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PHYS 30101 Quantum Mechanics
Lecture 21
Dr Jon Billowes
Nuclear Physics Group (Schuster Building, room 4.10)
[email protected]
These slides at: http://nuclear.ph.man.ac.uk/~jb/phys30101
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
Plan: Include coupling of orbital and spin
angular momenta in Hamiltonian for
hydrogen atom
S
L
-μ.B
The shift in energy of a state
is the eigenvalue of the
spin-orbit Hamiltonian:
j=3/2 (4 states)
l=1, s=1/2
ml=+1, 0, -1
ms=+1/2, -1/2
(6 states)
Aħ2/2
-Aħ2
j=1/2 (2 states)
The energy centroid is unchanged: 4 X A/2 = 2 X A
6.2(a) Weak-field Zeeman effect
L and S remain coupled to J.
Classically J precesses slowly
around field B, keeping
Jz= M a constant
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
B
B
S
L
S
ms
L
ml
Landé g-factor
The state
l=1, s=1/2
j=3/2
gJ= -4/3
j=1/2
gJ= -2/3
Aħ2/2
-Aħ2
(Spin-orbit splitting)
Zeeman structure
Strong field
for l = 1, s = 1/2 orbital
Weak field
(-1,+1/2)
(+1,-1/2)
For single-electron atom in an external magnetic field B applied
along z-axis the full Hamiltonian is
H =
H0
P2/2m + V
+
Hso
A/2(J2-L2-S2)
+
Hmag
-µB/ħ (glLz + gsSz) B
In strong external magnetic field this term is much greater
than the spin-orbit interaction Hso (which we now ignore).
Hmag does not commute with J so eigenfunctions
are no longer
x
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
6.3 Classical and QM description of precession
B
B
L
dφ
θ
dL
6.3 QM description of classical precession
Consider a state with angular momentum L in an external field B
Hmag = -gl μB lz B
Eigenstates are |l, mz >
B
These are stationary states.
L points with equal probability
everywhere on the surface of the cone.
Time evolution of a state (section 1.6(a))
is
| Ψ(t) > = e-iEt/ħ |l, mz >
The phase factor cannot be directly measured.
No precession observed.
Now consider a state with angular momentum L polarised along
the x-axis at t=0 in a magnetic field B applied along the z-axis
B
Eigenfunctions of spin-1/2 operators (from lecture 13)
x
Matrix representation: Eigenvectors of Sx, Sy, Sz
4.3.3 Example: description of spin=1 polarised along the x-axis
is
In Dirac notation:
Energies of |l, mz > states are E = +ħω,
0,
-ħω
where ω = -gl μB B / ħ (which actually equals the classical
Larmor precession frequency)
E = -μ.B
Time evolution of the initial state is
We are now able to observe interference between the different
phase factors – these are the “quantum beats” discussed in section
1.6(b) of this course.