## Scope of questions for State exams of Master's degree program

Branch: Mathematical Physics

Subject: Quantum Physics

Subjects regarding the questions:

- 02KFA Quantum Physics

1. Description of a quantum state –Hilbert space, states and observables, orthonormal basis, physical interpretation of state –de Broglie hypothesis, Born interpretation of the wave function, position, momentum, and energy representation, simple quantum systems –Cartesian position and momentum operators, correspondence principle –linear harmonic oscillator and its eigenstate basis.

2. Linear operators in quantum mechanics –self-adjoint operators, unitary operators, projectors, trace-class operators, spectral theorem, the significance of domain in solving eigenvalue equations, projector-valued measure and measuring observables with point and continuous spectra, direct sum and product of operators, complete sets of commuting observables.

3. Measurement postulate in quantum mechanics –prediction of measurement outcomes, probabilistic interpretation of the quantum state, transition probability between states, probability of yes-no measurements on Borel sets, expectation value, compatible observables, standard deviation of an observable, uncertainty relations.

4. Pure and mixed states, density matrix –physical reason for a using density matrix for statistical description, density matrix definition, mixed states after an unrecorded measurement, density matrix of a two-level system, Bloch sphere, density matrix time evolution, measurement probabilities with mixed states.

5. Particle in a spherically symmetric field –compatible observables, angular momentum component operators, spherical functions, Hamiltonian of a particle in spherically symmetric potential, effective potential, energy level degeneration, isotropic harmonic oscillator, particle in Coulomb field.

6. Particle in external electromagnetic field –hydrogen atom, Zeeman effect, electron spin, spin operators and their commutation relations, Pauli matrices, Stern–Gerlach experiment, static and dynamic Pauli equation.

7. Angular momentum quantization –algebraic angular momentum theory, ladder operators, orbital and spin angular momenta, angular momentum addition, irreducible tensor operators, Wigner–Eckart theorem.

8. Time evolution in quantum mechanics –unitary propagator, Schrödinger equation, stationary states, solving time evolution by expansion into stationary states, time evolution of expectation values, integrals of motion, Ehrenfest theorem, Schrödinger, Heisenberg, Dirac pictures of quantum mechanics and translations between them, time evolution of states and observables in the three pictures.

9. Feynman’s path integral and propagator –perturbative expansion of evolution operator, retarded and advanced propagator, their dynamical equations, Green’s function, obtaining propagator using path integral, description of scattering using path integral.

10. Composite systems –tensor product of spaces, states, and observables, statesof ensembles of undistinguishable identical particles, Pauli principle, occupation numbers, creation and annihilation operators, Hamiltonian of non-interacting particles, second quantization, Fock space.