Hyperfine Splitting In A Magnetic Field

[jkrivda]

1. Problem:
Consider Hydrogen in its ground state in a magnetic field of magnitude B.
Compute the effect of the magnetic field on the hyperfine structure of Hydrogen
(that is you should include the interaction of both the proton and electron magnetic
moments with the applied magnetic field). Give the values of the energy eigenvalues
for B = 1 gauss and for B = 10⁴ gauss.

Homework Equations

the hamiltonian for a dipole moment is H=-μB
μ_p=g_peS_p/2m_p
μ_e=-eS_e/m_e
F=J+I where F is the atom's total angular momentum

The Attempt at a Solution

well, i think that the Hamiltonian looks like:
H=μ_eB_p+B_ext(μ_e+μ_p)
so do we just act this H on the ground state wave function?

 

[TSny]

well, i think that the Hamiltonian looks like:
H=μ_eB_p+B_ext(μ_e+μ_p)
so do we just act this H on the ground state wave function?

Just a thought: From the wording of the problem, it seems to me that you want to find the effect of the external field on the hyperfine levels. So, I would think that you would assume the hyperfine levels are known and you just want to perturb those with the external field.

If so, wouldn't the perturbation part of the Hamiltonian just be due to the external field? Therefore you would have only the term B_ext(μ_e+μ_p) as the perturbation? [EDIT: I think the term should have an overall negative sign.]

[EDIT 2: My comments might be invalid. You can only treat B_ext(μ_e+μ_p) as a perturbation of the hyperfine levels if the perturbation is small compared to the hyperfine splitting. I think that might be true for an external field of 1 gauss, but not for 10,000 gauss. So, for 10,000 gauss I guess you have to treat both the hyperfine interaction and the external field interaction together as the perturbation of the unperturbed ground state level. Hope I'm not misleading you. Hopefully someone else will chime in.]

 

[andrien]

see 12.9 onwards,
http://books.google.co.in/books?id=ntMZQtSskBsC&pg=SA12-PA1&dq=hyperfine+splitting+of+hydrogen+feynman+lectures&hl=en#v=onepage&q=hyperfine%20splitting%20of%20hydrogen%20feynman%20lectures&f=false

 

[TSny]

see 12.9 onwards,
http://books.google.co.in/books?id=ntMZQtSskBsC&pg=SA12-PA1&dq=hyperfine+splitting+of+hydrogen+feynman+lectures&hl=en#v=onepage&q=hyperfine%20splitting%20of%20hydrogen%20feynman%20lectures&f=false

Thanks for the reference! Feynman has a very nice treatment.

 

[andrien]

I think the name of topic is somewhat awkward.From what I understand,hyperfine splitting arises because of interaction of magnetic moment of proton with the spin of electron in hydrogen.By considering proton's magnetic moment arising from a localized current density one can write for the current associated with it as,
J=c(∇×M)
now this J gives rise to a potential,which can be calulatd from
A=k∫j(r')/|r-r'|(k is just some constant),now assuming a delta function source A gives
A=k'(∇×m)/r,{M=mδ(r')}
So B is given by B=∇×A,m is for proton which conventionally written as m_p=(1/2)h^-σ_pμ_p
now plugging it into m and evaluating B one writes for the interaction as
H'=-μ_e.B,(HERE μe is different from upper one)where B is just calculated
after it one uses some vector identity to get,
H'=-μ_eμ_p{(σ.σ_p)∇²-(σ.∇)(σ_p.∇)}(1/2r)(apart from some constant)
spherical averaging over s tate gives(because s state is spherically symmetric)
(σ.∇)(σ_p.∇)=1/3(σ.σ_p)∇²(be sure to verify it in spherical coordinates)
After which one gets,
H'=-μ_eμ_p(2/3)(σ.σ_p)∇²(1/2r)(apart from some constant)

this is evaluatd for ground state using the wavefunction of hydrogen(non-relativistic) to get
∫∇²(1/r)|ψ(r)|² d³r=|ψ(0)|²(again some constant depending on unit chosen!)
After which (σ.σ_p)=-3 for singlet and 1 for triplet(easily seen by using S²=S₁²+S₂²+2S₁.S₂,IN TRIPLET S=1 and for singlet it is S=0)
the difference is taken for these two states for (σ.σ_p) will yield the required hyperfine splitting.

 

[andrien]

I just did not edit it,there are some mistakes.Like H' reads
H'=-μ_eσ.B,(I missed σ there)