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Équation de Pauli

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Cet article est une ébauche concernant la physique quantique.

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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 [1] .

Formulation

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En notant :

  • Ψ ( t , r → ) = ( Ψ + Ψ − ) {\displaystyle \Psi (t,{\vec {r}})={\binom {\Psi _{+}}{\Psi _{-}}}} {\displaystyle \Psi (t,{\vec {r}})={\binom {\Psi _{+}}{\Psi _{-}}}} la fonction d'état de la particule, où Ψ ± {\displaystyle \Psi _{\pm }} {\displaystyle \Psi _{\pm }} est l'amplitude de probabilité d'observer le spin ± 1 / 2 {\displaystyle \pm 1/2} {\displaystyle \pm 1/2},
  • q {\displaystyle \ q} {\displaystyle \ q} la charge de la particule, m {\displaystyle \ m} {\displaystyle \ m} sa masse,
  • A = ( U ( r → , t ) , A → ( r → , t ) ) {\displaystyle \mathbb {A} =\left(U({\vec {r}},t),{\vec {A}}({\vec {r}},t)\right)} {\displaystyle \mathbb {A} =\left(U({\vec {r}},t),{\vec {A}}({\vec {r}},t)\right)} le quadripotentiel du champ électromagnétique ambiant, B → = ∇ ×ばつ A → {\displaystyle {\vec {B}}=\nabla \times {\vec {A}}} {\displaystyle {\vec {B}}=\nabla \times {\vec {A}}} le champ magnétique,
  • σ → = ( σ 1 , σ 2 , σ 3 ) {\displaystyle {\vec {\sigma }}=\left(\sigma _{1},\sigma _{2},\sigma _{3}\right)} {\displaystyle {\vec {\sigma }}=\left(\sigma _{1},\sigma _{2},\sigma _{3}\right)} le vecteur des matrices de Pauli.

L'équation de Pauli est :

i ħ ∂ Ψ ( r → , t ) ∂ t = ( 1 2 m ( P → + q A → ( r → , t ) ) 2 + q U ( r → , t ) − q ħ 2 m σ → . B → ( r → , t ) ) Ψ ( r → , t ) {\displaystyle i\hbar {\partial \Psi ({\vec {r}},t) \over \partial t}=\left({1 \over 2m}\left({\vec {P}}+q{\vec {A}}({\vec {r}},t)\right)^{2}+qU({\vec {r}},t)-{q\hbar \over 2m}{\vec {\sigma }}.{\vec {B}}({\vec {r}},t)\right)\Psi ({\vec {r}},t)} {\displaystyle i\hbar {\partial \Psi ({\vec {r}},t) \over \partial t}=\left({1 \over 2m}\left({\vec {P}}+q{\vec {A}}({\vec {r}},t)\right)^{2}+qU({\vec {r}},t)-{q\hbar \over 2m}{\vec {\sigma }}.{\vec {B}}({\vec {r}},t)\right)\Psi ({\vec {r}},t)}

De l'expression précédente se déduit l'Hamiltonien de Pauli:

H = 1 2 m ( σ → . [ P → − q A → ( r → , t ) ] ) 2 + q U ( r → , t ) {\displaystyle H={1 \over 2m}\left({\vec {\sigma }}.[{\vec {P}}-q{\vec {A}}({\vec {r}},t)]\right)^{2}+qU({\vec {r}},t)} {\displaystyle H={1 \over 2m}\left({\vec {\sigma }}.[{\vec {P}}-q{\vec {A}}({\vec {r}},t)]\right)^{2}+qU({\vec {r}},t)}
Démonstration

On rappelle que: σ → . B → = σ → . ( ∇ → ×ばつ A → ) = σ x [ ∇ → ×ばつ A → ] x + σ y [ ∇ → ×ばつ A → ] y + σ z [ ∇ → ×ばつ A → ] z {\displaystyle {\vec {\sigma }}.{\vec {B}}={\vec {\sigma }}.({\vec {\nabla }}\times {\vec {A}})=\sigma _{x}[{\vec {\nabla }}\times {\vec {A}}]_{x}+\sigma _{y}[{\vec {\nabla }}\times {\vec {A}}]_{y}+\sigma _{z}[{\vec {\nabla }}\times {\vec {A}}]_{z}} {\displaystyle {\vec {\sigma }}.{\vec {B}}={\vec {\sigma }}.({\vec {\nabla }}\times {\vec {A}})=\sigma _{x}[{\vec {\nabla }}\times {\vec {A}}]_{x}+\sigma _{y}[{\vec {\nabla }}\times {\vec {A}}]_{y}+\sigma _{z}[{\vec {\nabla }}\times {\vec {A}}]_{z}}

