Lev Landau's "Theoretical Minimum"
The list of topics can be found here (in Russian, of course). Nowadays students are examined by collaborators of Landau Institute for Theoretical Physics. Each exam, as it was before, consists of problems solving. For every exam there is one or several examiners with whom you are supposed to contact with to inform that you're willing to pass this particular exam (they will make an appointment). Everyone can pass any exam in any order. Today Landau's theoretical minimum (not all 11 exams, but at least 6 of them) is included in the program for students of Department of General and Applied Physics (Moscow Institute of Physics and Technology).
The program for each exam, as you can see from the link above, corresponds to the contents of volumes in the Course of Theoretical Physics by L&L (usually you have to master almost all paragraphs in the volume to pass the exam).
- Mathematics I. Integration, ordinary differential equations, vector algebra and tensor analysis.
- Mechanics. Mechanics, Vol. 1, except §§ 27, 29, 30, 37, 51 (1988 russian edition)
- Field theory The Classical Theory of Fields, Vol. 2, except §§ 50, 54-57, 59-61, 68, 70, 74, 77, 97, 98, 102, 106, 108, 109, 115-119 (1973 russian edition)
- Mathematics II. The theory of functions of a complex variable, residues, solving equations by means of contour integrals (Laplace's method), the computation of the asymptotics of integrals, special functions (Legendre, Bessel, elliptic, hypergeometric, gamma function)
- Quantum Mechanics. Quantum Mechanics: Non-Relativistic Theory, Vol. 3, except §§ 29, 49, 51, 57, 77, 80, 84, 85, 87, 88, 90, 101, 104, 105, 106-110, 114, 138, 152 (1989 russian edition)
- Quantum electrodynamics. Relativistic Quantum Theory, Vol. 4, except §§ 9, 14-16, 31, 35, 38-41, 46-48, 51, 52, 55, 57, 66-70, 82, 84, 85, 87, 89 - 91, 95-97, 100, 101, 106-109, 112, 115-144 (1980 russian edition)
- Statistical Physics I. Statistical Physics, Vol. 5, except §§ 22, 30, 50, 60, 68, 70, 72, 79, 80, 84, 95, 99, 100, 125-127, 134-141, 150-153 , 155-160 (1976 russian edition)
- Mechanics of continua. Fluid Mechanics, Vol. 6, except §§ 11, 13, 14, 21, 23, 25-28, 30-32, 34-48, 53-59, 63, 67-78, 80, 83, 86-88, 90 , 91, 94-141 (1986 russian edition); Theory of Elasticity, Vol. 7, except §§ 8, 9, 11-21, 25, 27-30, 32-47 (1987 russian edition)
- Electrodynamics of Continuous Media. Electrodynamics of Continuous Media, Vol. 8, except §§ 1-5, 9, 15, 16, 18, 25, 28, 34, 35, 42-44, 56, 57, 61-64, 69, 74, 79-81 , 84, 91-112, 123, 126 (1982 russian edition)
- Statistical Physics II. Statistical Physics, Part 2. Vol. 9, only §§ 1-5, 7-18, 22-27, 29, 36-40, 43-48, 50, 55-61, 63-65, 69 (1978 russian edition)
- Physical Kinetics. Physical Kinetics. Vol. 10, only §§ 1-8, 11, 12, 14, 21, 22, 24, 27-30, 32-34, 41-44, 66-69, 75, 78-82, 86, 101.
Some real problems (Quantum Mechanics exam):
- The electron enters a straight pipe of circular cross section (radius $r$). The tube is bent at a radius $R \gg r$ by the angle $\alpha$ and then is aligned back again. Find the probability that the electron will jump out.
- A hemisphere lies on an infinite two-dimensional plane. The electron falls on the hemisphere, determine the scattering cross section in the Born approximation.
- The electron "sits" in the ground state in the cone-shaped "bag" under the influence of gravity. The lower end of the plastic bag is cut with scissors. Find the time for the electron to fall out (in the semi-classical approximation).
In the following arXiv article Landau's Theoretical minimum, the author details personal experiences on being one of 42 students ever who passed it, as well as some details of the examination. He also mentions some questions in passing, from which I gather most of them were similar to the exercises and examples of his famous books.
Quote:
Landau gave me the programs of eight courses on theoretical physics. Besides, there was one more examination on mathematics — complex variables, special functions, the Laplace transformation, etc. By that time only a few books of the famous Landau course were published: Mechanics, Classical Field Theory, The Theory of Continuous Media and the first (classical) part of Statistical Physics. One had to study all other courses by reading various books and original papers. For example, let me display the list of books/papers which we were supposed to study for the Quantum Mechanics course: