View source for Physics:Pauli equation

{{Short description|Quantum mechanical equation of motion of charged particles in magnetic field}}
{{Quantum mechanics|cTopic=Equations}}

In [[Physics:Quantum mechanics|quantum mechanics]], the '''Pauli equation''' or '''Schrödinger–Pauli equation''' is the formulation of the [[Physics:Schrödinger equation|Schrödinger equation]] for [[Physics:Spin-½|spin-½]] particles, which takes into account the interaction of the particle's [[Physics:Spin|spin]] with an external [[Physics:Electromagnetic field|electromagnetic field]]. It is the non-[[Physics:Special relativity|relativistic]] limit of the [[Physics:Dirac equation|Dirac equation]] and can be used where particles are moving at speeds much less than the [[Physics:Speed of light|speed of light]], so that relativistic effects can be neglected. It was formulated by [[Biography:Wolfgang Pauli|Wolfgang Pauli]] in 1927.<ref>{{Cite journal|last=Pauli|first=Wolfgang|year=1927|title=Zur Quantenmechanik des magnetischen Elektrons|url=http://link.springer.com/10.1007/BF01397326|journal=Zeitschrift für Physik|language=de|volume=43|issue=9–10|pages=601–623|doi=10.1007/BF01397326|bibcode=1927ZPhy...43..601P|s2cid=128228729|issn=0044-3328}}</ref>

== Equation ==

For a particle of mass <math>m</math> and electric charge <math>q</math>, in an [[Physics:Electromagnetic field|electromagnetic field]] described by the magnetic vector potential <math>\mathbf{A}</math> and the [[Physics:Electric potential|electric scalar potential]] <math>\phi</math>, the Pauli equation reads:

{{Equation box 1
|title='''Pauli equation''' ''(general)''
|indent =:
|equation = <math>\left[ \frac{1}{2m}(\boldsymbol{\sigma}\cdot(\mathbf{\hat{p}} - q \mathbf{A}))^2 + q \phi \right] |\psi\rangle = i \hbar \frac{\partial}{\partial t} |\psi\rangle  </math>
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Here <math>\boldsymbol{\sigma} = (\sigma_x, \sigma_y, \sigma_z)</math> are the [[Pauli matrices|Pauli operators]] collected into a vector for convenience, and  <math>\mathbf{\hat{p}} = -i\hbar \nabla</math> is the [[Physics:Momentum operator|momentum operator]] in position representation. The state of the system, <math>|\psi\rangle</math> (written in Dirac notation), can be considered as a two-component [[Physics:Spinor|spinor]] wavefunction, or a column vector (after choice of basis):
:<math> |\psi\rangle = \psi_+ |\mathord\uparrow\rangle + \psi_-|\mathord\downarrow\rangle \,\stackrel{\cdot}{=}\, \begin{bmatrix} 
\psi_+ \\
\psi_-
\end{bmatrix}</math>.

The [[Physics:Hamiltonian (quantum mechanics)|Hamiltonian operator]] is a 2 × 2 matrix because of the [[Pauli matrices|Pauli operators]].

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

Substitution into the [[Physics:Schrödinger equation|Schrödinger equation]] gives the Pauli equation. This Hamiltonian is similar to the classical Hamiltonian for a charged particle interacting with an electromagnetic field. See [[Physics:Lorentz force#Lorentz force and analytical mechanics|Lorentz force]] for details of this classical case. The [[Physics:Kinetic energy|kinetic energy]] term for a free particle in the absence of an electromagnetic field is just  <math>\frac{\mathbf{p}^2}{2m}</math> where <math>\mathbf{p}</math>  is the [[Physics:Momentum#Particle in a field|''kinetic'' momentum]], while in the presence of an electromagnetic field it involves the [[Physics:Minimal coupling|minimal coupling]] <math>\mathbf{\Pi} = \mathbf{p} - q\mathbf{A}</math>, where now <math>\mathbf{\Pi}</math> is the kinetic momentum and <math>\mathbf{p}</math> is the [[Physics:Canonical momentum|canonical momentum]].

