an electron bound in a 3D harmonic oscillator? Give the energy shifts and and draw a diagram
for the 0sand 1pstates.V=^12 mω^2 r^2
dV
dr=mω(^2) r
〈HSO〉= ̄h
2
2 m^2 c^2
1
2 [j(j+ 1)−l(l+ 1)−s(s+ 1)]mω
2
〈HSO〉=h ̄
(^2) ω 2
4 mc^2 [j(j+ 1)−l(l+ 1)−s(s+ 1)]
for the 0S 12 , ∆E= 0,
for the 1P 12 , ∆E=− 2 ̄h
(^2) ω 2
4 mc^2 ,
for the 1P 32 , ∆E= +1 ̄h
(^2) ω 2
4 mc^2.
- We computed that the energies after the fine structure corrections to the hydrogen spectrum
areEnlj=−α
(^2) mc 2
2 n^2 +
α^4 mc^2
8 n^4 (3−
4 n
j+^12 ). Now a weak magnetic fieldBis applied to hydrogen
atoms in the 3dstate. Calculate the energies of all the 3dstates (ignoring hyperfine effects).
Draw an energy level diagram, showing the quantum numbers of thestates and the energy
splittings.
- In Hydrogen, then= 3 state is split by fine structure corrections into states of definitej,mj,ℓ,
ands. According to our calculations of the fine structure, the energy only depends onj. We
label these states in spectroscopic notation:N^2 s+1Lj. Draw an energy diagram for then= 3
states, labeling each state in spectroscopic notation. Give the energy shift due to the fine
structure corrections in units ofα^4 mc^2. - The energies of photons emitted in the Hydrogen atom transitionbetween the 3S and the 2P
states are measured, first with no external field, then, with the atoms in a uniform magnetic
field B. Explain in detail the spectrum of photons before and after the field is applied. Be sure
to give an expression for any relevant energy differences.