*Michaelmas Term,* 24 Lectures – Prof. A. Davis

*Further information about this course is available on the Department of Mathematics course pages. Examples papers are available on the DAMTP Examples page.*

**Dirac formalism**

Bra and ket notation, operators and observables, probability amplitudes, expectation values, complete commuting sets of operators, unitary operators. Schrödinger equation, wave functions in position and momentum space. [3]

Time evolution operator, Schrödinger and Heisenberg pictures, Heisenberg equations of motion. [2]

**Harmonic oscillator**

Analysis using annihilation, creation and number operators. Significance for normal modes in physical examples. [2]

**Multiparticle systems**

Composite systems and tensor products, wave functions for multiparticle systems. Symmetry or antisymmetry of states for identical particles, Bose and Fermi statistics, Pauli exclusion principle. [3]

**Perturbation theory**

Time-independent theory; second order without degeneracy, first order with degeneracy. [2]

**Angular momentum**

Analysis of states |*jm*〉 from commutation relations. Addition of angular momenta, calculation of Clebsch–Gordan coefficients. Spin, Pauli matrices, singlet and triplet combinations for two spin half states. [4]

**Translations and rotations**

Unitary operators corresponding to spatial translations, momenta as generators, conservation of momentum and translational invariance. Corresponding discussion for rotations. Reflections, parity, intrinsic parity. [3]

**Time-dependent perturbation theory**

Interaction picture. First-order transition probability, the golden rule for transition rates. Application to atomic transitions, selection rules based on angular momentum and parity, ¤absorption, stimulated and spontaneous emission of photons¤. [3]

**Quantum basics**

Quantum data, qubits, no cloning theorem. Entanglement, pure and mixed states, density matrix. Classical determinism versus quantum probability, Bell inequality for singlet two-electron state, GHZ state. [2]

**Books**

† E. Merzbacher *Quantum Mechanics*, 3rd edition. Wiley 1998

† B.H. Bransden and C.J. Joachain *Quantum Mechanics*, 2nd edition. Pearson

J. Binney and D. Skinner *The Physics of Quantum Mechanics*. Cappella Archive, 3rd edition

P.A.M. Dirac *The Principles of Quantum Mechanics*. Oxford University Press 1967, reprinted 2003

C.J. Isham *Lectures on Quantum Theory*: Mathematical and Structural Foundations. Imperial College Press 1995

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