Avec les matrices de Pauli: σ x = ( 0 1 1 0 ) σ y = ( 0 − i i 0 ) σ z = ( 1 0 0 − 1 ) {\displaystyle \sigma _{x}={\begin{pmatrix}0&1\1円&0\end{pmatrix}}\quad \sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}\quad \sigma _{z}={\begin{pmatrix}1&0\0円&-1\end{pmatrix}}} {\displaystyle \sigma _{x}={\begin{pmatrix}0&1\1円&0\end{pmatrix}}\quad \sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}\quad \sigma _{z}={\begin{pmatrix}1&0\0円&-1\end{pmatrix}}}
Le rotationnel a pour expression: ∇ → ×ばつ A → = ( ∇ y A z − ∇ z A y ∇ z A x − ∇ x A z ∇ x A y − ∇ y A x ) {\displaystyle {\vec {\nabla }}\times {\vec {A}}={\begin{pmatrix}\nabla _{y}A_{z}-\nabla _{z}A_{y}\\\nabla _{z}A_{x}-\nabla _{x}A_{z}\\\nabla _{x}A_{y}-\nabla _{y}A_{x}\end{pmatrix}}} {\displaystyle {\vec {\nabla }}\times {\vec {A}}={\begin{pmatrix}\nabla _{y}A_{z}-\nabla _{z}A_{y}\\\nabla _{z}A_{x}-\nabla _{x}A_{z}\\\nabla _{x}A_{y}-\nabla _{y}A_{x}\end{pmatrix}}}
⇒ σ → . B → = ( ( ∇ → ×ばつ A → ) z ( ∇ → ×ばつ A → ) x − i ( ∇ → ×ばつ A → ) y ( ∇ → ×ばつ A → ) x + i ( ∇ → ×ばつ A → ) y − ( ∇ → ×ばつ A → ) z ) {\displaystyle \Rightarrow {\vec {\sigma }}.{\vec {B}}={\begin{pmatrix}({\vec {\nabla }}\times {\vec {A}})_{z}&({\vec {\nabla }}\times {\vec {A}})_{x}-i({\vec {\nabla }}\times {\vec {A}})_{y}\\({\vec {\nabla }}\times {\vec {A}})_{x}+i({\vec {\nabla }}\times {\vec {A}})_{y}&-({\vec {\nabla }}\times {\vec {A}})_{z}\end{pmatrix}}} {\displaystyle \Rightarrow {\vec {\sigma }}.{\vec {B}}={\begin{pmatrix}({\vec {\nabla }}\times {\vec {A}})_{z}&({\vec {\nabla }}\times {\vec {A}})_{x}-i({\vec {\nabla }}\times {\vec {A}})_{y}\\({\vec {\nabla }}\times {\vec {A}})_{x}+i({\vec {\nabla }}\times {\vec {A}})_{y}&-({\vec {\nabla }}\times {\vec {A}})_{z}\end{pmatrix}}}
= ( ∇ x A y − ∇ y A x ∇ y A z − ∇ z A y − i ( ∇ z A x − ∇ x A z ) ∇ y A z − ∇ z A y + i ( ∇ z A x − ∇ x A z ) − ∇ x A y + ∇ y A x ) {\displaystyle ={\begin{pmatrix}\nabla _{x}A_{y}-\nabla _{y}A_{x}&\nabla _{y}A_{z}-\nabla _{z}A_{y}-i(\nabla _{z}A_{x}-\nabla _{x}A_{z})\\\nabla _{y}A_{z}-\nabla _{z}A_{y}+i(\nabla _{z}A_{x}-\nabla _{x}A_{z})&-\nabla _{x}A_{y}+\nabla _{y}A_{x}\end{pmatrix}}} {\displaystyle ={\begin{pmatrix}\nabla _{x}A_{y}-\nabla _{y}A_{x}&\nabla _{y}A_{z}-\nabla _{z}A_{y}-i(\nabla _{z}A_{x}-\nabla _{x}A_{z})\\\nabla _{y}A_{z}-\nabla _{z}A_{y}+i(\nabla _{z}A_{x}-\nabla _{x}A_{z})&-\nabla _{x}A_{y}+\nabla _{y}A_{x}\end{pmatrix}}}

