Time Evolution of Hydrogen Atom in a Magnetic Field

[Xyius]

Homework Statement

A hydrogen atom is prepared in its ground state with spin up along the z-direction.
At time t = 0 a constant magnetic field $\vec{B}$
(pointing in an arbitrary direction determined
by $\theta$ and $\phi$) is turned on. Neglecting the fine structure and terms proportional to $\vec{A}^2$
Compute the probability that the atom will be found in the ground state with spin down as
a function of time.
(i) Solve this problem exactly
(ii) Use first order perturbation theory
(iii) Expand your solution in (i) in Taylor series for short times and compare you result
with the one obtained in (ii).
3

Homework Equations

So to me this seems exactly like the Zeeman effect. So naturally I am using the Hamiltonian from this analysis (with the $\vec{A}^2$ or $\vec{B}^2$ term neglected.)

$$\hat{H}=\frac{\hat{\vec{p}}^2}{2m_e}+V_C(r)-\frac{e}{2m_e c}B(\hat{L}_z+2\hat{S_z})$$

Where $V_C(r)$ is the coulomb potential, and is part of the unperturbed Hamiltonian.

The exact solution for the time evolution of a ket of the form..

$$|\alpha,t_0,t>_I=\sum_n C_n(t)|n>$$

is found by finding the expansion coefficients $C_n(t)$ using..

$$i\hbar\frac{\partial}{\partial t}C_n(t)=\sum_m e^{i(E_n-E_m)/\hbar}V_{nm}(t)C_m(t)$$

And the approximate form is found by the Dyson series, with first order being equal to..
$$C_n^1=-\frac{i}{\hbar}\int_{t_0}^{t} e^{i(E_n-E_i)\\hbar}V_{ni}(t')dt'$$

The Attempt at a Solution

There is one major hurdle that is stopping me from making any progress on this problem, and that is the fact that if I write the perturbation term in its matrix form, from the Zeeman effect, the elements are equal to..

$$-\frac{e\hbar Bm}{2m_e c}\left[ 1 \pm \frac{1}{(2l+1)} \right]$$

Where this matrix is diagonal in the |n,l,j,m> basis. Thus, when I apply the third equation (the exact solution for the expansion coefficients) from section 2 above, I get uncoupled equations for the expansion coefficients, meaning a diagonal matrix, meaning, when I try to find any transition probabilities, I get zero from orthogonality!

I have realized that this comes from the fact that my hamiltonian is no different than the Zeeman effect and goes not depend on time. I feel like this has to be my source of error.

Where am I going wrong??

 

[Xyius]

Okay so I realized that the magnetic field is not only in the Z direction as the Zeeman effect was. So now I redid the problem and got this as the perturbation.

$$-\frac{eB}{2m_e c}\left[ \hat{L}_y-\hat{L}_x-\hat{L}_z-2\hat{S}_z \right]$$

I am still having problems because he doesn't specify what n has to be. I am assuming the electron is bound and can only go from spin up to spin down, making this a two state problem. Is this assumption wrong? Because I still get a diagonal matrix even with the above Hamiltonian. I can scan my work if anyone needs to see it.