PHYS 30101 Quantum Mechanics

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PHYS 30101 Quantum Mechanics
Lecture 20
Dr Jon Billowes Nuclear Physics Group (Schuster
Building, room 4.10) j.billowes_at_manchester.ac.uk
These slides at http//nuclear.ph.man.ac.uk/jb/p
hys30101
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6. The hydrogen atom revisited - Reminder of
eigenfunctions, eigenvalues and quantum numbers
n, l, ml of hydrogen atom. 6.1 Spin-orbit
coupling and the fine structure. 6.2 Zeeman
effect for single electron atoms in (a) a
weak magnetic field (b) a strong
magnetic field 6.3 Spin in magnetic field QM
and classical descriptions
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RECAP 6. 1 Spin-orbit coupling and fine
structure
Classically an electron orbiting a nucleus of
charge Ze sees the nucleus in orbit around it
(a current loop) which produces a field at the
electron of
Putting I Ze/T where T is period of orbit
(obtained from the classical angular momentum
expression Lme?r2 ) we get
Electron magnetic moment is in direction of its
intrinsic spin
Thus interaction energy and corresponding
Hamiltonian can be written
(relativistic effect of Thomas precession)
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The shift in energy of a state
is the eigenvalue of the spin-orbit Hamiltonian
j3/2 (4 states)
l1, s1/2
Ah2/2
ml1, 0, -1 ms1/2, -1/2 (6 states)
-Ah2
j1/2 (2 states)
The energy centroid is unchanged 4 X A/2 2 X A
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6.2(a) Weak-field Zeeman effect
6.2(b) Strong-field Zeeman
For electron, B is much greater than the field it
sees due to its orbital motion. S and L
independently precess around B keeping ms and ml
constants of motion
L and S remain coupled to J. Classically J
precesses slowly around field B, keeping
Jz M a constant
B
B
S
S
ms
L
ml
L
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Landé g-factor
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The state
gJ -4/3
j3/2
l1, s1/2
Ah2/2
-Ah2
gJ -2/3
j1/2
(Spin-orbit splitting)
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Zeeman structure for l 1, s 1/2 orbital
Strong field
Weak field
(-1,1/2) (1,-1/2)