| Lecturer: | Paul Hoyer | C321 | 191 50681 |
| Assistant: | Matti Järvinen | C311 | 191 50702 |
| Lectures: | Mo 12-14 | A315 | |
| We 12-14 | A315 | ||
| Exercises: | Mo 16-18 | E205 |
| Exams: | 1. | Friday 5 March | 14-18 | E204 |
| 2. | Tuesday 18 May | 10-14 | D101 |
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] |
Table of Clebsch-Gordan coefficients, spherical harmonics and gradients.
The first partial examination on Friday 5.3 covers the material that was presented at the lectures up to and including Wednesday 25 February.
Problems of partial exam 1/2003
1/2004
Note that problem set #7 covers material which is included in the first partial examination.
Problems of partial exam 2/2003
2/2004
Problems of final exams:
16.5.2003
4.6.2003
13.8.2003
2.4.2004
2.6.2004
16.6.2004
11.8.2004
NOTE: You are encouraged to give feedback on the course on this form.
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26.1 Exercise_1.pdf Exercise_1.ps |
2.2 Exercise_2.pdf Exercise_2.ps |
9.2 Exercise_3.pdf Exercise_3.ps |
16.2 Exercise_4.pdf Exercise_4.ps |
23.2 Exercise_5.pdf Exercise_5.ps |
1.3 Exercise_6.pdf Exercise_6.ps |
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15.3 Exercise_7.pdf Exercise_7.ps Workload.pdf |
22.3 Exercise_8.pdf Exercise_8.ps |
29.3 Exercise_9.pdf Exercise_9.ps |
5.4 Exercise_10.pdf Exercise_10.ps |
13.4 D106 at 16-18 Exercise_11.pdf Exercise_11.ps |
19.4 Exercise_12.pdf Exercise_12.ps |
|
26.4 Exercise_13.pdf Exercise_13.ps |
3.5. Exercise_14.pdf Exercise_14.ps |
| Lecture notes: | p1-146.pdf |
21.1
Applications to the spin 1/2 system using the Pauli matrix representation. 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.
26.1
Applications to solar neutrinos and the neutral kaon system. The time-energy uncertainty relation.
Quantum information. Private key cryptography. Secure transmission using random polarization axes.
Principles of public key (RSA) cryptography. Quantum computing.
28.1
Quantum teleportation. The Heisenberg picture. 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).
2.2
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.
4.2
The Feynman path integral: Intuitive derivation and check of the Schrödinger equation.
Quantum interference in a gravitational field.
9.2
Application of path integral to neutron interference in a gravitational field.
Gauge symmetry in electrodynamics. Kinetic momentum. Gauge transformations.
11.2
The Aharonov-Bohm effect. Magnetic monopoles and the quantization of electric charge.
16.2
Theory of angular momentum. Basics of group theory. SU(2) vs. SO(3).
18.2
Euler angles, rotation operators and matrix elements. Orbital angular momentum. Relation between spherical harmonics Y_\ell^m and rotation matrices D_{m_1 m_2}^{(j)}. Addition of angular momenta.
23.2
Identities following from the completeness and orthogonality of the D_{m_1 m_2}^{(j)} and Y_\ell^m. Transformation of operators under rotations. Cartesian vs spherical tensors. Commutation rules of spherical tensors with the generators J_i of rotations.
25.2
Multiplication of spherical tensors. The Wigner-Eckart theorem.
The projection theorem. Spin correlations and Bell's inequality.
1.3
Symmetries in quantum mechanics. Energy degeneracy. Parity transformation of operators, states and matrix elements. The double well potential and handed molecules.
3.3
Parity selection rule. Parity violation in the weak interactions: left-handed neutrinos. Time reversal symmetry. Antiunitary operators. Transformation of operators, states and matrix elements under time reversal.
No lectures or problem sessions during week 11 (8.3 and 10.3)
15.3
Time reversal (cont). Selection rule for matrix elements of spherical tensors. Neutron dipole moment. Kramers degeneracy.
17.3
Density matrix. [Sakurai, Sec. 3.4] Pure and mixed ensembles. Ensemble average. Time evolution. Thermal equilibrium.
22.3
Wave functions of identical bosons and fermions. Statistical quantum mechanics for distinguishable particles vs. identical bosons and fermions. The Fermi-Dirac and Bose-Einstein distributions.
24.3
The Fermi-Dirac and Bose-Einstein distributions. Black-body spectrum.
Introduction to quantum fields. Creation and annihilation operators for bosons and fermions. Fock states. Commutation and anticommutation relations.
29.3
Normal ordering. Wick's theorem. Field operators. Commutation and anticommutation of field operators.
31.3
Hamiltonian and density operators. Time dependence. Propagator.
Introduction to relativistic quantum mechanics. [Niskanen 4, Aitchison and Hey 4]
5.4
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.
Plane wave solutions to the Dirac equation.
7.4
Lie algebra of the Lorentz group and its equivalence to SU(2) x SU(2). Probability density and current for the Dirac equation. Spin, precession and helicity.
14.4
Lorentz covariant notation. Dirac matrices and their anticommutation relatios. The u and v Dirac spinors and their normalization. Lorentz transformation of the Dirac wave function.
19.4
Lorentz transformation of the Dirac bilinears. Interaction with an electromagnetic potential. Gauge invariance.
21.4
Magnetic moment of the electron. The Dirac sea and negative energy solutions as backward moving antiparticles. Crossing symmetry. Lagrangian density and equations of motion for relativistic scalar fields. The QED lagrangian. The Dirac and Maxwell equations. U(1) gauge invariance with derivatives: The connection (gauge) field.
26.4
The uniqueness of the QED lagrangian, given locality, U(1) gauge and lorentz invariance, P and T invariance and renormalizability. Non-abelian gauge invariance. The Yang-Mills lagrangian for SU(2) and SU(N). QCD.
28.4
Symmetries of the lagrangian: Chiral symmetry and its spontaneous breaking. Isospin invariance. Scale invariance broken by renormalization. The Higgs mechanism.
Quantization using functional integrals vs. the Hamilotonian formulation using operators.
The free scalar field operator. Its conjugate momentum and their canonical commutation relation. The hamiltonian.
3.5
The propagator. The Dirac lagrangian and field operator. The conjugate Dirac field and hamiltonian. Interactions.
5.5
Review of the course. Please prepare questions!