En appliquant cet opérateur au spineur [ ψ ] ( r → ) = ( ψ + ( r → ) ψ − ( r → ) ) {\displaystyle [\psi ]({\vec {r}})={\begin{pmatrix}\psi _{+}({\vec {r}})\\\psi _{-}({\vec {r}})\end{pmatrix}}} {\displaystyle [\psi ]({\vec {r}})={\begin{pmatrix}\psi _{+}({\vec {r}})\\\psi _{-}({\vec {r}})\end{pmatrix}}} on obtient:

♦ Pour le terme [ σ → . B → ] 11 : {\displaystyle [{\vec {\sigma }}.{\vec {B}}]_{11}:} {\displaystyle [{\vec {\sigma }}.{\vec {B}}]_{11}:}
( ∇ x A y − ∇ y A x ) ψ + = ∇ x A y ψ + − ∇ y A x ψ + {\displaystyle (\nabla _{x}A_{y}-\nabla _{y}A_{x})\psi _{+}=\nabla _{x}A_{y}\psi _{+}-\nabla _{y}A_{x}\psi _{+}} {\displaystyle (\nabla _{x}A_{y}-\nabla _{y}A_{x})\psi _{+}=\nabla _{x}A_{y}\psi _{+}-\nabla _{y}A_{x}\psi _{+}}
= ∇ x ( A y ψ + ) − A y ∇ x ψ + − ∇ y ( A x ψ + ) + A x ∇ y ψ + {\displaystyle =\nabla _{x}(A_{y}\psi _{+})-A_{y}\nabla _{x}\psi _{+}-\nabla _{y}(A_{x}\psi _{+})+A_{x}\nabla _{y}\psi _{+}} {\displaystyle =\nabla _{x}(A_{y}\psi _{+})-A_{y}\nabla _{x}\psi _{+}-\nabla _{y}(A_{x}\psi _{+})+A_{x}\nabla _{y}\psi _{+}}
= [ ∇ x A y − ∇ y A x + A x ∇ y − A y ∇ x ] ψ + = [ [ ∇ → ×ばつ A → ] z + [ A → ×ばつ ∇ → ] z ] ψ + {\displaystyle =[\nabla _{x}A_{y}-\nabla _{y}A_{x}+A_{x}\nabla _{y}-A_{y}\nabla _{x}]\psi _{+}=[[{\vec {\nabla }}\times {\vec {A}}]_{z}+[{\vec {A}}\times {\vec {\nabla }}]_{z}]\psi _{+}} {\displaystyle =[\nabla _{x}A_{y}-\nabla _{y}A_{x}+A_{x}\nabla _{y}-A_{y}\nabla _{x}]\psi _{+}=[[{\vec {\nabla }}\times {\vec {A}}]_{z}+[{\vec {A}}\times {\vec {\nabla }}]_{z}]\psi _{+}}
♦ Pour le terme [ σ → . B → ] 12 : {\displaystyle [{\vec {\sigma }}.{\vec {B}}]_{12}:} {\displaystyle [{\vec {\sigma }}.{\vec {B}}]_{12}:}

∇ y A z ψ − − ∇ z A y ψ − − i ( ∇ z A x ψ − − ∇ x A z ψ − ) =: ∇ y ( A z ψ − ) − A z ∇ y ψ − − ∇ z ( A y ψ − ) {\displaystyle \nabla _{y}A_{z}\psi _{-}-\nabla _{z}A_{y}\psi _{-}-i(\nabla _{z}A_{x}\psi _{-}-\nabla _{x}A_{z}\psi _{-})=:\nabla _{y}(A_{z}\psi _{-})-A_{z}\nabla _{y}\psi _{-}-\nabla _{z}(A_{y}\psi _{-})} {\displaystyle \nabla _{y}A_{z}\psi _{-}-\nabla _{z}A_{y}\psi _{-}-i(\nabla _{z}A_{x}\psi _{-}-\nabla _{x}A_{z}\psi _{-})=:\nabla _{y}(A_{z}\psi _{-})-A_{z}\nabla _{y}\psi _{-}-\nabla _{z}(A_{y}\psi _{-})}

