PH3061 Quantum Mechanics 1

Further information on which modules are specific to your programme.

Module description

This module introduces the main features of quantum mechanics. The syllabus includes: the emergence of the Schrödinger equation, the interpretation of the wave function and Heisenberg's uncertainty relation. The concepts of eigenfunctions and eigenvalues. Simple one-dimensional problems including potential wells and the harmonic oscillator. Solution of the Schrödinger equation for central forces, the radial Schrödinger equation, and the hydrogen atom.

Additional information from school

Overview

Quantum Mechanics is that description of physical phenomena in which the wave and particle aspects of matter and radiation are reconciled in a unified theory. As such it is one of the most fundamental topics in physics. It has widespread applicability in virtually every area of physics from the solid-state to fundamental particles, and is hence an essential item in the 'toolbox' of any practitioner in modern physics. As a theory it has never been shown to be incorrect in any of its predictions if properly employed. However, at a fundamental level it poses many challenges to our understanding, and this is an area of much current research. As well as playing a key role in describing many traditional areas of physics, deeper insights continue to emerge, leading to new applications such as quantum computing, quantum cryptography and quantum information processing. It is hence a topic of continuing fundamental interest and study in its own right as well as being essential for describing many areas of applied physics. The present course is the first in a sequence of four courses that progressively develop the topic to an advanced level.

The present course is based on the Schrödinger equation description of quantum mechanics (often referred to as wave mechanics), and is limited to non-relativistic situations. A more formal approach to describing quantum systems based on operator methods is also developed. Essential mathematical background is developed throughout the course.

Aims & Objectives

To present an introductory account of quantum mechanics (wave mechanics), in particular:

• To develop an intuitive understanding of such basic concepts as the wave function, probability density, operators, eigenfunctions and eigenvalues.
• To introduce both the time-independent and time-dependent Schrödinger equations and to develop an understanding of their meaning and how they are utilised.
• To apply the Schrödinger equation to a range of important physical situations, develop solutions and discuss their implication.
• To introduce the operator formalism and consider a number of its applications.

Learning Outcomes

You will have acquired the ability to:

• Write down and solve the Schrödinger equation for simple 1D, 2D and 3D systems, and use the wave functions to calculate expectation values and measurement probabilities for observables such as energy, position and momentum using Cartesian and spherical polar coordinates.
• Compare and contrast classical and quantum behaviour for simple 1D, 2D and 3D systems.
• Contrast eigenfunctions of an operator with superposition states with respect to that operator.
• Explain the relevance of Hermitian and non-Hermitian operators in quantum mechanics, and determine whether or not a given operator is Hermitian.
• State the measurement problem in quantum mechanics.
• Determine degeneracies of energy levels for simple 2D and 3D systems.
• Use properties of ladder operators for the 1D harmonic oscillator to determine wave functions and expectation values.
• State properties of the spherical harmonics and their relation to angular momentum.
• Author and critique high-quality quantitative, conceptual and interpretive multiple choice questions focusing on quantum mechanics developed in the course.

Synopsis

Uncertainty principle, time-dependent Schrödinger equation, momentum operator and Hamilton operator, time-independent Schrödinger equation, probability interpretation, normalization, expectation values, probability current

Wave packets and their propagation; group velocity and spreading

The infinite potential well: Eigenfunctions and energy eigenvalues, orthonormality and completeness of eigenfunctions, superposition of states, expansion theorem, Bohr's correspondence principle

Dirac notation, properties of Hermitian operators, the commutator, maximal set of mutually commuting operators

The 1- and 3-dimensional quantum harmonic oscillator: solution of the Schrödinger equation, discussion of eigenfunctions and eigenenergies, superposition state and wave packet oscillations, solution via ladder operators

The hydrogen atom: Schrödinger equation for the hydrogen atom, discussion of the hydrogen atom solutions, angular momentum

Additional information on continuous assessment etc.

Please note that the definitive comments on continuous assessment will be communicated within the module.  This section is intended to give an indication of the likely breakdown and timing of the continuous assessment.

This module is part of the core JH programme, and as such there is a summary of deadlines etc on the School’s Students and Staff web pages.   There is one class test, contributing 14% to the module mark, likely to be in week eight.   Successful engagement with a web-based question system counts for 6% of the module mark.  Students have a total of four compulsory tutorials.

Accreditation Matters

This module contains material that is or may be part of the IOP “Core of Physics”.  This includes

Heisenberg’s uncertainty principle

Wave function and its interpretation

Standard solutions and quantum numbers, to the level of the hydrogen atom

Quantum structure and spectra of simple atoms

Recommended Books

Please view University online record: https://sta.rl.talis.com/index.html