Equation de Pauli

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L'équation de Pauli est une équation non-relativiste de la mécanique quantique qui correspond à 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 son équation. En 1969, Jean-Marc Lévy-Leblond l'a redémontrée en linéarisant l'équation de Schrödinger<ref>Walter Greiner, Mécanique quantique : Une introduction , Springer éditeur, 1999, ISBN 3540643478 ; ISBN 978-3540643470.</ref>.


Formulation

En notant :

L'équation de Pauli est :

LaTeX: 
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})
</p>

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.


LaTeX: \breve \sigma = \begin{bmatrix}0 &\sigma_3 &-\sigma_2\\-\sigma_3 &0 &\sigma_1\\\sigma_2 &-\sigma_1 &0 \end{bmatrix} le tenseur antisymétrique des matrices de Pauli ?

LaTeX: 
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})
</p>
- - - - - - -

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

LaTeX:  \bold{A} = (A_x,A_y,A_z) \ \

and (scalar) electric potential ϕ, the Pauli equation reads:

Modèle:Equation box 1

where:

LaTeX:  \bold{\sigma} = (\sigma_x,\sigma_y,\sigma_z) \ \

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

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

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

LaTeX:  |\psi\rangle = \begin{pmatrix} 
\psi_0 \\
\psi_1
\end{pmatrix}

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

More explicitly in full notation, the Pauli equation is:

LaTeX: \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}.

The Hamiltonian (the expression between square brackets) is a two-by-two matrix operator, because of the Pauli LaTeX:  \sigma 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:

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:

Modèle:Equation box 1

where

LaTeX:  |\varphi_\pm\rangle = \begin{pmatrix} 
|\varphi_+\rangle \\
|\varphi_-\rangle 
\end{pmatrix},

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

LaTeX:  \hat 1 = \begin{pmatrix}
1 & 0 \\
0 & 1 \\
\end{pmatrix}

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

The 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:


LaTeX: 
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)

where

LaTeX: \boldsymbol{\Pi} = \bold p - q \bold A

is the kinetic momentum, and the following approximations are used:

  • Simplification of the equation through following ansatz
LaTeX: \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)

  • Eliminating the rest energy through an Ansatz with slow time dependence
LaTeX: \partial_t \vec{\tilde{\varphi_i}} \ll \frac{mc^2}{\hbar} \vec{\tilde{\varphi_i}}
  • weak coupling of the electric potential
LaTeX: q \phi \ll mc^2

References

Notes and references

Modèle:Reflist