PHYS 30101 Quantum MechanicsPHYS 30101 Quantum Mechanics“the dreams stuff is made of”“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

http://www.hqrd.hitachi.co.jp/em/doubleslit.cfm

Wave-particle duality applies to all objects:

particle

screen

Particle detected at single point on the screen – the probability wave 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:

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 quantum-fluctuations in energy could allow the car through the energy-forbidden region and appear at E.

This quantum process controls the rates of alpha-decay and spontaneous fission in nuclear physics.Application: Scanning tunnelling microscope

Iron atoms on copper

Useful formulae

TDSE – time dependent Schrödinger Equation TISE – time independent S.E.

Vector operators in spherical polar coordinates

Angular momentum operators in spherical polars

Plan of action

1. Basics of QM

2. 1D QMWill be covered in the following order:

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