Home: Quantum Mechanics I, 2014

Purpose of the course

This is the first course in quantum mechanics. The purpose of this course is to introduce basic quantum phenomena and understand its ramifications. At the end of this course you should be able to reason about any quantum system, and to solve simple systems.

Many of you may know most of this material. You can skip the drudgery of sitting through a course by opting to take a drop test. Please ask the course coordinator about how to register for this test.

Prerequisites

It is assumed that you know the basic physics of waves and can think of classical mechanics in terms of phase space and Hamilton's equations. Everything else you need, you will learn during the course.

You will have to learn to use Mathematica. Here is a quick tutorial introduction to Mathematica: download it and try it out. In order to do this you will need an account on the servers in the computer center.

You may have to do a lot of searching and reading on the web. Read this web page on how to begin to check your search results.

Method of teaching

The method I will use to teach may not be something you are used to. I plan to have no lectures. Instead, "lecture notes" are already put on the web; you will find them as you scroll down this page. Before each lecture you are expected to read the notes and the chapters of books suggested in them. During the contact hours we will only discuss the problems which you have had in working by yourself.

Evaluation

Your performance will be continually evaluated by assessments and tests. Instead of a term paper, you will be required to edit and continuously maintain a page on Wikipedia. For more details see the course page on assignments.

Course material

Setting up quantum mechanics

Lecture 1: States, Amplitudes and Interference

The classical mechanics of a point particle, classical waves, the double slit experiment and quantum particles. [Animated, printable].

Classical mechanics, Hamiltonian formulation, phase space, state, classical waves, wave train, amplitude, wave-number, Fourier transforms, power spectrum, position-wavenumber duality, double slit experiment, Born waves, Planck hypothesis, de Broglie hypothesis, Planck's constant, Heisenberg uncertainty relations, the old quantum theory.

Lecture 2: Quanta, Waves and Vector Spaces

Exploring the double-slit experiments, vector spaces and the addition of waves. [Animated, printable].

Interference, waves, which-path information, polarizers, quantum eraser, delayed-choice quantum eraser, vector space, inner product, square-intgrable function, linear independence, basis, orthonormal basis, Kronecker delta, coordinates, rotations.

Lecture 3: Vector spaces and Operators

Setting up the mathematical formalism, exploring the formalism. [Animated, printable].

Wave function, quantum state, ket, bra, bracket, c-number, linear operators, orthogonal operators, unitary operators, Hermitean operators, rays, projective space, eigenvalue, eigenvector, Gram-Schmidt process, commutator, vector space, Hilbert space, completeness, Cauchy sequence.

Lecture 4: The mathematics of 2-state systems

Examples of 2-state systems, the 2-dimensional complex vector space, time evolution and Schrödinger's equation. [Printable].

Double-slit experiment, H₂⁺, NH₃, benzene, K₀-K₀bar oscillations, 2-state systems, Pauli matrices, exponentials of matrices, Planck's hypothesis, evolution operator, Schroedinger's equation.

Lecture 5: The physics of 2-state systems

Basics of symmetries, Rabi oscillations, density matrices. [Animated, printable].

Unperturbed energy, mixing parameter, resonant stabilization, avoided level crossings, Rabi formula, gyromagnetic ratio, Larmor precession, Bloch sphere, canonical ensemble, heat bath, Curie-Weiss law.

Lecture 6: Systems with continuous infinity of states

Infinite dimensional Hilbert space, a free particle. [Printable].

Wave function, Fourier transformations, Baker-Campbell-Hausdorff formula, group, Abelian group, continuous group, isomorphism, translation invariance, free particle Hamiltonian, parity, projection operators, wavepackets, lattice, quantization.

Lecture 7: Units and dimensions

Universal constants, quantum mechanics, atomic physics. [Printable].

Gaussian units, Systeme Internationale, natural units, black body radiation, Stefan-Boltzmann law, Rayleigh-Jeans law, ultraviolet catastrophe, Bohr radius, Rydberg energy, Compton wavelength, spherical harmonics.

