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L''''équation de Pauli''' est une équation non-relativiste de la [[mécanique quantique]] qui correspond à [[Equation de Schrödinger|celle de Schrödinger]] pour les particules de [[spin]] 1/2 dans un [[champ électromagnétique]].

En 1927, [[Wolfgang Pauli]] a postulé cette équation comme étant l'équation de l'[[électron]], puis, en 1928, elle a été démontrée par [[Paul Dirac]] comme approximation non-relativiste de [[équation de Dirac|son équation]]. En 1969, [[Jean-Marc Lévy-Leblond]] l'a redémontrée en linéarisant l'[[Equation de Schrödinger|équation de Schrödinger]]<ref>Walter Greiner, ''Mécanique quantique : Une introduction''
, Springer éditeur, 1999, <small>ISBN 3540643478 ; ISBN 978-3540643470</small>.</ref>.

== Formulation ==

En notant :
*<math>\Psi(t,\vec{r}) = \binom{\Psi_+}{\Psi_-}</math> la [[fonction d'état]] de la particule, où <math>\Psi_{\pm }</math> est l'amplitude de probabilité d'observer le spin <math>\pm 1/2</math>,
*<math>\ e</math> la charge de la particule, <math>\ m</math> sa masse,
*<math>\bold A = \left( \Phi , \vec A \right)</math> le [[Équations de Maxwell#Quadri-potentiel|quadri-potentiel]] du champ électromagnétique ambiant, <math>\breve B = \breve{rot} \vec A</math> le [[champ magnétique]],
*<math>\vec \sigma = \left( \sigma_1 , \sigma_2 , \sigma_3 \right)</math> le vecteur des [[matrices de Pauli]].

L'équation de Pauli est :

<center><math>
i\hbar{\partial\Psi(t,\vec{r})\over\partial t}= \left( {1\over 2m}\left( i \hbar \overrightarrow{\nabla} + \vec A \right)^2 + e \Phi - {e\hbar\over2mc} \vec \sigma . \breve B \right)\Psi(t,\vec{r})
</math></center>

Et l'incohérence héritée saute aux yeux. Il est considéré traditionnellement que le champ magnétique est '''encore''' un vecteur, et du coup les matrices de Pauli sont présentées sous la même convention, qui évidemment ne respecte pas les symétries du champ magnétique. Nous devons résoudre cette contradiction en réexprimant les matrices de Pauli avant d'aller plus loin.

<math>\breve \sigma = \begin{bmatrix}0 &\sigma_3 &-\sigma_2\\-\sigma_3 &0 &\sigma_1\\\sigma_2 &-\sigma_1 &0 \end{bmatrix}</math> le tenseur antisymétrique des matrices de Pauli ?

<center><math>
i\hbar{\partial\Psi(t,\vec{r})\over\partial t}= \left( {1\over 2m}\left( i \hbar \overrightarrow{\nabla} + \vec A \right)^2 + e \Phi - {e\hbar\over2mc} \breve \sigma . \breve B \right)\Psi(t,\vec{r})
</math></center>

<center>- - - - - - -</center>

For a particle of mass ''m'' and charge ''q'', in an electromagnetic field described by the three-component [[vector potential]]

:<math> \bold{A} = (A_x,A_y,A_z) \ \ </math>

and ([[scalar potential|scalar]]) [[electric potential]] ''ϕ'', the Pauli equation reads:

{{Equation box 1
|title='''Pauli equation''' ''(General)''
|indent =:
|equation = <math>\left[ \frac{1}{2m}(\boldsymbol{\sigma}\cdot(\bold{p} - q \bold{A}))^2 + q \phi \right] |\psi\rangle = i \hbar \frac{\partial}{\partial t} |\psi\rangle </math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4}}

where:

:<math> \bold{\sigma} = (\sigma_x,\sigma_y,\sigma_z) \ \ </math>

is a three-component vector of the two-by-two [[Pauli matrices]], i.e. that each component of the vector is a Pauli matrix,

:<math> \bold{p} = -i\hbar\nabla = -i\hbar\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right)\ \ </math>

is the three-component vector of the [[momentum operator]], in turn ∇ denotes the [[gradient operator]], and

:<math> |\psi\rangle = \begin{pmatrix}
\psi_0 \\
\psi_1
\end{pmatrix} </math>

is the two component [[spinor]] [[wavefunction]], a [[column vector]] written in [[Dirac notation]]

More explicitly in full notation, the Pauli equation is:

: <math>\left[ \frac{1}{2m} \left( \sum_{n=1}^3 \left(\sigma_n \left( - i \hbar \frac{\partial}{\partial x_n} - q A_n\right)\right) \right) ^2 + q \phi \right]
\begin{pmatrix} \psi_0 \\ \psi_1 \end{pmatrix}
= i \hbar \begin{pmatrix} \frac{\displaystyle \partial \psi_0 }{\displaystyle \partial t} \\[6pt] \frac{\displaystyle \partial \psi_1 }{\displaystyle \partial t}
\end{pmatrix}. </math>

The [[hamiltonian (quantum mechanics)|Hamiltonian]] (the expression between square brackets) is a two-by-two matrix operator, because of the Pauli <math> \sigma </math> matrices.

