PHYS 30101 Quantum Mechanics
“the dreams stuff is made of”
Dr Jon Billowes
Nuclear Physics Group (Schuster Building, room 4.10)
j.billowes@manchester.ac.uk
These slides at: www.man.ac.uk/dalton/phys30101
PC3101 Quantum Mechanics
Recommended texts:
A.I.M Rae, Quantum Mechanics (4th edition, IOP)*
F. Mandl, Quantum Mechanics
P.C.W. Davies, Quantum Mechanics
A.P. French & E.F. Taylor, An Intro. To Quantum Mechanics
Beyond level of course:
S. Gasiorowicz, Quantum Mechanics
(but recommended for PC3602)
*supplementary material has moved to
www.crcpress.com/e_products/downloads/webdownload/IP609%5CRAEextra.pdf
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.
The Schrödinger Equation was guessed by induction:
Seemed plausible
test
works
OK, accept until falsified
Classical plane wave (sound or light)
obeys the wave equation
The solution requires
But does not work for matter waves where we want
Requires 2nd derivative of x but only 1st derivative of t
Idea! Let’s try
(TDSE: Time-dependent
Schrödinger equation)
No prediction of quantum mechanics has ever been experimentally falsified
Wave-particle duality applies to all objects:
screen
Particle detected
at single point on
the screen – the
probability wave
particle
instantaneously
collapses to zero
everywhere else.
If undisturbed, the particle propagates as a
(probability) wave. Development of the wave
with time is exactly described by the TDSE
Electron interference:
http://www.hqrd.hitachi.co.jp/em/doubleslit.cfm
Quantum interference experiments:
Single photons  electrons  neutrons  atoms  Buckminster fullerene (C60)

Conceptual problems with quantum mechanics
Quantum mechanics works - but there are many worries on
interpretation that tend to become matters of opinion; debate
enters realm of philosophy.
Conceptual basis of QM is fundamental to our understanding of the
nature of the physical universe – so we should try and learn more
by experiment, not debate (EPR paradox and Bell’s inequalities).
Alastair Rae: (in 4th edition of his text book) “my own
understanding continues to grow…”
Richard Feynman: “I think I can safely say that nobody
understands quantum mechanics.”
Niels Bohr: “Anyone who is not shocked by quantum mechanics
has not understood it.”
Feynman (again): “…shut up and calculate.”
After revision of basics, first new topic will be
Quantum Mechanical Tunnelling
Consider a roller-coaster…
Classically the car can only
go as far As C before rolling
back – but quantumfluctuations in energy could
allow the car through the
energy-forbidden region
and appear at E.
This quantum process controls the rates of alphadecay and spontaneous fission in nuclear physics.
Application: Scanning tunnelling microscope
Iron atoms on copper
Useful formulae
TDSE – time dependent
Schrödinger Equation
Vector operators in spherical polar
coordinates
Angular momentum
operators in spherical polars
TISE – time independent S.E.
Plan of action
1. Basics of QM
Will be covered in the following order:
2. 1D QM
1.1 Some light revision and reminders
1.2 TISE applied to finite wells
1.3 TISE applied to barriers – tunnelling phenomena
1.4 Postulates of QM
(i) What Ψ represents
(ii) Hermitian operators for dynamical variables
(iii) Operators for position, momentum, ang. mom.
(iv) Result of measurement
1.5 Commutators, compatibility, uncertainty principle
1.6 Time-dependence of Ψ