How do you apply perturbation theory to a magnetic field Hamiltonian?

[Danielk010]

TL;DR

Given,
Take with . Determine the energy eigenvalues exactly and
compare with the results of perturbation theory through second order in .

I understand that the first order results would be $\langle \phi_n^0 | -\mu * B | \phi_n^0 \rangle$ = $\langle \phi_n^0 | -\frac{gq}{2mc}\hat{S} * B | \phi_n^0 \rangle$, the second order results would be $\sum_{k \ne n} \frac{|\langle \phi_n^0 | -\mu * B | \phi_n^0 \rangle|^2} {E_n^{(0)} - E_k^{(0)}}$, and the unperturbed hamiltonian would $E_n^{(0)} = (n + \frac{1}{2})\hbar\omega$.

I am confused on how to evaluate braket of phi as the problem is asking for the exact energy eigenvalues. I tried plugging in $B = \frac{-Zev \times r}{cr^3}$, but then I would have the Z, v and r term, which would not give me an exact solution.


Am I on the right track? Is there an equation I am missing? Thank you for any help on this problem. Also do any of you all know how to add equations to the TL;DR?

 

[renormalize]

TL;DR: Given,
Take with . Determine the energy eigenvalues exactly and
compare with the results of perturbation theory through second order in .

Can you edit your post to very clearly state the problem you are trying to solve?

 

[Danielk010]

Sorry, I tried to insert LaTeX in the TL;DR but it did not display correctly. I am also past the time limit to edit a post.

Here is the problem:
The spin Hamiltonian for a spin-1/2 particle in an external magnetic field is
$$\hat{H} = -\hat{\mu} * B = - \frac{gq}{2mc}\hat{S} * B$$
Take $B = B_0k + B_2j\text{, with }B_2 \ll B_0$. Determine the energy eigenvalues exactly and compare with the results of perturbation theory through second ordder in $\frac{B_2}{B_0}$


I made some progress and I asked the TA for my class for help on this problem. For my initial attempt for the first order differential, I did this:

In the end, I got $\frac{\omega_0 \hbar}{2} sin w_0t$ where $\omega_1 = -\frac{gq}{2mc}(B_0 + B_2)$.
I started by setting the hamiltonian to be: $\textbf{Equation 1} = H = -\frac{gq}{2mc} * \hat{S_z}B_0 + \hat{S_y}B_2$.

Assuming, $\omega_0 = -\frac{gq}{2mc}$, we can set the energy to $E_n^1 = \omega_0(<n|\hat{S_z}B_0|n> + <n|\hat{S_yB_1}|n>)$ given the first-order P.T equation.

Since $B_0$ and $B_2$ are constants, you can get $w_1(<S_z> + <S_y>)$. From equations (4.23 $<S_z> = 0$) and (4.30 $<S_y> = \frac{hbar}{2}sin\omega_0t$)

I get $E_n^1 = \frac{\omega_0 \hbar}{2} sin w_0t$. According to the TA, it should be 0, which I am confused on how they got that.

For the exact energy eigenvalues, he mentioned to use the Pauli matrices.
I used Equation 1 and by plugging in the Pauli matrices, I got $\begin{pmatrix} \omega_0 B_0 & -i\omega_0 B_2 \\ i\omega_0 B_2 & -\omega_0 B_0 \end{pmatrix}$. By taking the det. I got $-B_0^2\omega_0^2 + \lambda^2 - B_2^2\omega_0^2$ => $\lambda = \omega_0 \sqrt{B_0^2 + B_2^2}$. According to the TA, it is supposed to be $\pm \hbar \omega [1 + ...]$, which is not what I got.

Sorry for the lengthy post, but what did I do wrong? I am using the A Modem Approach to Quantum Mechanics Second Edition textbook. Please let me know if there is anything else I need to provide.