\}+\backslash mathbf\{S\}$ and Wigner-Eckart theorem. Thus we find

:$\backslash left[\backslash frac\{|\backslash mathbf\{p\}|^2\}\{2m\}\; +\; \backslash mu\_\{\backslash rm\; B\}\; g\_J\; m\_j|\backslash mathbf\{B\}|\; -\; e\; \backslash phi\backslash right]|\backslash psi\backslash rangle\; =\; i\; \backslash hbar\; \backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; |\backslash psi\backslash rangle$

where $\backslash mu\_\{\backslash rm\; B\}=\backslash frac\{e\backslash hbar\; \}\{2m\}$ is the Bohr magneton and $m\_j$ is the magnetic quantum number related to $\backslash mathbf\{J\}$. The term $g\_J$ is known as the Landé g-factor, and is given here by

:$g\_J\; =\; \backslash frac\{3\}\{2\}+\backslash frac\{\backslash frac\{3\}\{4\}-\backslash ell(\backslash ell+1)\}\{2j(j+1)\},${{efn|The formula used here is for a particle with spin-1/2, with a g-factor $g\_S=2$ and orbital g-factor $g\_L=1$. More generally it is given by:$g\_J\; =\; \backslash frac\{3\}\{2\}+\backslash frac\{m\_s(m\_s+1)-\backslash ell(\backslash ell+1)\}\{2j(j+1)\}.$ where $m\_s$ is the spin quantum number related to $\backslash hat\{S\}$.}}

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

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.{{Cite book|last=Greiner|first=Walter|url=https://books.google.com/books?id=a6_rCAAAQBAJ|title=Relativistic Quantum Mechanics: Wave Equations|date=2012-12-06|publisher=Springer|isbn=978-3-642-88082-7|language=en}}

The Dirac equation can be written as:

$$i\; \backslash hbar\backslash ,\; \backslash partial\_t\; \backslash begin\{pmatrix\}\; \backslash psi\_1\; \backslash \backslash \; \backslash psi\_2\backslash end\{pmatrix\}\; =\; c\; \backslash ,\; \backslash begin\{pmatrix\}\; \backslash boldsymbol\{\; \backslash sigma\}\backslash cdot\; \backslash boldsymbol\; \backslash Pi\; \backslash ,\backslash psi\_2\; \backslash \backslash \; \backslash boldsymbol\{\backslash sigma\}\backslash cdot\; \backslash boldsymbol\; \backslash Pi\; \backslash ,\backslash psi\_1\backslash end\{pmatrix\}\; +\; q\backslash ,\; \backslash phi\; \backslash ,\; \backslash begin\{pmatrix\}\; \backslash psi\_1\; \backslash \backslash \; \backslash psi\_2\backslash end\{pmatrix\}\; +\; mc^2\backslash ,\; \backslash begin\{pmatrix\}\; \backslash psi\_1\; \backslash \backslash \; -\backslash psi\_2\backslash end\{pmatrix\}\; ,$$

where $\backslash partial\_t=\backslash frac\{\backslash partial\}\{\backslash partial\; t\}$ and $\backslash psi\_1,\backslash psi\_2$ are two-component spinor, forming a bispinor.

Using the following ansatz:

$$\backslash begin\{pmatrix\}\; \backslash psi\_1\; \backslash \backslash \; \backslash psi\_2\; \backslash end\{pmatrix\}\; =\; e^\{-\; i\; \backslash tfrac\{mc^2t\}\{\backslash hbar\}\}\; \backslash begin\{pmatrix\}\; \backslash psi\; \backslash \backslash \; \backslash chi\; \backslash end\{pmatrix\}\; ,$$

