Phys 500: Quantum mechanics I

Course outline

Phys 500 is a required course for all incoming graduate students in Physics, Medical Physics and Astrophysics. Its purpose is two-fold, namely

Material we will cover: Fundamental concepts (pure and mixed quantum states, observables, measurement, uncertainty relations), quantum dynamics, theory of angular momentum, symmetry and conservation laws, perturbation theory, identical particles, quantum mechanics in medicine - medical imaging, quantum mechanics in astrophysics, quantum information and computation, foundations of quantum mechanics.

Schedule and practical information

Special announcements:

Prerequisites: One of PHYS 450, PHYS 402.

Time and location: MWF 1-2PM in Hebb 12

Credits: 3.

Grading: Homework: 40%, Midterm: 20%, Final: 40%.

Homework assignments: All done.

Office hour: My office hour is Tuesdays, 3:30-4:30 PM in September, and 4-5 PM thereafter.

Book: J.J. Sakurai, Modern Quantum mechanics, Addison Wesley (1994, 2010 [with Napolitano]).

Additional source: C.J. Isham, Quantum Theory - Mathematical and Structural Foundations, Imperial College Press (1995).

Topics covered

  1. Fundamental concepts
    • The Stern-Gerlach experiment (1922)
    • Hilbert spaces
    • Quantum meaurement: observables, uncertainty relations
  2. Quantum dynamics
    • The Schroedinger Equation (1926)
    • Schroedinger vs Heisenberg picture
    • Nuclear magnetic resonance and medical imaging
    • The simple harmonic oscillator
    • Density operators
  3. The theory of angular momentum
    • Rotations and the angular momentum commutation relations
    • Spin-1/2 systems
    • SO(3) vs SU(2)
    • Addition of angular momenta
  4. Symmetries in quantum mechanics
    • Continuous symmetries
    • Discrete symmetries: translation (by Delta), parity, time-reversal
  5. Approximation methods
    • The WKB-method
    • Time-independent perturbation theory
    • Time-dependent perturbation theory
  6. Identical particles
    • Permutation symmetry and the symmetrization postulate
    • Two-electron systems
    • The Helium-atom
  7. Scattering theory
    • The Lippmann-Schwinger Equation
    • Born approximation
  8. Foundations of quantum mechanics
    • ``Can quantum mechanics be considered complete?'' (Einstein, Podolsky and Rosen, 1935)
    • The Bell inequalities
    • The Kochen-Specker theorem
  9. Quantum information
    • Quantum cryptography - BB84 and Ekert protocol

Past & future lectures

Wednesday, Sept 5. First lecture. Course outline. The Stern-Gerlach experiment. Sequential Stern-Gerlach experiments. Measurement changes the quantum state. Reading assignment: Sakurai, Section 1.1.

Monday, Sept 10. Hilbert spaces: Vector spaces, inner product, closedness. The analogy between sequential Stern-Gerlach apparata and polarized light. (Sakurai, Section 1.2)

Wednesday, Sept 12. Hilbert spaces continued: Orthonormal bases, matrix representations. (Sakurai, Section 1.3)

Friday, Sept 14. Measurement in quantum mechanics: ``collapse of the quantum state (wavefunction)''. Born rule for the non-degenerate and the degenerate case. (Sakurai, Section 1.4)

Monday, Sept 17. Lecture 1: Measurement, continued: compatible and incompatible observables. The uncertainty relation. (Sakurai, Section 1.4). Lecture 2: Continuous spectra and infinite-dimensional Hilbert spaces.

Wednesday, Sept 19. Defining the momentum operator: Momentum as generator of translations. Position-momentum commutation relation. The Heisenberg uncertainty relation. The momnetum operator in the position eigenbasis and plane waves as momentum eigenstates. Gaussian wave packets.

Friday, Sept 21. No lecture.

Monday, Sept 24. No lecture.

Wednesday, Sept 26. Quantum dynamics: The Schroedinger equation (for time-evolution operators, states and wave-functions). Expectation values.