Simple applications

Lecture 8: Simple one-dimensional potentials

Quantum mechanics in one space dimension, a potential step, a potential barrier. [Printable].

Matching conditions, transfer matrix, reflected wave, transmitted wave, reflection coefficient, transmission coefficient, relative phase, sub-barrier transmission, resonances, universal behaviour.

Lecture 9: Simple one-dimensional potentials

Energy bands in periodic potentials, the harmonic oscillator, a charged particle in a magnetic field, the isotropic two-dimensional harmonic oscillator. [Printable].

Bloch's theorem, lattice vector, finite range potential, scattering matrix, transmission matrix, resonances, harmonic oscillator, Rayleigh coefficient, ground state, generating function, Hermite polynomials, recurrence relation, Schroedinger representation, Heisenberg picture, cyclotron frequency, Landau levels, the group SU(2), representations of SU(2), generators of SU(2).

Lecture 10: Rotations and Angular momentum

The algebra of angular momentum operators, rotations, properties of operators under rotations. [Printable].

Angular momentum operators, raising and lowering operators, scalar representation, spinor representation, translation operator, rotation operator, Euler angles, Wigner's D matrices, Wigner's d matrices, scalar operator, vector operator, Wigner-Eckart theorem, reduced matrix element.

Lecture 11: Adding angular moments

Some definitions, the simplest example, interesting physics. Printable.

Irreducible representations, reducible representations, direct products, tensor products, direct sums, block-diagonal matrices, two particle system, Clebsch-Gordan series, Clebsch-Gordan coefficients.

Lecture 12: Spherical symmetry

Rotationally invariant potentials, the free particle, the Coulomb problem. [Printable].

Rotationally invariant potential, reduced mass, hidden symmetry, radial momentum, effective potential, centrifugal barrier, free particle, short-ranged potentials, phase shifts, spherical square well, Coulomb potential, elliptic orbits, hyperbolic orbits, Laguerre polynomials, spin, Lorentz group, Coulomb scattering states, the Runge-Lenz vector.

Lecture 13: The use of symmetries

The use of symmetries in quantum mechanics. [Printable].

Hilbert space, group, transformation of coordinates, group multiplication table, degenerate eigenvalues, invariants, symmetry groups, symmetries of Hamiltonians, symmetries under interchange, permutation groups, degeneracies of energy levels, irreducible representations, Clebsch-Gordan series, Clebsch-Gordan coefficients, isotropic harmonic oscillator, Lie group, orthogonal group, symplectic group, unitary group, SO(3), Sp(6,R), U(3), bilinear forms.

Many-particle systems

Lecture 14: Identical particles

The problem of identical particles and its resolution, examples, second quantization. [Printable].

Intrinsic properties, interchange operator, bosons, fermions, spin-statistics theorem, permutation, Slater determinant, Pauli's exchange principle, exchange effects, shell model, colour quantum number, Fock space, quantum field theory, many-body theory.

Lecture 15: Path integrals

The time-evolution operator, Feynman's path integral, the harmonic oscillator. [Printable].

Time evolution, evolution operator, path integral, Euclidean time, transfer matrix, diagrammatic method, action, Feynman's path integral, Gaussian integrals, classical limit, quantum fluctuations.

Lecture 16: Entanglement

Seperating particles, entanglement, Bell's inequalities. [Printable].

Preparin a state, quantum entanglement, entangled pair, correlation function, hidden variable theory, Bell's inequality.

Solving unsolvable problems: the easy techniques

Lecture 17: Perturbation Theory

The set up, first order perturbation theory, second order perturbation theory. [Printable].

Lorentz force, cyclotron frequency, magnetic moment, Bohr magneton, gyromagnetic ratio, hydrogenic atom, Magnetars, Pascal's triangle, renormalization constant, null vector, matrix deflation, projection operator, null spave, pseudo-inverse, spin, Lande factor, spin-orbit coupling, fine structure of spectra, magic numbers, nuclear shell model, semiconductor heterostructures, Rashba effect, tensor potential, hyperfine structure, nucleon-nucleon potentials.