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QUANTUM MECHANICS
Prof. Gary Efimov (Bogoliubov Laboratory of Theoretical Physics)
6th Semester. Lectures: 34 hours, Seminars: 34 hours
7th Semester. Lectures: 32 hours, Seminars: 30 hours
The course “Quantum mechanics” consists of two parts, which are given in the spring term of the 3rd
year of study and in the autumn term of the 4th year of study. Students are supposed to have taken a
basic course in mathematical analysis and mathematical physics equations.
Lectures and seminars are given. Students are supposed to master the studied material on their own. As
a result, a student should know basic ideas and methods of quantum mechanics and be able to solve
different quantum-mechanical tasks.
Part I
1. Classical and quantum mechanics. Description of a state. Wave function. Atomism, ensemble,
quantum numbers. Indeterminacy relation.
2. Equilibrium radiation. Classical radiation. Rayleigh-Jeans equation. Quantum radiation. Planck
formula.
3. Canonical quantization. Schrodinger equation. Task setting.
4. Boundary conditions. Wave function. Eigenvalues and eigenfunctions. Discrete and continuous
eigenvalues. Normalization of eigenfunctions. Probability density and current. Continuity equation.
5. Classical limit. Wavepackage broadening. Condition of classical motion.
6. One-dimensional motion. Oscillator. One-dimensional pits. Energy spectrum. Reflection and
passing through a potential barrier.
7. Quasi-stationary states. Scattering amplitude. Resonances.
8. Perturbation theory. Degenerated states. Degeneracy release.
9. Orbital moment in 𝑅 3 . Commutation relations. Conserved quantities. 𝐿0 and 𝐿∓ -operators.
Eigenvalues and eigenfunctions. Representation: symbolic, spherical functions, oscillator.
10. Spin. Pauli matrixes.
11. Quantum mechanical sum of momentum. Making of orthogonal basis. Klebsch-Gordon
coefficients.
12. Centrally-symmetric field. Two-body problem. Center-of-inertia system. Radial equation.
Boundary conditions.
13. Oscillator in 𝑅 3 . Coulomb potential. Transition to oscillatory basis. Yukawa potential.
14. Quasiclassics.
15. Electron behaviour in the magnetic field.
16. Mean field approximation. Thomas-Fermi equation.
Part II
17. Scattering theory. Scattering in classical mechanics. Scattering in quantum mechanics. Scattering
amplitude.
18. General scattering theory. Amplitude and S-matrix. Analytic properties of S-matrix.
19. Scattering in Coulomb field.
20. Born approximation.
21. Scattering phases.
22. Scattering of slow and fast particles. Phase behaviour at 𝑙 → ∞.
23. Scattering by square potential. S-wave, scattering in the presence of a shallow level.
24. Eikonal approximation.
25. Resonance scattering.
26. Quasistationary (unstable) states and their decay.
27. Indeterminacy relation for Fermi systems.
28. K-meson regeneration.
29. Canonical quantization in quantum theory of scalar and Dirac fields.
Methodical recommendations for the tutor
The programme is based on the book “Quantum mechanics” by Landau and Lifshits. The focus is
made on the methods of solving different tasks of quantum mechanics.
Methodical recommendations for the students
Students should understand that knowledge of quantum mechanics presupposes practical ability to
solve different tasks of quantum mechanics. That is why students should solve a few tasks on each
topic by themselves.