+ A y ∇ z ψ − − i ( ∇ z ( A x ψ − ) − A x ∇ z ψ − ∇ x ( A z ψ − ) + A z ∇ x ψ − ) {\displaystyle +A_{y}\nabla _{z}\psi _{-}-i(\nabla _{z}(A_{x}\psi _{-})-A_{x}\nabla _{z}\psi _{-}\nabla _{x}(A_{z}\psi _{-})+A_{z}\nabla _{x}\psi _{-})} {\displaystyle +A_{y}\nabla _{z}\psi _{-}-i(\nabla _{z}(A_{x}\psi _{-})-A_{x}\nabla _{z}\psi _{-}\nabla _{x}(A_{z}\psi _{-})+A_{z}\nabla _{x}\psi _{-})}
= [ ( ∇ → ×ばつ A → ) x + ( A → ×ばつ ∇ → ) x ] ψ − − i [ ( ∇ → ×ばつ A → ) y + ( A → ×ばつ ∇ → ) y ] ψ − {\displaystyle =[({\vec {\nabla }}\times {\vec {A}})_{x}+({\vec {A}}\times {\vec {\nabla }})_{x}]\psi _{-}-i[({\vec {\nabla }}\times {\vec {A}})_{y}+({\vec {A}}\times {\vec {\nabla }})_{y}]\psi _{-}} {\displaystyle =[({\vec {\nabla }}\times {\vec {A}})_{x}+({\vec {A}}\times {\vec {\nabla }})_{x}]\psi _{-}-i[({\vec {\nabla }}\times {\vec {A}})_{y}+({\vec {A}}\times {\vec {\nabla }})_{y}]\psi _{-}}
Ce qui permet de conclure: σ → . B → = σ → ( ∇ → ×ばつ A → ) + σ → ( A → ×ばつ ∇ → ) {\displaystyle {\vec {\sigma }}.{\vec {B}}={\vec {\sigma }}({\vec {\nabla }}\times {\vec {A}})+{\vec {\sigma }}({\vec {A}}\times {\vec {\nabla }})} {\displaystyle {\vec {\sigma }}.{\vec {B}}={\vec {\sigma }}({\vec {\nabla }}\times {\vec {A}})+{\vec {\sigma }}({\vec {A}}\times {\vec {\nabla }})}

On rappellera: P → = ħ i ∇ → ⇒ {\displaystyle {\vec {P}}={\frac {\hbar }{i}}{\vec {\nabla }}\Rightarrow } {\displaystyle {\vec {P}}={\frac {\hbar }{i}}{\vec {\nabla }}\Rightarrow } σ → . B → = i q ħ . σ → [ ( P → ×ばつ q A → ) + ( q A → ×ばつ P → ) ] = − i q ħ . σ → [ ( P → − q A → ) ×ばつ ( P → − q A → ) ] {\displaystyle {\vec {\sigma }}.{\vec {B}}={\frac {i}{q\hbar }}.{\vec {\sigma }}[({\vec {P}}\times q{\vec {A}})+(q{\vec {A}}\times {\vec {P}})]=-{\frac {i}{q\hbar }}.{\vec {\sigma }}[({\vec {P}}-q{\vec {A}})\times ({\vec {P}}-q{\vec {A}})]} {\displaystyle {\vec {\sigma }}.{\vec {B}}={\frac {i}{q\hbar }}.{\vec {\sigma }}[({\vec {P}}\times q{\vec {A}})+(q{\vec {A}}\times {\vec {P}})]=-{\frac {i}{q\hbar }}.{\vec {\sigma }}[({\vec {P}}-q{\vec {A}})\times ({\vec {P}}-q{\vec {A}})]}

