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4C26 Advanced Quantum Theory

PHYS4226 Advanced Quantum Theory

Prerequisites
To have attended and passed the department's introductory quantum mechanics courses, PHYS2B22: Quantum Physics and the intermediate course, PHYS3C26: Quantum Mechanics, or equivalent courses in another department. The following topics will be assumed to have been covered:

Introductory material: states, operators and the Born interpretation of the wave function, probability current density, transmission and reflection coefficients;
Harmonic oscillator: by the differential equation approach giving the energy eigenvalues and wave functions;

Angular momentum: angular momentum operators and the spectrum of eigenvalues, the spherical harmonics and hydrogenic wave functions;
Time-independent perturbation theory: including the non-degenerate and degenerate cases and its application to the helium atom ground state, Zeeman effect and spin-orbit interactions;

Aims of the Course

This course aims to:

review the basics of quantum mechanics so as to establish a common body of knowledge for the students from the different Colleges on the Intercollegiate M.Sci. programme; extend this by discussing these basics in more formal mathematical terms;
develop techniques for non-perturbative solutions of the time-dependent Schrodinger equation.

Introduce time-dependent perturbation theory
introduce the JWKB method for non-perturbative approximations;
discuss the addition of angular momentum and Clebsch-Gordan coefficients;
discuss the quantum mechanical description of the non-relativistic potential scattering of spinless particles;

provide the students with basic techniques in these areas which they can then apply in specialist physics courses.

Objectives
After completing the module the student should be able to:

state mathematically the expansion postulate and to give a physical interpretation to the quantities and explain what is meant by compatible/commuting observables;
understand and use the Dirac notation for quantum states;
generalize the definition of angular momentum to include spin and solve the generalized angular momentum eigenvalue problem employing raising and lowering operator techniques;
discuss the properties of spin-1/2 systems and use the Pauli matrices to solve simple problems;

state the rules for the addition of angular momenta and to outline the underlying general, mathematical arguments, applying them in particular to two spin-1/2 particles; discuss and apply the JWKB approximation;
formulate first-order time-dependent perturbation theory and extend the method to second-order theory. Show, as an example, how it can lead to Fermi's Golden Rule; apply the theory to harmonic perturbations (e.g. quantum system interacting with an electromagnetic wave);

define cross section and scattering amplitude;
give a quantum mechanical description of the scattering process via the partial wave expansion and phase shifts and to be able to apply it to the cases of low-energy scattering of spinless particles from simple potentials;
develop and apply the first Born approximation for the cross section.

Methodology and Assessment
The module consists of 30 lectures. These will be used to cover the syllabus material and to discuss problem sheets as the need arises. The assessment is based on an unseen written examination paper (90%) and (approximately) weekly coursework questions (10%) applying the methods discussed in the lectures. The overall coursework mark is derived from the best 80% of the solutions submitted by each student to the questions set. Sample answers are provided for all questions attempted.

Textbooks:
The one closest to the material and level of the course is

Introduction to Quantum Mechanics, B.H. Bransden and C.J.Joachain, Longman (2nd Ed, 2000).

Others which are also close to the course are (in alphabetical order):

Quantum Mechanics, (2 Vols) C.Cohen-Tannoudji, B.Diu and F.Laloe, Wiley,
Quantum Physics, S.Gasiorowicz, Wiley (1996),
Quantum Mechanics, F.Mandl , Wiley (1992),
Quantum Mechanics, E.Merzbacher, (3rd Ed.) Wiley, (1998)

Syllabus

(The approximate allocation of lectures to topics is shown in brackets below. Basic ideas of quantum mechanics (partly revision) and formal quantum mechanics[2] (Formal aspects of quantum theory are distributed throughout the course and introduced as needed.) Bras and kets, states, operators, Born interpretation of the wave function, continuous and discrete eigenvalues, Dirac delta function, compatible observables, Hermitian and unitary operators, Dirac notation, closure relation, transformation brackets, momentum representation.

Non-perturbative solutions of the time-dependent Schrodinger equation [6]
The time-evolution operator. Applications : 1) to conservative systems and free wavepackets 2) using matrix form of the operator. 3) using split-operator method 4) to time-periodic systems: Floquet theory.

Time-dependent perturbation theory [6]
First-order time-dependent perturbation theory. Harmonic perturbations and other applications of time-dependent perturbation theory. Second-order perturbation theory and energy denominators. Fermi's Golden Rule.

Non-perturbative approximations [5]
The JWKB approximation. Examples.
Scattering [6]

Currents and cross sections; the scattering amplitude and the optical theorem. Partial wave expansion of wave function and scattering amplitude. Phase shifts. Low-energy scattering from square well potential and scattering length expansion, bound states and resonances. First Born approximation from the time-independent approach. Integral equation for potential scattering.

Angular momentum (partly revision) [5]
Angular momentum operators, commutation algebra, raising and lowering operators,
spectrum of angular momentum eigenvalues, combination of angular momenta treating the simplest case of two spin-1/2 particles, notation of Clebsch-Gordan coefficients, spin-1/2 angular momentum and Pauli matrices.

Other advanced QM Lecture notes:

Lecture notes for the 4C26 Advanced QM MSci Physics module:

  • Sec. 1 Floquet states
  • Section 2: Time-Dependent Perturbation Theory
  • CST Perturbation
  • Sec. 3a WKB Approximation
  • Sec. 3b Application of WKB to bound states
  • Sec. 4a Revision of Angular Momentum
  • Sec. 4b Pauli matrices
  • Sec. 4c Addition of AM
  • Sec. 5 Quantum scattering

    Revision questions for each section:

    Revision for Sec. 1
    Revision for Sec. 2
    Revision for Sec. 3
    Revision for Sec. 4
    Revision for Sec. 5

    4C26 Exam 2005 - Section B

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