The Pauli operators can be removed from the kinetic energy term using the [[Pauli matrices#Relation to dot and cross product|Pauli vector identity]]:

:<math>(\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)</math>

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

:<math>\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</math>

where <math>\mathbf{B} = \nabla \times \mathbf{A}</math> is the magnetic field.

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

{{Equation box 1
|title='''Pauli equation''' ''(standard form)''
|indent =:
|equation = <math>\hat{H} |\psi\rangle = \left[\frac{1}{2m}\left[\left(\mathbf{\hat{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</math>
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|background colour = #ECFCF4}}for which only a few analytic results are known, e.g., in the context of [[Physics:Landau quantization|Landau quantization]] with homogenous magnetic fields or for an idealized, Coulomb-like, inhomogeneous magnetic field.<ref>{{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 }}</ref>

=== Weak magnetic fields ===
For the case of where the magnetic field is constant and homogenous, one may expand  <math display="inline">(\mathbf{\hat{p}}-q\mathbf{A})^2</math> using the [[Physics:Landau quantization#In the symmetric gauge|symmetric gauge]] <math display="inline">\mathbf{\hat{A}}=\frac{1}{2}\mathbf{B}\times\mathbf{\hat{r}}</math>, where <math display="inline">\mathbf{r}</math> is the [[Physics:Position operator|position operator]] and A is now an operator. We obtain

:<math>(\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\,, </math>

where <math display="inline">\mathbf{\hat{L}}</math> is the particle [[Physics:Angular momentum|angular momentum]] operator and we neglected terms in the magnetic field squared <math display="inline">B^2</math>. Therefore, we obtain
{{Equation box 1
|title='''Pauli equation''' ''(weak magnetic fields)''
|indent =:
|equation = <math> \left[\frac{1}{2m}\left[|\mathbf{\hat{p}}|^2 - q (\mathbf{\hat{L}}+2\mathbf{\hat{S}})\cdot\mathbf{B}\right] + q \phi\right]|\psi\rangle = i \hbar \frac{\partial}{\partial t} |\psi\rangle</math>
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where  <math display="inline">\mathbf{S}=\hbar\boldsymbol{\sigma}/2</math> is the [[Physics:Spin|spin]] of the particle. The factor 2 in front of the spin is known as the Dirac [[Physics:G-factor|''g''-factor]]. The term in <math display="inline">\mathbf{B}</math>, is of the form <math display="inline">-\boldsymbol{\mu}\cdot\mathbf{B}</math> which is the usual interaction between a magnetic moment <math display="inline">\boldsymbol{\mu}</math> and a magnetic field, like in the [[Physics:Zeeman effect|Zeeman effect]].

For an electron of charge <math display="inline">-e</math> in an isotropic constant magnetic field, one can further reduce the equation using the total angular momentum <math display="inline">\mathbf{J}=\mathbf{L}+\mathbf{S}</math> and Wigner-Eckart theorem. Thus we find
:<math> \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</math>
where <math display="inline">\mu_{\rm B}=\frac{e\hbar }{2m}</math> is the [[Physics:Bohr magneton|Bohr magneton]] and <math display="inline">m_j</math> is the [[Physics:Magnetic quantum number|magnetic quantum number]] related to <math display="inline">\mathbf{J}</math>. The term <math display="inline">g_J</math> is known as the [[Physics:Landé g-factor|Landé g-factor]], and is given here by
:<math>g_J = \frac{3}{2}+\frac{\frac{3}{4}-\ell(\ell+1)}{2j(j+1)},</math>{{efn|The formula used here is for a particle with spin ½, with a ''g''-factor <math display="inline">g_S=2</math> and orbital ''g''-factor <math display="inline">g_L=1</math>. More generally it is given by:<math>g_J = \frac{3}{2}+\frac{m_s(m_s+1)-\ell(\ell+1)}{2j(j+1)}.</math> where <math>m_s</math> is the [[Physics:Spin quantum number|spin quantum number]] related to <math>\hat{S}</math>.}}
where <math>\ell</math> is the orbital quantum number related to <math>L^2</math> and <math>j</math> is the total orbital quantum number related to <math>J^2</math>.

