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Electron in B Field - Quantum

by JesseC
Tags: electron, field, quantum
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JesseC
#1
Nov21-10, 11:24 AM
P: 281
1. The problem statement, all variables and given/known data

Magnetic field in xz plane.

[tex] \vec{B}=\hat{i}B_x+\hat{k}B_z [/tex]

Write down the hamiltonian operator for the interaction of the electron's intrinsic magnetic moment with this field and express it in matrix form. Find its eigenvalues and sketch these as a function of Bz, for fixed, nonzero Bx. How would the picture differ if Bx were zero.

3. The attempt at a solution

So I got the hamiltonian looking like this:

[tex] \hat{H}= \frac{e g_s}{2m_e} \hat{S} \cdot \vec{B} [/tex]

I'm not sure about the form of [tex]\hat{S}[/tex] in this case? Is it a combination of z and x components?
Normally if the field is just constant in the z-direction we could write B as a scalar and we'd just find the eigenvalues of the third pauli matrix.
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vela
#2
Nov21-10, 12:35 PM
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It's all three components

[tex]\hat{S} = \hat{i}S_x + \hat{j}S_y + \hat{k}S_z[/tex]

Take the dot product as usual and then you can express the Hamiltonian as a linear combination of the Pauli matrices.
JesseC
#3
Nov21-10, 06:24 PM
P: 281
cheers for that, cleared it up.


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