September 26, 2012

This is the web page for Advanced Quantum Mechanics (PHYS30201) for the session 2012/13.

The notes are not intended to be fully self-contained, but summarise lecture material and give pointers to textbooks.

NEW: The full pdf file of the notes is here.

These notes have been prepared with TEX4ht, and use MathML to render the equations. Try the link to “This course and other resources” to see if your browser is compatible. (In particular there is a problem with square roots of fractions that you should check is OK.) If you are using Internet Explorer, you may need to download “MathPlayer” from here.

Please report errors to Judith McGovern.

This course and other resources.

Operator methods in Quantum Mechanics

1.1 Postulates of Quantum Mechanics

1.2 Position and Momentum Representations

1.3 The Stern-Gerlach experiment

1.4 Ehrenfest’s Theorem and the Classical Limit

Approximate methods I: variational method and WKB

2.1 Variational methods: ground state

2.2 Variational methods: excited states

2.3 Variational methods: the helium atom

2.4 WKB approximation

2.4.1 WKB approximation for bound states

2.4.2 WKB approximation for tunnelling

Approximate methods II: Time-independent perturbation theory

3.1 Formalism

3.1.1 Simple examples of perturbation theory

3.2 Example of degenerate perturbation theory

3.3 The ﬁne structure of hydrogen

3.4 The Zeeman eﬀect: hydrogen in an external magnetic ﬁeld

3.5 The Stark eﬀect: hydrogen in an external electric ﬁeld

Approximate methods III: Time-dependent perturbation theory

4.1 Formalism

4.1.1 Perturbation which is switched on slowly

4.1.2 Sudden perturbation

4.2 Oscillatory perturbation and Fermi’s golden rule

4.3 Emission and absorption of radiation

4.3.1 Einstein’s $A$ and $B$ coeﬃcients

4.4 Radiative decay of $2p$ state of hydrogen

4.5 Finite width of excited state

4.6 Selection rules

4.7 Heisenberg vs Schrödinger pictures

Approximate methods IV: Scattering theory

5.1 Preliminaries

5.2 The Born approximation

5.3 Phase Shifts

5.3.1 Hard sphere scattering

5.3.2 Scattering from a ﬁnite square barrier or well

Quantum Measurement

6.1 The Einstein-Poldosky-Rosen “paradox” and Bell’s inequalities

Mathematical background and revision

A.1 Vector Spaces

A.1.1 Direct Products

A.2 Angular Momentum

A.3 Hydrogen wave functions

A.4 Harmonic oscillators, creation and annihilation operators

A.5 The Helium atom

A.6 Spherical Bessel functions

A.7 Airy functions

A.8 Properties of $\delta $-functions

A.9 Gaussian integrals

A.10 Density of states, periodic boundary conditions and black-body radiation

A.11 Checking units and scaling

A.12 Units in EM

Problem Sheets

Operator methods in Quantum Mechanics

1.1 Postulates of Quantum Mechanics

1.2 Position and Momentum Representations

1.3 The Stern-Gerlach experiment

1.4 Ehrenfest’s Theorem and the Classical Limit

Approximate methods I: variational method and WKB

2.1 Variational methods: ground state

2.2 Variational methods: excited states

2.3 Variational methods: the helium atom

2.4 WKB approximation

2.4.1 WKB approximation for bound states

2.4.2 WKB approximation for tunnelling

Approximate methods II: Time-independent perturbation theory

3.1 Formalism

3.1.1 Simple examples of perturbation theory

3.2 Example of degenerate perturbation theory

3.3 The ﬁne structure of hydrogen

3.4 The Zeeman eﬀect: hydrogen in an external magnetic ﬁeld

3.5 The Stark eﬀect: hydrogen in an external electric ﬁeld

Approximate methods III: Time-dependent perturbation theory

4.1 Formalism

4.1.1 Perturbation which is switched on slowly

4.1.2 Sudden perturbation

4.2 Oscillatory perturbation and Fermi’s golden rule

4.3 Emission and absorption of radiation

4.3.1 Einstein’s $A$ and $B$ coeﬃcients

4.4 Radiative decay of $2p$ state of hydrogen

4.5 Finite width of excited state

4.6 Selection rules

4.7 Heisenberg vs Schrödinger pictures

Approximate methods IV: Scattering theory

5.1 Preliminaries

5.2 The Born approximation

5.3 Phase Shifts

5.3.1 Hard sphere scattering

5.3.2 Scattering from a ﬁnite square barrier or well

Quantum Measurement

6.1 The Einstein-Poldosky-Rosen “paradox” and Bell’s inequalities

Mathematical background and revision

A.1 Vector Spaces

A.1.1 Direct Products

A.2 Angular Momentum

A.3 Hydrogen wave functions

A.4 Harmonic oscillators, creation and annihilation operators

A.5 The Helium atom

A.6 Spherical Bessel functions

A.7 Airy functions

A.8 Properties of $\delta $-functions

A.9 Gaussian integrals

A.10 Density of states, periodic boundary conditions and black-body radiation

A.11 Checking units and scaling

A.12 Units in EM

Problem Sheets