== From Dirac equation ==

The Pauli equation can be inferred from the non-relativistic limit of the [[Physics:Dirac equation|Dirac equation]], which is the relativistic quantum equation of motion for  spin-½ particles.<ref name=":0">{{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}}</ref>

=== Derivation ===
Dirac equation can be written as:
<math display="block">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} ,
</math>

where <math display="inline">\partial_t=\frac{\partial}{\partial t}</math> and <math>\psi_1,\psi_2</math> are two-component [[Physics:Spinor|spinor]], forming a [[Physics:Bispinor|bispinor]].

Using the following ansatz:
<math display="block">\begin{pmatrix} \psi_1 \\ \psi_2 \end{pmatrix} = e^{- i \tfrac{mc^2t}{\hbar}} \begin{pmatrix} \psi \\ \chi \end{pmatrix} ,</math>
with two new spinors <math>\psi,\chi</math>, the equation becomes
<math display="block">
 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} .
</math>

In the non-relativistic limit, <math>\partial_t \chi</math> and the kinetic and electrostatic energies are small with respect to the rest energy <math>mc^2</math>.

Thus
<math display="block">\chi \approx \frac{\boldsymbol \sigma \cdot \boldsymbol{\Pi}\,\psi}{2\,mc}\,.</math>

Inserted in the upper component of Dirac equation, we find Pauli equation (general form):
<math display="block">i \hbar\, \partial_t \, \psi=  \left[\frac{(\boldsymbol \sigma \cdot \boldsymbol \Pi)^2}{2\,m}
+q\, \phi\right] \psi.</math>

=== From a Foldy–Wouthuysen transformation ===
The rigorous derivation of the Pauli equation follows from Dirac equation in an external field and performing a [[Physics:Foldy–Wouthuysen transformation|Foldy–Wouthuysen transformation]]<ref name=":0" /> considering terms up to order  <math>\mathcal{O}(1/mc)</math>.  Similarly, higher order corrections to the Pauli equation can be determined giving rise to [[Physics:Spin–orbit interaction|spin-orbit]] and [[Physics:Fine structure|Darwin]] interaction terms, when expanding up to order <math>\mathcal{O}(1/(mc)^2)</math> instead.<ref>{{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}}</ref>

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

:<math>\gamma^{\mu}p_\mu\to \gamma^{\mu}p_\mu-q\gamma^{\mu}A_\mu +a\sigma_{\mu\nu}F^{\mu\nu}</math>

where <math>p_\mu</math> is the [[Physics:Four-momentum|four-momentum]] operator, <math>A_\mu</math> is the [[Physics:Electromagnetic four-potential|electromagnetic four-potential]], <math>a</math> is proportional to the [[Physics:Anomalous magnetic dipole moment|anomalous magnetic dipole moment]], <math>F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}</math> is the [[Physics:Electromagnetic tensor|electromagnetic tensor]], and <math display="inline">\sigma_{\mu\nu}=\frac{i}{2}[\gamma_{\mu},\gamma_{\nu}]</math> are the Lorentzian spin matrices and the commutator of the [[Gamma matrices|gamma matrices]] <math>\gamma^{\mu}</math>.<ref>{{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}}</ref><ref>{{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}}</ref> 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 [[Physics:Zeeman energy|Zeeman energy]]) for an arbitrary ''g''-factor.

==See also==
* [[Physics:Semiclassical physics|Semiclassical physics]]
* [[Physics:Atomic, molecular, and optical physics|Atomic, molecular, and optical physics]]
* [[Group contraction]]
*[[Physics:Gordon decomposition|Gordon decomposition]]

==Footnotes==
{{notelist}}

== References ==
{{reflist}}

===Books===
* {{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}}

{{Quantum mechanics topics}}

{{DEFAULTSORT:Pauli Equation}}
[[Category:Eponymous equations of physics]]
[[Category:Quantum mechanics]]

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