Magnetic Moment and Spacing of Adjacent Magnetic Substates

[njdevils45]

Homework Statement

[/B]
Consider the hydrogen atom in the 4²F_5/2 state. Take into account the effects of finestructure (spin-orbit coupling).
(a) Write down the spectroscopic notation of the state that the 4²F_5/2 is degenerate with, in the absence of an external magnetic field.
(b) Calculate the magnitude of the magnetic moment of the hydrogen atom in the 4²F_5/2 state, in units of the Bohr magneton µ_B.
(c) Suppose the hydrogen atom in the 4²F_5/2 state is placed in an external magnetic field B. What will be the spacing in energy between adjacent magnetic substates, in terms of µ_BB, where again µ_B is the Bohr magneton?

The Attempt at a Solution

So I actually managed to do all three of them. But the solutions I got for B and C were so simple I feel like I did something wrong like maybe used the wrong equation of forgot to add something.

A) Degenerate with 4²D_5/2

B) I said that j =5/2, l = 3, s = 1/2. With this info I found the lande g factor to be = 0.857
Then I used μ_J = |ge√(j(j+1) ħ/2m = 0.857µ_B√(5/2(5/2+1) = 2.54µ_B

C) I said ΔE = Δm_jgµ_BB where Δm_j = 1 between adjacent substates. So

ΔE = 0.857µ_BB

Was just hoping I could get a double check on these solutions and any tips if they were wrong.

 

[DrClaude]

Then I used μ_J = |ge√(j(j+1) ħ/2m = 0.857µ_B√(5/2(5/2+1) = 2.54µ_B

I'm not sure what you are doing here. What equation is this based on?

The rest appears correct.

 

[njdevils45]

I'm not sure what you are doing here. What equation is this based on?

The rest appears correct.

I'm not sure. I just found this,

On the lecture slides.

I used the equation for "THE" Magnetic Moment, which I think takes into account fine structure. I think I just wasn't able to write it out using keyboard. Did I miss something or use the wrong equation?