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PHYS 30101 Quantum Mechanics
Lecture 18
Dr Jon Billowes
Nuclear Physics Group (Schuster Building, room 4.10)
[email protected]
These slides at: http://nuclear.ph.man.ac.uk/~jb/phys30101
Syllabus
1. Basics of quantum mechanics (QM)
Postulate, operators,
eigenvalues & eigenfunctions, orthogonality & completeness, time-dependent
Schrödinger equation, probabilistic interpretation, compatibility of
observables, the uncertainty principle.
2. 1-D QM Bound states, potential barriers, tunnelling phenomena.
3. Orbital angular momentum
Commutation relations, eigenvalues
of Lz and L2, explicit forms of Lz and L2 in spherical polar coordinates, spherical
harmonics Yl,m.
4. Spin
Noncommutativity of spin operators, ladder operators, Dirac notation,
Pauli spin matrices, the Stern-Gerlach experiment.
5. Addition of angular momentum
Total angular momentum
operators, eigenvalues and eigenfunctions of Jz and J2.
6. The hydrogen atom revisited
Spin-orbit coupling, fine structure,
Zeeman effect.
7. Perturbation theory
First-order perturbation theory for energy levels.
8. Conceptual problems
The EPR paradox, Bell’s inequalities.
Coupling two angular momenta
When M (= m1 + m2) is a constant of motion, m1 and m2 are
not well defined
S
L
We shall try and follow this convention:
Capitals J, L, S indicate angular momentum vectors
with magnitudes that can be expressed in units of
ħ:
L2 = l ( l + 1) ħ2
Lower case j, l, s indicate quantum numbers that
are integer or half-integer:
l = 0, 1, 2, 3…
s = 1/2
j = 1/2 , 3/2, 5/2
Lower case vectors j, l, s indicate vectors whose
components along a quantization axis are integer or halfinteger values (ie not in units of ħ).
We’ll do this now
Choice of basis for atomic electrons
Clebsch-Gordan coefficients
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