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

Branch: Quantum Technologies

Subject: Quantum Mechanics

Subjects regarding the questions:

- 02KVAN1 Quantum Mechanics 1
- 02KVAN2 Quantum Mechanics 2

- States and observables, simple quantum systems – de Broglie hypothesis, Born’s interpretation of the wave-function, Hilbert space, position and momentum operators, correspondence principle, hamiltonian, linear harmonic oscillator, raising and lowering operators, x-and p-representation
- Prediction of measurement outcomes – measurement in quantum mechanics, probabilistic interpretation of the quantum state, transition probability between two states, probability of measuring a given value of an observable, average value of an observable, standard deviation, uncertainty relations.
- Pure and mixes states, density matrix – definition of a density matrix, statistical description with the density matrix, mixed state after measurement, density matrix of a two-level system, Bloch sphere, time evolution of a density matrix, measurement predictions for mixed states.
- Particle in a spherically symmetric field – compatible observables, orbital angular momentum, spherical functions, hamiltonian of a particle in a spherically symmetric potential, degeneracy of energy levels, isotropic oscillator, particle in a Coulomb field.
- Particle in an electromagnetic field – charged particle in an external electromagnetic field, hydrogen atom in an external homogeneous magnetic field, Zeeman effect, spin of an electron, spin operators and their commutation relations, Pauli matrices, Stern-Gerlach experiment, Pauli equation.
- Quantization of angular momentum – algebraic theory of angular momentum, raising and lowering operators, addition of angular momentum, irreducible tensor operators, Wigner-Eckart theorem.
- Time evolution in quantum mechanics – Schrödinger equation, stationary states, expansion into stationary states, time evolution of the average value of an observable, integrals of motion, Ehrenfest theorems, Schrödinger’s, Heisenberg’s and Dirac’s picture of quantum mechanics, time evolution of states and observables in respective pictures.
- Approximation methods in quantum mechanics – perturbation theory for simple and degenerate eigenvalue, JWKB approximation, Ritz variation method, time-dependent perturbation theory, time-ordering operator.
- Feynman’s path integral and the propagator –perturbation expansion of the evolution operator, retarded and advanced propagator, Green’s function, propagator in terms of the path integral, scattering described with the path integral.
- Identical particles –states of identical particles, Slater’s determinant, Pauli principle, occupation numbers, creation and annihilation operators, Hamiltonian of non-interacting particles, second quantization, Fock’s space.