Friday, Sept 28. Example for unitary evolution: precession of spin in a static magnetic field. Application: Nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI).

Monday, Oct 1. First lecture: The Heisenberg picture and Ehrenfest's theorem. Second lecture: The simple harmonic oscillator (SHO). Algebraic method to obtain the energy spectrum. Energy eigenfunctions in position space. Reading assignment: Time evolution of the SHO (Section 2.3 in the Sakurai).

Wednesday, Oct 3. Density operators. The no-cloning theorem. Impossibility of superluminal communication in QM.

Friday, Oct 5. No lecture.

Monday, Oct 8. Thanksgiving.

Wednesday, Oct 10. The theory of angular momentum (Ch. 3 in Sakurai, will take several lectures). Symmetry reduces complexity. Example from classical mechancis: The Kepler problem. Noether's theorem. Rotational symmetry and angular momentum.

Friday, Oct 12. The commutation relations of angular momentum. Example & application: Rotating angular momentum itself. The action of rotations on states in Hilbert space: the spin 1/2-case.

Monday, Oct 15. Groups and representations. SU(2) vs. SO(3).

Wednesday, Oct 17. Angular momentum eigenstates and eigenvalues.

Friday, Oct 19. Orbital angular momentum: definition. Solving the Schroedinger Equation for potentials with a rotation symmetry. Spherical harmonics.

Monday, Oct 22. Addition of angular momenta. Clebsch-Gordan coefficients - meaning and calculation. Reading assignment: derivation of the recursion formula for Clebsch-Gordon coefficients.

Wednesday, Oct 24. MIDTERM EXAM

Friday, Oct 26. Discrete symmetries (Sakurai Ch. 4). (i) Parity (space inversion). The parity operator and its properties. Non-degenerate eigenstates of a parity-symmetric Hamiltonian are parity eigenstates. The symmetric double-well potential, and the NH_3 molecule. Is parity a symmetry of nature?

Monday, Oct 29. Discrete symmetries (ii): Lattice translation invariance. Bloch's theorem.

Wednesday, Oct 31. Discrete symmetries (iii): Time-reversal.

Friday, Nov 2. Approximation Methods: I. WKB. Exploiting the ``smallness'' of h. WKB in one dimension: bound and unbound case. Range of validity for the approximation.

Monday, Nov 5. Approximation Methods: II. Perturbation theory. II a: Time-ndependent perturbation theory. The non-degenerate case. (Ch. 5)

Wednesday, Nov 7. Time-independent perturbation theory continued. The degenerate case. A physical example: The Stark effect - an atom in a strong electrostatic field.

Friday, Nov 9. Approximation Methods: I. WKB. Exploiting the ``smallness'' of h. WKB in one dimension: bound and unbound case. Range of validity for the approximation.

Monday, Nov 12. Remembrance day - no lecture.

Wednesday, Nov 14. Approximation Methods II b: Time-dependent perturbation theory. The interaction picture. Perturbation expansion - the Dyson series.

Friday, Nov 16. Time-dependent perturbation theory continued. Fermi's golden rule.

Monday, Nov 19. Identical particles (Ch. 6). The symmetrization postulate. Bosons and Fermions. Pauli's exclusion principle for fermions. Bose-Einstein condensation. The spin-statistics theorem.

Wednesday, Nov 21. The two-electron system. The exchange density - when does it matter? The helium atom: eigenenergies are spin-dependent even if the Hamiltonian isn't (consequence of the symmetrization postulate).

Friday, Nov 23. Scattering theory. The experimental setting. Scattering amplitude and differential cross section. Lippmann-Schwinger equation.

Monday, Nov 26. Scattering theory continued. Solving the Lippmann-Schwinger equation for the scattering amplitude. Physical application of Cauchy's residue theorem.

Wednesday, Nov 28. Scattering theory continued. Born approximation. Physical example: a proton scattering off an atomic nucleus. Summary of the course material.

Friday, Nov 30. Foundations of quantum mechanics. The Einstein-Podolsky-Rosen paradox. Hidden-variable models. Bell's ineaquality.