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PHYS 30101 Quantum Mechanics
Lecture 22
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.
Einstein, Podolski, Rosen (EPR) problem and
Bell’s theorem
What is the correct interpretation of quantum
mechanics?
System itself does not
know position of particle
or direction of spin until it
is forced to decide by our
measurement
(our present QM opinion)
System has “hidden
variables” which we do not
know but which gives the
system an underlying
deterministic structure. We
hide our ignorance by
describing the “most
probable” outcomes of
measurement
John Bell proposed a theorem “it is impossible for any local hidden
variable theory to reproduce all predictions of QM” which could be
tested experimentally by comparing the outcomes of spin-polarization
measurements of pairs of “entangled” particles.
Systems exist which emit pairs of particles in an overall spin = 0
state although each particle has non-zero spin
Examples:
Low-energy proton s-wave
scattering from hydrogen
π0
(e-
e +)
Back-to-back emission of photons
Electron-positron annihilation from 1S0 state
Consider a source that emits two photons in mutually perpendicular
polarization states. What happens when the polarization of one
photon is measured
A
B
source
QM: Particles do not decide on polarization direction until one (A)
is measured. This instantaneously (faster than light) forces B into
the orthogonal state. This can be confirmed by a polarization
measurement.
Local hidden variables: Assumes mutually orthogonal polarization
directions are determined as photons are emitted.
QM prediction confirmed by series of experiments by Alain Aspect
(France) testing polarizations against Bell’s Inequalities relations
So, either a problem with quantum mechanics or relativity
(instantaneous collapse of wavefunction is a problem)
But so far NO QM prediction has been experimentally falsified.
(ditto relativity)