(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The Hamiltonian of an electron with mass m, electric charge q and spin

of [tex]\frac{\hbar }{2}\vec{\sigma}[/tex] in a magnetic field described by the

potential vector [tex]\vec{A}\left( \vec{r},t\right) [/tex] and a scalar potential [tex]U\left( \vec{r},t\right) [/tex] is given by

[tex]\[H=\frac{1}{2m}\left[ \vec{P}-q\vec{A}\right] ^{2}+qU-\frac{q\hbar }{2m}\vec{

\sigma}.\vec{B}\][/tex]

where [tex]\vec{B}=\vec{\nabla}\times \vec{A}[/tex]. Show that this Hamiltonian can

also be obtained from Pauli Hamiltonian:

[tex]\[H=\frac{1}{2m}\left\{ \vec{\sigma}.\left[ \vec{P}-q\vec{A}\right] \right\}^{2}+qU\][/tex]

2. Relevant equations

I belive this one is useful here:

[tex]\[\left( \vec{\sigma}.\vec{A}\right) \left( \vec{\sigma}.\vec{B}\right) =\vec{A}.\vec{B}I+i\vec{\sigma}.\left( \vec{A}\times \vec{B}\right) \][/tex]

Wich in our case, we can rewrite it as

[tex]\[\left( \vec{\sigma}.\vec{A}\right) ^{2}=A^{2}I+i\vec{\sigma}.\left( \vec{A}\times \vec{A}\right) \][/tex]

(it's not the same vector A of the problem statement, of course)

3. The attempt at a solution

Using the above identity, we end up with a term like this:

[tex]\[\left[ \vec{P}-q\vec{A}\left( \vec{R},t\right) \right] \times \left[ \vec{P}-q\vec{A}\left( \vec{R},t\right) \right] =\vec{P}\times \vec{P}-\vec{P}\times q\vec{A}\left( \vec{R},t\right) -q\vec{A}\left( \vec{R},t\right)\times \vec{P}+q^{2}\vec{A}\left( \vec{R},t\right) \times \vec{A}\left( \vec{R},t\right) \][/tex]

Wich is... almost nice. If I knew what to do with all of these guys! I can see that if we consider only the second term we can solve the problem. What does this mean?..

Using

[tex]\[\vec{P}\rightarrow i\hbar \vec{\nabla}\]\[-\vec{P}\times q\vec{A}\left( \vec{R},t\right) =-i\hbar q\vec{\nabla}\times \vec{A}\left( \vec{R},t\right) =-i\hbar q\vec{B}\][/tex]

And using this result in the Hamiltonian..

[tex]\[H=\frac{1}{2m}\left\{ \left[ \vec{P}-q\vec{A}\left( \vec{R},t\right) \right]

^{2}+i\vec{\sigma}.\left[ -i\hbar q\vec{B}\right] \right\} +qU\left( \vec{R},t\right) \]\[[/tex]

[tex]H=\frac{1}{2m}\left\{ \left[ \vec{P}-q\vec{A}\left( \vec{R},t\right) \right]^{2}+\hbar q\vec{\sigma}.\vec{B}\right\} +qU\left( \vec{R},t\right) \][/tex]

[tex]\[H=\frac{1}{2m}\left[ \vec{P}-q\vec{A}\left( \vec{R},t\right) \right] ^{2}+

\frac{\hbar q}{2m}\vec{\sigma}.\vec{B}+qU\left( \vec{R},t\right) \][/tex]

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# Homework Help: Pauli Hamiltonian

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