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Quantum Mechanics
Yanko Todorov
Syllabus
1. Basic Quantum Mechanical Concepts
- Wavefunction and probabilistic interpretation, interference, position and momentum measurement and representation.
- Stability of the atom, wavefunctions and energy levels in hard wall potentials (through standing wave condition)
- Introduction to Hilbert space; measurement as projection; introduction to Schrodinger equation
2. Dirac formalism
- Observables, Hilbert space, Probability amplitude (bra, ket product), position & momentum operators, Hamiltonian, Commutator and Heisenberg uncertainty relation, The postulates of quantum mechanics, Evolution operator, Time evolution of variables, Schrödinger and Heisenberg picture, Tensor product of Hilbert spaces, The Einstein- Podolsky-Rosen “paradox”.
3. Some 1D problems in Quantum Mechanics
- Conservation law for probability current, Transmission, Three “classic” 1D problems in Quantum Mechanics (step, barrier, well), Numerical resolution of 1D Schrödinger equation
- The Quantum Harmonic Oscillator: Hamiltonian, spectrum, ladder operators, eigenstates, coherent states, Translation operator.
4. Spin and Two-level systems
- Stern and Gerlach experiment, Pauli matrices & their properties, Spin in an arbitrary direction, Spin and 3D rotation, Coherent control of a spin, Spin as a Q-bit: single bit quantum gates, Two Q-bits quantum gates, Intro: Spatial rotations and angular momentum operator
5. Open systems and density matrix
- Case study: quantum well coupled to a continuum, Wigner-Weisskopf approach & derivations of the Fermi’s golden rule
- Density matrix operator: Definition, properties, examples, application, elementary introduction to Lindblad equation
Prerequisites
None
Evaluation
Written exam (2 h)
