How Does an Electron's Spin Interact with a Non-Uniform Magnetic Field?

[JesseC]

Homework Statement

Magnetic field in xz plane.

\vec{B}=\hat{i}B_x+\hat{k}B_z

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.

The Attempt at a Solution

So I got the hamiltonian looking like this:

\hat{H}= \frac{e g_s}{2m_e} \hat{S} \cdot \vec{B}

I'm not sure about the form of \hat{S} 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.

 

[vela]

It's all three components

\hat{S} = \hat{i}S_x + \hat{j}S_y + \hat{k}S_z

Take the dot product as usual and then you can express the Hamiltonian as a linear combination of the Pauli matrices.

 

[JesseC]

cheers for that, cleared it up.