== Relationship to the Schrödinger equation and the Dirac equation ==

The Pauli equation is non-relativistic, but it does predict spin. As such, it can be thought of as occupying the middle ground between:
* The familiar Schrödinger equation (on a complex scalar [[wavefunction]]), which is non-relativistic and does not predict spin.
* The Dirac equation (on a [[dirac spinor|complex four-component spinor]]), which is fully [[special relativity|relativistic]] (with respect to [[special relativity]]) and predicts spin.

Note that because of the properties of the Pauli matrices, if the magnetic vector potential '''A''' is equal to zero, then the equation reduces to the familiar Schrödinger equation for a particle in a purely electric potential ϕ, except that it operates on a two component spinor. Therefore, we can see that the spin of the particle only affects its motion in the presence of a magnetic field.

== Special Cases ==

Both spinor components satisfy the Schrödinger equation. This means that the system is degenerated as to the additional degree of freedom.

For an external magnetic field '''B''' the Pauli equation reads:

{{Equation box 1
|title='''Pauli equation''' ''(B-field)''
|indent =:
|equation = <math>
\underbrace{i \hbar \frac{\partial}{\partial t} |\varphi_\pm\rangle = \left( \frac{( \bold{p} -q \bold A)^2}{2 m} + q \phi \right) \hat 1 \bold |\varphi_\pm\rangle }_\mathrm{Schr\ddot{o}dinger~equation} - \underbrace{\frac{q \hbar}{2m}\boldsymbol{\sigma} \cdot \bold B \bold |\varphi_\pm\rangle }_\text{Stern Gerlach term}</math>
|cellpadding
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

where

:<math> |\varphi_\pm\rangle = \begin{pmatrix}
|\varphi_+\rangle \\
|\varphi_-\rangle
\end{pmatrix}</math>,

are the Pauli [[spinor]] components, '''B''' is the external [[magnetic field]], and

:<math> \hat 1 = \begin{pmatrix}
1 & 0 \\
0 & 1 \\
\end{pmatrix} </math>

is the 2 × 2 [[identity matrix]], which acts as an [[identity operator]].

The [[Stern–Gerlach experiment|Stern–Gerlach term]] can obtain the spin orientation of atoms with one [[valence electron]], e.g. silver atoms which flow through an inhomogeneous magnetic field.

Analogously, the term is responsible for the splitting of spectral lines (corresponding to energy levels) in a magnetic field as can be viewed in the [[anomalous Zeeman effect]].

== Derivation of the Pauli equation by Schrödinger ==

The Dirac equation for weak electromagnetic interactions is the starting point:



:<math>
i \hbar \frac{\partial}{\partial t} \left( \begin{array}{c} \vec \varphi_1\\\vec \varphi_2\end{array} \right) = c \left( \begin{array}{c} \vec{\hat \sigma} \vec \pi \vec \varphi_2\\\vec{\hat \sigma} \vec \pi \vec \varphi_1\end{array} \right)+q \phi \left( \begin{array}{c} \vec \varphi_1\\\vec \varphi_2\end{array} \right) + mc^2 \left( \begin{array}{c} \vec \varphi_1 \\-\vec \varphi_2\end{array} \right)
</math>

where

:<math>\boldsymbol{\Pi} = \bold p - q \bold A </math>

is the [[momentum#Particle in field|kinetic momentum]], and the following approximations are used:

* Simplification of the equation through following ansatz
:: <math>\left( \begin{array}{c} \vec \varphi_1 \\ \vec \varphi_2 \end{array} \right) = e^{-i \frac{mc^2t}{\hbar}} \left( \begin{array}{c} \vec{\tilde{\varphi_1}} \\ \vec{\tilde{\varphi_2}} \end{array} \right)
</math>
* Eliminating the rest energy through an Ansatz with slow time dependence
:: <math>\partial_t \vec{\tilde{\varphi_i}} \ll \frac{mc^2}{\hbar} \vec{\tilde{\varphi_i}}</math>
* weak coupling of the electric potential
:: <math>q \phi \ll mc^2 </math>

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

== Notes and references ==
{{reflist}}

Equation de Pauli - Historique des versions

Jacques Lavau : Page créée avec « Ebauche en cours d'élaboration. L''''équation de Pauli''' est une équation non-relativiste de la mécanique quantique qui correspond à Equation de Schrödinger... »