Quantum Mechanics — Lecture notes for PHYS223

Henning Schomerus, Lancaster University

Abstract

These lecture notes lay out the mathematical and interpretational framework of quantum mechanics and describe how this theory can be utilised to describe common physical systems and phenomena. Starting from the Schrödinger equation, we study a number of exactly solvable problems, including one-dimensional potentials, angular momentum quantization, spin, and the hydrogen atom, and also provide systematic approximations for not exactly solvable problems. The physical content of the mathematical solutions is discussed in terms of probabilities and expectation values. The latter parts of the notes concern temporal dynamics and systems composed of more than one particle. Non-examinable sections are marked by an asterisk ( ${}^{*}$ ).

I Waves are particles, particles are waves
- I.1 Compton scattering: waves are particles
- I.2 The double-slit experiment with electrons: particles are waves
II The Schrödinger equation
- II.1 Wave equation for photons
- II.2 The Schrödinger equation
- II.3 Stationary Schrödinger equation
III Mathematical interpretation of the Schrödinger equation
- III.1 The Schrödinger equation as a differential equation
- III.2 The Schrödinger equation as an eigenvalue problem
- III.3 Momentum eigenfunctions
- III.4 Position eigenfunctions and the Dirac delta-function
IV A first example: particle in the square well
- IV.1 The particle in the one-dimensional square well
- IV.2 Discussion: The probability interpretation of the wave function
- IV.3 Momentum and energy
V Further examples: bound states, extended states, and tunneling
- V.1 Constant potential
- V.2 Constant potential, terminated by a hard wall
- V.3 Constant potential, terminated by a soft wall
- V.4 Tunnelling through a potential barrier
- V.5 Two hard walls
- V.6 General classification of states
VI The harmonic oscillator
- VI.1 Classical harmonic oscillator
- VI.2 Schrödinger equation of the quantum harmonic oscillator
- VI.3 Mathematical solution
- VI.4 Ground state and first excited state
- VI.5 Normalisation of the ground state wave function
- VI.6 General form of the bound states
VII Momentum probabilities and the uncertainty principle
- VII.1 Momentum wave function
- VII.2 Expectation values of position and momentum
- VII.3 Uncertainty
- VII.4 Heisenberg’s uncertainty relation
- VII.5 Example I: Minimal-uncertainty wave packet
- VII.6 Example II: ground state of the particle in the square well
- VII.7 Energy-time uncertainty relation
VIII Mathematical interlude: Hilbert spaces and linear operators
- VIII.1 Vectors
- VIII.2 Scalar product, norm and orthogonality
- VIII.3 Basis
- VIII.4 Linear operators
- VIII.5 Eigenvalues and eigenvectors
- VIII.6 Common types of operators
IX General principles of quantum mechanics
- IX.1 Quantum states
- IX.2 Observables and operators
- IX.3 Dynamics
- IX.4 Measurements
X Consequences of the measurement postulate
- X.1 Expectation values
- X.2 Consecutive measurements
- X.3 Commutator
- X.4 Generalised uncertainty principle
- X.5 Simultaneous and incompatible variables
- X.6 Complete determination of a state by measurements
XI Perturbation theory
- XI.1 Objective
- XI.2 Method
- XI.3 Summary of the result
- XI.4 Example I: worn out harmonic oscillator
- XI.5 Example II: harmonic oscillator exposed to a constant force
- XI.6 Example III: anharmonic oscillator
XII Quantum mechanics in three dimensions
- XII.1 Coordinates and wavefunction
- XII.2 Position operators
- XII.3 Momentum operators
- XII.4 Commutators between position and momentum
- XII.5 Momentum eigenstates
- XII.6 Dirac notation
- XII.7 Schrödinger equation in three dimensions
XIII Three-dimensional examples and applications
- XIII.1 Example: free particle in three dimensions
- XIII.2 Separation of variables in cartesian coordinates
- XIII.3 Example: particle in the box
- XIII.4 Degeneracy
- XIII.5 Density of states
- XIII.6 Example: harmonic oscillator
XIV Central potentials
- XIV.1 Spherical polar coordinates
- XIV.2 Separation of variables in spherical polar coordinates
- XIV.3 Angular part
- XIV.4 Quantisation of angular momentum
- XIV.5 Degeneracy of the radial part
XV Angular momentum
- XV.1 Classical angular momentum and magnetic moment
- XV.2 Angular momentum operators
- XV.3 Angular momentum eigenfunctions
- XV.4 Angular momentum in Dirac notation
- XV.5 Magnetic moment
XVI Spin
- XVI.1 Stern-Gerlach experiment
- XVI.2 Spinor
- XVI.3 Spin operators
- XVI.4 Eigenvalues and eigenvectors of $\hat{\bf S}^{2}$ and $\hat{S}_{z}$
- XVI.5 Eigenvalues and eigenvectors of $\hat{S}_{x}$ and $\hat{S}_{y}$
- XVI.6 Expectation values
- XVI.7 Polarisation vector and Bloch sphere
- XVI.8 Stationary Schrödinger equation
- XVI.9 Spin in Dirac notation
- XVI.10 Mathematical appendix: two-dimensional linear algebra
XVII Hydrogen atom
- XVII.1 Separation of relative and centre-of-mass motion
- XVII.2 Separation of radial and angular motion
- XVII.3 Radial motion
- XVII.4 Energies
- XVII.5 Spectral lines
- XVII.6 Degeneracy
- XVII.7 Atomic orbitals
- XVII.8 Application: The Zeeman effect
XVIII Dynamics for stationary Hamiltonians
- XVIII.1 General solution via eigenstates
- XVIII.2 General solution via the time-evolution operator
- XVIII.3 Example I: Dynamics of the free particle ${}^{*}$
- XVIII.4 Detour: the time of flight ${}^{*}$
- XVIII.5 Example II: coherent state dynamics in the harmonic oscillator ${}^{*}$
- XVIII.6 Example III: Spin precession (dynamics of a two-state system)
XIX Dynamics for time-dependent Hamiltonians
- XIX.1 Rabi oscillations
- XIX.2 General solution via the time-evolution operator
- XIX.3 Ehrenfest theorem and Heisenberg picture
- XIX.4 Time-dependent perturbation theory and Fermi’s golden rule
  - XIX.4.1 Sudden perturbation
  - XIX.4.2 Harmonic perturbation
- XIX.5 Radiative transitions
XX Dirac notation for composite systems
- XX.1 Many degrees of freedom
  - XX.1.1 Factorisation and entanglement
- XX.2 Density matrix ${}^{*}$
- XX.3 Example: Two-state systems ${}^{*}$
- XX.4 Composite systems ${}^{*}$
- XX.5 Entanglement in composite systems ${}^{*}$
XXI Many particles
- XXI.1 Distinguishable particles
- XXI.2 Indistinguishable particles
- XXI.3 Pauli exclusion principle
- XXI.4 Two-electron orbitals
- XXI.5 Two electrons in a harmonic-oscillator potential
- XXI.6 The periodic table of chemical elements
- XXI.7 Quantum statistical mechanics ${}^{*}$

Quantum Mechanics — Lecture notes for PHYS223

Abstract

Contents