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Course: Quantum Mechanics

Department/Abbreviation: SLO/KM

Year: 2021

Guarantee: 'prof. RNDr. Jan Peřina, Ph.D.'

Annotation: Basic principles of quantum mechanics.

Course review:
1. Photoeffect, Compton scattering, de Broglie hypothesis, plane waves 2. Schrödinger equation a. Postulates in v x-representation b. Free particle, group velocity, wave packet. 3. Time dependent and independent Schr. equation, separation of variables, stationary states a. Normalizacion, probabilistic interpretation, probability density conservation, continuity equation. b. Principle superposition 4. Simple systems and their solutions a. Infinite 1D well, generalization to 3D b. Finite square well, reflection, transmission, resonance energy. c. Finite barrier, tunneling d. Delta function potential, bound state 5. Postulates of quantum mechanics in bracket formalism, state, operators of observables, quantization. a. Observables, operators, Poisson brackets, commutation relations, superposition. b. Eigenstate and eigenvalues of position and momentum operators. X and P representations and their correspondence. c. Formal construction of QM, brackets, states, Hilbert space d. Postulates in brackets. e. Operators expectation values, matrix elements. f. Eigenstates of Hamiltonian g. Discrete and continuous spectra, normalization of momentum eigenstates to a delta function. h. Collapse of the wave function, measurement, probability amplitude. i. Ehrenfest theorems, virial theorem. 6. The uncertainty principle a. expectation values and quadratic fluctuations in statistics b. the unc. principle for non-commuting observables c. applications on a wave packed 7. Schroedinger equation revisited in brackets a. representations b. general state as a superposition c. system and observables in t > 0 8. Harmonic oscillator a. algebraic method, ladder operators, their commutators b. analytical method of Frobenius c. coherent states 9. WKM method, classical limit, alpha decay 10. Angular momentum and its addition a. commutation relations, ladder operators, the algebraic method b. spherical harmonics, parity 11. Central potential a. Correspondence of the angular momentum operator and Laplace operator b. Radial and angular equations, energy quantization c. Spherically symmetric infinite well, Bessel function, 3D harmonic oscillator. d. Hydrogen atom 12. Particle in a homogeneous electric field a. harmonic oscillator in a homog. el. field b. hydrogen atom in homog. field, Stark effect 13. Spin, particle in an electromagnetic field, Pauli equation a. Pauli matrices, spinors, spin operators to a given direction b. Larmore precession 14. Perturbation methods a. time-independent perturbation theory b. Stark effect, Zeeman effect c. correction to energy and wave function