Hydrogen Atom in Magnetic Field

[antibrane]

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

[tex]
\left|n,l,s,j,m_j\right\rangle \longrightarrow \left|n',l',s,j',m_j'\right\rangle
[/tex]

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

What I thought I should do was use the Hamiltonian

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

and then I get the matrix element for the transition,

[tex]
\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
[/tex]

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

I really appreciate any comments.

 

[eljose79]

Thank you for sharing your thoughts and approach to finding the probability of a hydrogen atom transition in a magnetic field. Your use of the Hamiltonian and matrix element is a good starting point. However, there are a few things to consider in your calculation.

First, the matrix element you have written is for a spin-orbit interaction, which is not the only interaction present in a hydrogen atom in a magnetic field. There is also the Zeeman effect, which arises from the coupling between the electron spin and the magnetic field. The total Hamiltonian for a hydrogen atom in a magnetic field would be the sum of these two interactions:

\hat{H}=-\vec{\boldsymbol{\mu}}\cdot\vec{\mathbf{B}}+\hat{H}_{SO}+\hat{H}_{Z}

where \hat{H}_{SO} is the spin-orbit interaction and \hat{H}_{Z} is the Zeeman interaction.

Second, the matrix element you have written is only valid for transitions between states with the same total angular momentum j. For transitions between states with different values of j, the matrix element would involve a Clebsch-Gordan coefficient and would not be as simple as the one you have written.

In order to properly calculate the probability of a transition between states with different values of j, you would need to use time-dependent perturbation theory and take into account both the spin-orbit and Zeeman interactions. This would involve solving the time-dependent Schrödinger equation for the system and calculating the transition probability from the time evolution of the wavefunction.

I hope this helps guide you in the right direction for your calculation. If you have any further questions, please don't hesitate to ask. Best of luck with your research!