PHY2002 Quantum Physics I

2007-2008

Aims

Quantum Mechanics is one of the fundamental building-blocks of Physics. It affects profoundly the way we think about the universe and is the basis for much of condensed-matter, nuclear and statistical physics. It also has a strong influence on technological developments, for instance in optical and electronic devices. The purpose of this module is to introduce students to the basic principles of quantum mechanics and to the solutions of simple one-dimensional problems, which permit simple mathematical analysis. These solutions demonstrate many of the general features of the subject and prepare the students for subsequent courses (such as PHY3201) in which three-dimensional problems are introduced and more sophisticated methods employed.

Intended Learning Outcomes

Students will be able:

• to describe of the concepts and roles of the wavefunction and of operators in quantum mechanics,
• to discuss the origin of energy quantisation and quantum tunnelling effects,
• to describe the general properties of the stationary states of quantum particles confined to simple symmetric potentials,
• to perform calculations on wavefunctions, and in the solution of the Schrödinger equation for a range of one-dimensional problems.

Transferable Skills

• Application of the principles of quantum mechanics to unfamiliar problems,
• Problem-solving, in lecture examples, problems classes and tutorials,
• Students are required to meet deadlines for completion of work for problems classes and must therefore develop appropriate time-management strategies.

Learning and Teaching Methods

Lectures, tutorials and problems classes, e-learning resources. Self-Study software to demonstrate many of phenomena described in this module is available in the folder 'Applications:McMurry QM' in the Computer Laboratory.

Assignments

Exercise sheets handed out in lectures. Problem assignments for problem classes.

Assessment

Problems-class assignments (10%), one 30-minute mid-semester test (20%) and one 90-minute examination (70%)

Syllabus Plan and Content

Note: references to sections in the recommend texts are given in square brackets.

1. Wave Description of Matter
   1. Introduction: recap of key quantum phenomena described in PHY1118 [M ch 1, R ch 1]
   2. The wave function [M2.1, R2.1]
   3. Motion of a free particle [M2.4, R2.1]
   4. The free-particle wave equation [M3.2, R2.1]
   5. Quantum motion in a potential [M3.2, R2.1]
   6. The time-dependent Schrödinger equation [M3.2, R2.1]
2. Interpretation of the Wave Function
   1. The Born probability interpretation [M2.2, R2.1]
   2. Normalization of the wavefunction [M2.2,2.3,2.4, R2.1,4.1]
   3. Localization of a particle; wave packets [M9.4]
   4. The uncertainty principle again [M9.4]
   5. Time evolution of wave-packets [M9.4]
3. Dynamical variables
   1. Variables as operators [M3.1, R4.2]
   2. Introduction to eigenvalue equations [M3.1, R4.2]
   3. The Schrödinger equation again [M3.2, R4.2,4.6]
   4. The second postulate [M3.1, R 4.2]
   5. Summary: the quantum recipe [M3.2, R 4.2, 4.6]
4. Stationary States and the Time-Independent Schrödinger Equation [M3.2]
   1. Time-independent probability distributions [R2.2]
   2. The time-independent Schrödinger equation [R2.2]
   3. Stationary states R2.2,4.6]
5. Solution of the Time-independent Schrödinger Equation [M3.3]
   1. Independent solutions
   2. Method of solution
   3. Boundary conditions [R2.3]
   4. Example: region of constant potential
6. Particle in a Box (Infinite Square Well) [M3.3, R 2.4]
   1. Internal solution
   2. Boundary conditions
   3. Energy quantization
   4. Normalized wave functions
   5. Wave-function forms
   6. Nodes: symmetry about centre
7. Finite Square Potential Well [M3.3, R 2.4]
   1. Symmetric and antisymmetric solutions
   2. Interior and exterior solutions
   3. Boundary conditions
   4. Symmetric solutions - eigenvalue equations
   5. Wave functions and normalization
   6. Antisymmetric solutions - eigenvalue equations
   7. Pictorial forms of wave functions
   8. Qualitative conclusions
8. Flux of Particles [M3.2, R 9.1]
   1. Probability flux
   2. The continuity equation
   3. Persistence of normalization
9. Barrier Problems [M3.4]
   1. Boundary conditions at potential discontinuity
   2. The potential step
   3. Tunnelling; reflection and transmission by a barrier
   4. Practical examples of transmission resonances and tunnelling
10. Quantum Measurement and the Structure of Quantum Mechanics
    1. Eigenstates and eigenvalue equations [M3.1, R4.2]
    2. Properties of Hermitian operators [R4.3]
    3. The superposition principle [M9.1, R4.4]
    4. Measurements on a general quantum state [M3.1,9.2]
    5. Commutation relations and simultaneous observables [M4.1, R4.5]
    6. The uncertainty principle [R4.5]
    7. Commutation with the Hamiltonian [M12.1]
11. Quantum harmonic oscillator
    1. The Hamiltonian in operator form [M5.1]
    2. Eigenvalues and eigenfunctions [M5.2]

Core Text

Supplementary Text(s)

Formative Mechanisms

This module is supported by problems classes and tutorials. Students are able to monitor their own progress by attempting problems sheets provided in the lectures. The graded mid-semester test scripts are discussed by tutors. Students with specific problems should first approach their tutor, and if the problem persists, the lecturer.

Evaluation Mechanisms

The module will be evaluated using information gathered via the student representation mechanisms, the staff peer appraisal scheme, and measures of student attainment based on summative assessment.