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.