No. of possible electron transitions in hydrogen in magnetic field

[Newtonz]

Homework Statement

A hydrogen atom is in the n=3 state. Given that a magnetic field is present, how many photons of different energy can we observe when the atom de-excites to the ground state?

Homework Equations

quantum number, angular momentum quantum number, orbital magnetic quantum number

The Attempt at a Solution

So far, I am aware in the absence of a magnetic field, there should be three possible transitions: n=3 to n=2, n=3 to n=1, and n=2 to n=1. I know that a magnetic field will cause there to be orbital magnetic quantum number. However, may I know whether electrons in the same n-state but with different angular momentum quantum number and orbital magnetic quantum number have different energy levels? For example, are energy levels of electron in the state: [n=3, l=2, m$_{l}$=1] different from [n=3, l=2, m$_{l}$=-1]?

 

[Simon Bridge]

Welcome to PF;
Magnetic fields interact with the electron angular momenta ... thus splitting the energy levels.
So yes - each angular momentum state now has a unique energy where before you had degenerate states.
The question is basically testing your knowledge of this ...

Look up "splitting" and "hyperfine splitting".
But what I think you really want is "Zeeman Effect".
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/zeeman.html