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Atomic and Molecular Physics January -May 2010
Problem Sheet IV
30th January 2010
due 11th February 2010
21. (Hyperfine structure) Consider the dipole associated with the nucleus (~µN ) which
leads to vector potential
~ = µ0 ~µN × ~r/r3 .
A
4π
This causes an additional hamiltonian for the hydrogen like atom
e ~
~ e .B
~
HN0 M =
p~.A − 2µB S
me
~ is the magnetic field produced by the nucleus and S
~e is the spin of the electron.
where B
~e /h̄ is the electron magnetic moment operator.) Show that this can be written
(2 µB S
as
~ I~
µ0 2gN µB µN L.
~e .I∇
~ 2 1 − (S
~e .∇)(
~ I.
~ ∇)
~ 1 ])
HN0 M = −
+
[
S
(
2
4π
r3
r
r
h̄
~
Here the spin of the nucleus is I~ and its magnetic moment operator is µN gN I.
22. (Hyperfine structure) Use the above expression to calculate the hyperfine splitting for the ground state of the hydrogen atom.Calculate the numerical value and show
the separation leads to an absorption/emission of a photon of wavelength 21cm.
23.(linear Stark effect) The hamiltonian of a hydrogen like atom in the presence of
~ has an extra term
an electric field E
~ r = eE.~
~ r
Hel = −q E.~
where −e is the charge of the electron.
(a) show that n = 1 state has no linear Stark effect.
(b) Calculate the effect for n = 2.
(c) Estimate numerically the shift in electron volts for electric fields of the order of
kilovolts per metre.
24. (Quadratic Stark effect) Calculate the shift in energy of the ground state of
hydrogen like atom in the presence of an electric field upto second order in Hel given in
problem 23 in terms of an infinte sum involving all the eigenstates of the unperturbed
hydrogen atom. Estimate the shift using only the n = 2 term ( n is the principal
quantum number)in electron volts for the same value of the electric field used earlier in
problem 23(c).
25.(Doppler broadening)An atom emits light at a frequency ωo when it is at rest.
Consider light emitted by the atomswhen they form a gas of temperature T . Assuming
the molecules obey Maxwell’s distribution law. Show that the intensity of light I(ω)
emitted has the distribution
mc2
I(ω) = I(ω0 )exp[−
2kB T
ω − ω0 )
ω0
!2
]
m is the mass of the atom and kB is the Boltzmann’s constant.