# Hydrogen Atom in Magnetic Field

1. Aug 11, 2011

### antibrane

I am attempting to find the probability, after time $t$, of a hydrogen atom in a magnetic field $\vec{\mathbf{B}}=B_0\hat{\mathbf{z}}$ to go from

$$\left|n,l,s,j,m_j\right\rangle \longrightarrow \left|n',l',s,j',m_j'\right\rangle$$

where $j=l+\frac{1}{2}$ and $j'=l'+\frac{1}{2}$ or $j'=l'-\frac{1}{2}$. Since it is hydrogen then $s=s'=\frac{1}{2}$.

What I thought I should do was use the Hamiltonian

$$\hat{H}=-\vec{\boldsymbol{\mu}}\cdot\vec{\mathbf{B}}=\frac{e}{2m}(\vec{\mathbf{L}}+2\vec{\mathbf{S}})$$

and then I get the matrix element for the transition,

$$\left\langle n',l',s',j',m_j'\right|\hat{H}\left|n,l,s,j,m_j \right\rangle=\frac{\hbar e}{2m} B_0g_Jm_j\langle n',l',s',j',m_j'|n,l,s,j,m_j\rangle$$

where $g_F$ is the Lande g-factor. I would have used this in time-dependent perturbation theory to get the probability after time $t$, however, while this does give me a non-zero element for $j'=l'+\frac{1}{2}$ (where we would have $l'=l$), it is zero due to orthogonality for the other case ($j'=l'-\frac{1}{2}$). Is there something wrong with the way I am going about this?