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Zeeman effect and defining the g_F Factor

  • Thread starter TFM
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  • #1
TFM
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Homework Statement



A hydrogen atom is interacting with an external magnetic field.
1. Derive the equation for the [tex]g_F[/tex]-factor of the hyperfine states.

Homework Equations





The Attempt at a Solution



Okay, so the question asks to define the gF factor, however, I am not quite sure where to start.

I know firstly that it is based on the diagram of the vector arrows, as (crudely drawn) attached:

I also know the answer I need to get (it is mentioned in the notes):

[tex] g_F = g_J\frac{F(F + 1) + j(j + 1) - I(I +1)}{2F(F + 1)} + \frac{\mu_N}{\mu_B}g_I \frac{F(F + 1) + I(I + 1) - j(j +1)}{2F(F + 1)}[/tex]

Also, for the gJ, it starts with:

[tex] H_J = \frac{\mu_N}{\hbar}(\hat{L} + 2\vec{S})\cdot \vec{B} = \frac{\mu_N}{\hbar}(\hat{J} + \vec{S})\cdot \vec{B} [/tex]

and I know for gF, we have:

[tex] H_J = \frac{\mu_B}{\hbar}(\hat{L} + 2\vec{S})\cdot \vec{B} - g_I ({\frac{\mu_N}{\hbar} \vec{I}\cdot \vec{B})[/tex]

Anyone got any suggestions about what I should do first?

TFM
 

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Answers and Replies

  • #2
Hao
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I suspect the approach would be to treat the external magnetic field as a perturbation to the internal magnetic field of the Hydrogen atom.

In other words, [tex]\vec{B} = \vec{B}_{Internal} + \vec{B}_{External}[/tex].

Considering [tex]\vec{B}_{Internal}[/tex] first, we can find the orbital angular momentum (l), spin angular momentum (s),ml,ms (or J,m?) eigenstate of the Hamiltonian (possibly after some simplifying assumptions?).

We then apply first order perturbation theory for [tex]\vec{B}_{External}[/tex].

Since you know the energy, you should be able to get the g factor.

Hopefully, this is the way to go.
 

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