Quantum Mechanics I Content
Quantum Mechanics I
1. Introduction
Brief outline of the history of quantum mechanics. The need for quantum mechanics: the double slit experiment.
2. Mathematical Apparatus
Vector spaces. Linear independence. Bases. Tensor products. Dual spaces. Linear operators. Isomorphisms. Inner product spaces. Orthonormal bases. Dual vectors. Adjoint operators. Hilbert spaces. Eigenvectors and Eigenvalues. Unitary operators.
Dirac notation (bra and kets). Resolution of the identity. Basis transformation. Operators in Dirac notation.
3. The Postulates of Quantum Mechanics
State vectors. Observables. Probabilities of measurement outcomes. Expectation values. Reduction of the state vector. Canonical quantization. Density matrices. Examples.
4. Canonical Quantization
Review of classical mechanics. The Hamiltonian. Coordinates and conjugate momenta. Poisson brackets. Canonical commutation relations.
5. Symmetries and Conservation Laws
Time translations and the Hamiltonian. Heisenberg and Schroedinger representations. Commutators and conservation laws. Space translations and the momentum operator. Wigner's theorem: anti-unitary operators and time reversal.
6. Wave functions
Position and momentum operators in position and momentum representation. Heisenberg's uncertainty relation. Gaussian wave packets. Energy-time uncertainty relation (reading).
7. Conceptual Issues
Ensembles. Determinism and reversibility. The measurement problem. Hidden variable theories. EPR paradox and Bell's inequalities (reading). Reduction of the state vector. Relative state formulation of quantum mechanics (reading).
8. Motion in one dimension
Schroedinger equation as eigenvalue problem. Free propagation. Wave packets and Gaussian integrals. Particle in a well. Parity. Bound states. Potential step. Transmission and Reflection coefficients. Resonances. Tunneling .The probability current. Analytical properties of the scattering amplitude.
9. The Harmonic Oscillator
Hamiltonian. Creation and annihilation operators. Vacuum and energy eigenstates. Zero point energy. Coherent states and the classical limit. Ehrenfest's theorem.
10. Path Integrals
The propagator for a time-dependent Hamiltonian. Dyson's time ordering. Derivation of the path integral formulation. Wick rotation and the Euclidean action. Relation to diffusion. Justification of the principle of least action. The Aharonov-Bohm effect.
Web page last updated December 1, 2005 by CAP.