On rappelle la relation, avec M → , N → {\displaystyle {\vec {M}},{\vec {N}}} {\displaystyle {\vec {M}},{\vec {N}}} deux opérateurs quelconques et I l'opérateur unitaire:
( σ → . M → ) ( σ → . N → ) = M → . N → I + i σ → ( M → ×ばつ N → ) {\displaystyle ({\vec {\sigma }}.{\vec {M}})({\vec {\sigma }}.{\vec {N}})={\vec {M}}.{\vec {N}}I+i{\vec {\sigma }}({\vec {M}}\times {\vec {N}})} {\displaystyle ({\vec {\sigma }}.{\vec {M}})({\vec {\sigma }}.{\vec {N}})={\vec {M}}.{\vec {N}}I+i{\vec {\sigma }}({\vec {M}}\times {\vec {N}})}
⇒ i σ → [ ( P → − q A → ) ×ばつ ( P → − q A → ) ] = ( σ → . ( P → − q A → ) ) 2 − ( P → − q A → ) 2 I {\displaystyle \Rightarrow i{\vec {\sigma }}[({\vec {P}}-q{\vec {A}})\times ({\vec {P}}-q{\vec {A}})]=({\vec {\sigma }}.({\vec {P}}-q{\vec {A}}))^{2}-({\vec {P}}-q{\vec {A}})^{2}I} {\displaystyle \Rightarrow i{\vec {\sigma }}[({\vec {P}}-q{\vec {A}})\times ({\vec {P}}-q{\vec {A}})]=({\vec {\sigma }}.({\vec {P}}-q{\vec {A}}))^{2}-({\vec {P}}-q{\vec {A}})^{2}I}
⇒ σ → B → = − 1 q ħ [ σ → ( P → − q A → ) ] 2 + 1 q ħ [ P → − q A → ] 2 I {\displaystyle \Rightarrow {\vec {\sigma }}{\vec {B}}=-{\frac {1}{q\hbar }}[{\vec {\sigma }}({\vec {P}}-q{\vec {A}})]^{2}+{\frac {1}{q\hbar }}[{\vec {P}}-q{\vec {A}}]^{2}I} {\displaystyle \Rightarrow {\vec {\sigma }}{\vec {B}}=-{\frac {1}{q\hbar }}[{\vec {\sigma }}({\vec {P}}-q{\vec {A}})]^{2}+{\frac {1}{q\hbar }}[{\vec {P}}-q{\vec {A}}]^{2}I}
Il vient alors:
H = 1 2 m [ P → − q A → ] 2 + q U − q ħ 2 m σ → . B → {\displaystyle H={\frac {1}{2m}}[{\vec {P}}-q{\vec {A}}]^{2}+qU-{\frac {q\hbar }{2m}}{\vec {\sigma }}.{\vec {B}}} {\displaystyle H={\frac {1}{2m}}[{\vec {P}}-q{\vec {A}}]^{2}+qU-{\frac {q\hbar }{2m}}{\vec {\sigma }}.{\vec {B}}}
= 1 2 m [ P → − q A → ] 2 + q U + 1 2 m [ σ → ( P → − q A → ) ] 2 − 1 2 m [ P → − q A → ] 2 I {\displaystyle ={\frac {1}{2m}}[{\vec {P}}-q{\vec {A}}]^{2}+qU+{\frac {1}{2m}}[{\vec {\sigma }}({\vec {P}}-q{\vec {A}})]^{2}-{\frac {1}{2m}}[{\vec {P}}-q{\vec {A}}]^{2}I} {\displaystyle ={\frac {1}{2m}}[{\vec {P}}-q{\vec {A}}]^{2}+qU+{\frac {1}{2m}}[{\vec {\sigma }}({\vec {P}}-q{\vec {A}})]^{2}-{\frac {1}{2m}}[{\vec {P}}-q{\vec {A}}]^{2}I}
= 1 2 m [ σ → ( P → − q A → ( r → , t ) ) ] 2 + q U ( r → , t ) {\displaystyle ={\frac {1}{2m}}[{\vec {\sigma }}({\vec {P}}-q{\vec {A}}({\vec {r}},t))]^{2}+qU({\vec {r}},t)} {\displaystyle ={\frac {1}{2m}}[{\vec {\sigma }}({\vec {P}}-q{\vec {A}}({\vec {r}},t))]^{2}+qU({\vec {r}},t)}




  1. ↑ Walter Greiner, Mécanique quantique – Une introduction, Springer éditeur, 1999 (ISBN 3540643478 et 978-3540643470).

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