Quantum Mechanics II

Exams:	1. We 12.3. at 10-14 in aud E204
	2. Fr  16.5. at 10-14 in aud E204

Course description:

An advanced course in quantum mechanics. Review of fundamental formalism. Quantum dynamics. Symmetries, representations of the rotation group and theory of angular momentum. Many-particle systems. Relativistic quantum mechanics. [Quantum information]. Recommended for 3rd year theoretical physics students, 4th-5th year experimental physics students. The course Quantum mechanics I (or equivalent) is a prerequisite.

Textbooks:	J. Niskanen: Kvanttimekaniikka II (Limes 2003)  [N]
	J. J. Sakurai: Modern Quantum Mechanics (Addison Wesley 1994)  [S]
	B. H. Bransden and C. J. Joachain: Quantum Mechanics (Prentice Hall 2000)  [BJ]
	D. J. Griffiths: Introduction to Quantum Mechanics (Prentice Hall)  [G]
	I. J. R. Aitchison and A. J. G. Hey: Gauge Theories in Particle Physics, Vol. I: From Relativistic Quantum Mechanics to QED (IOP Publishing, 3rd Edition, 2003)  [AH]

Lecture topics:

22.1
The time evolution operator. Electron spin precession in a magnetic field. General solution for the eigenstates and eigenvalues of a two-state system. The non-crossing of levels and adiabatic mixing of states. Applications to solar neutrinos and the neutral kaon system. The time-energy uncertainty relation. The Heisenberg picture.

29.1
The time dependence of position and momentum operators in the Heisenberg picture. Non-commutation of an operator taken with itself at different times. Classical equations of motion for the expectation values of operators (Ehrenfest's theorem). Definition of the propagator K and its differential equation. The trace of the propagator and its connection with statistical physics in imaginary time (Euclidean space). The Fourier transformation of the propagator trace. The propagator as a transition amplitude.

3.2
The Feynman path integral: Intuitive derivation and check of the Schrödinger equation. Application to neutron interference in a gravitational field. Gauge symmetry in electrodynamics. Gauge transformations.

5.2
Gauge symmetry in electrodynamics. Gauge transformations. The Aharonov-Bohm effect.

10.2
Theory of angular momentum. [Lecture held by J. Niskanen.]

12.2
Theory of angular momentum. [Lecture held by J. Niskanen.]

17.2
Magnetic monopoles and the quantization of electric charge. SU(2) vs. SO(3). Euler rotations and rotation operators and matrix elements. Transformation of operators under rotations.

19.2
Cartesian vs spherical tensors. Multiplication of spherical tensors. The Wigner-Eckart theorem.

24.2
The projection theorem. The Bell inequality. Symmetries in quantum mechanics. Parity.

26.2
Parity (continued). Handed molecules. Discrete lattice translational symmetry.

NOTE: The first partial exam on Wednesday 12 March at 10-14 in aud E204 will comprise the material covered in the lectures up to and including parity symmetry.
      The problem session will be held on Tuesday 11 March at 14-16 in aud D112.

3.3
Discrete lattice translational symmetry. Bloch's theorem. Time reversal.

5.3
Time reversal (cont). Selection rules.

10.3
Review of material for first partial exam.

12.3
No lecture: First examination.

17.3
Kramers degeneracy. Density matrix. [Sakurai, Sec. 3.4]

19.3
Density matrix. Canonical ensemble. Identical particles [Niskanen 3.1]

24.3
Wave functions of identical bosons and fermions. Statistical quantum mechanics for distinguishable particles vs. identical bosons and fermions. [Griffiths, sec.5.4]

26.3
The Fermi-Dirac and Bose-Einstein distributions. Black-body spectrum. [Griffiths, sec.5.4]
Introduction to quantum fields. [Niskanen 3.2]

31.3
Creation and annihilation operators for bosons and fermions. Fock states. Commutation and anticommutation relations. Normal ordering. Wick's theorem. Field operators. [Niskanen 3.2]

2.4
Commutation and anticommutation of field operators. Hamiltonian and density operators. Time dependence. Propagator.
Introduction to relativistic quantum mechanics [Niskanen 4, Aichison and Hey 4]

7.4
Introduction to relativistic quantum mechanics [Niskanen 4, Aichison and Hey 4]
Lorentz transformations and connection to the SL(2,C) group. The Klein-Gordon equation, density and current. The Dirac equation, the alpha and beta matrices.

9.4
Plane wave solutions to the Dirac equation. Probability density and current. Spin, precession and helicity. Lorentz transforming the Dirac wave function.

14.4
The u and v Dirac spinors and their normalization. Lorentz covariant notation. Anticommutation relation of the Dirac matrices. Dirac bilinears.

16.4
The Dirac sea. Negative energy solutions as backward moving antiparticles. Interaction with an electromagnetic potential. Gauge invariance. The magnetic moment. Relativistic fields.

23.4 No lecture.

28.4
The magnetic moment from the Dirac equation. Precision measurements and calculations of the anomalous magnetic moment of the electron and neutrino. Relativistic fields.

30.4
The free relativistic scalar field. The classical equations of motion from the lagrangian density. The conjugate momentum and hamiltonian. Quantization. Field operators in terms of creation and annihilation operators. The propagator. The Dirac lagrangian and field operator.

5.5
The conjugate Dirac field and hamiltonian. Interactions. The Schrödinger, Heisenberg and Interaction pictures.

7.5
Review of the course. Please prepare questions!

Problems