- #1
diegzumillo
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Homework Statement
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]
Homework Equations
I believe 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)
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]