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

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, $\backslash partial\_t\; \backslash chi$ and the kinetic and electrostatic energies are small with respect to the rest energy $mc^2$, leading to the Lévy-Leblond equation.{{Cite book |last=Greiner |first=Walter |url=https://books.google.com/books?id=7qCMUfwoQcAC |title=Quantum Mechanics: An Introduction |date=2000-10-04 |publisher=Springer Science & Business Media |isbn=978-3-540-67458-0 |language=en}} Thus$$\backslash chi\; \backslash approx\; \backslash frac\{\backslash boldsymbol\; \backslash sigma\; \backslash cdot\; \backslash boldsymbol\{\backslash Pi\}\backslash ,\backslash psi\}\{2\backslash ,mc\}\backslash ,.$$

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

$$i\; \backslash hbar\backslash ,\; \backslash partial\_t\; \backslash ,\; \backslash psi=\; \backslash left[\backslash frac\{(\backslash boldsymbol\; \backslash sigma\; \backslash cdot\; \backslash boldsymbol\; \backslash Pi)^2\}\{2\backslash ,m\}$$

+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 considering terms up to order $\backslash mathcal\{O\}(1/mc)$. 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 $\backslash mathcal\{O\}(1/(mc)^2)$ instead.{{Cite journal |last1=Fröhlich |first1=Jürg |last2=Studer |first2=Urban M. |date=1993-07-01 |title=Gauge invariance and current algebra in nonrelativistic many-body theory |url=https://link.aps.org/doi/10.1103/RevModPhys.65.733 |journal=Reviews of Modern Physics |language=en |volume=65 |issue=3 |pages=733–802 |doi=10.1103/RevModPhys.65.733 |bibcode=1993RvMP...65..733F |issn=0034-6861|url-access=subscription }}

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

:$\backslash gamma^\{\backslash mu\}p\_\backslash mu\backslash to\; \backslash gamma^\{\backslash mu\}p\_\backslash mu-q\backslash gamma^\{\backslash mu\}A\_\backslash mu\; +a\backslash sigma\_\{\backslash mu\backslash nu\}F^\{\backslash mu\backslash nu\}$

where $p\_\backslash mu$ is the four-momentum operator, $A\_\backslash mu$ is the electromagnetic four-potential, $a$ is proportional to the anomalous magnetic dipole moment, $F^\{\backslash mu\backslash nu\}=\backslash partial^\{\backslash mu\}A^\{\backslash nu\}-\backslash partial^\{\backslash nu\}A^\{\backslash mu\}$ is the electromagnetic tensor, and $\backslash sigma\_\{\backslash mu\backslash nu\}=\backslash frac\{i\}\{2\}[\backslash gamma\_\{\backslash mu\},\backslash gamma\_\{\backslash nu\}]$ are the Lorentzian spin matrices and the commutator of the gamma matrices $\backslash gamma^\{\backslash mu\}$.{{Cite book|last=Das|first=Ashok|url=https://books.google.com/books?id=HFFkDQAAQBAJ|title=Lectures on Quantum Field Theory|date=2008|publisher=World Scientific|isbn=978-981-283-287-0|language=en}}{{Cite journal|last1=Barut|first1=A. O.|last2=McEwan|first2=J.|date=January 1986|title=The four states of the Massless neutrino with pauli coupling by Spin-Gauge invariance|url=http://link.springer.com/10.1007/BF00417466|journal=Letters in Mathematical Physics|language=en|volume=11|issue=1|pages=67–72|doi=10.1007/BF00417466|bibcode=1986LMaPh..11...67B|s2cid=120901078|issn=0377-9017|url-access=subscription}} 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

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{{reflist}}

• {{cite book | author=Schwabl, Franz| title=Quantenmechanik I | publisher=Springer |year=2004 |isbn=978-3540431060}}
• {{cite book | author=Schwabl, Franz| title=Quantenmechanik für Fortgeschrittene | publisher=Springer |year=2005 |isbn=978-3540259046}}
• {{cite book |author1=Claude Cohen-Tannoudji |author2=Bernard Diu |author3=Frank Laloe | title= Quantum Mechanics 2| publisher=Wiley, J |year=2006 |isbn=978-0471569527}}

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Category:Eponymous equations of physics

Category:Quantum mechanics