| # | Date | Read | Topic |
|---|---|---|---|
| 1. | Tue, Sep 1 | KKL 1.1 RLL 2 JJS 1.1 |
Introduction. Physics at 1900. "Little problems" with classical physics. Experimental motivations for quantum mechanics. Black body radiation. The photoelectric effect. The Stern-Gerlach experiment. Analogy with the polarization of light. |
| 2. | Thu, Sep 3 | JJS 1.2-1.3 MIT part I |
Classical systems. Postulates of quantum mechanics. Mathematical preliminaries. Vector spaces. Linear independence and bases. Dual spaces. Inner product spaces. |
| 3. | Tue, Sep 8 | JJS 1.5 MIT part I |
Homework 1 is given. Hilbert spaces. Basis and completeness relation. Linear operators. Hermitian operators. Unitary operators. |
| 4. | Thu, Sep 10 | JJS 1.5, 1.4 MIT part I |
Projection operators. Eigenvalues and eigenvectors of Hermitian operators. Spectrum. Completeness relation. Matrix representation. Unitary transformations. Diagonalization and simultaneous diagonalization of Hermitian operators. |
| 5. | Tue, Sep 15 | JJS 1.5, 1.4 MIT part I |
Homework 2 is given. Back to the postulates of quantum mechanics. Measurements. Expectation value. The example of spin 1/2 system (any two-level system). Hilbert space, operators. Algebra of Pauli matrices. |
| 6. | Thu, Sep 17 | JJS 1.4 MIT part I |
Stern-Gerlach measurements in spin 1/2 system. Compatible and incompatible observables. A complete set of commuting observables. Quantum "behavior" and incompatibility of observables. The uncertainty relation. |
| 7. | Tue, Sep 22 | JJS 1.4 MIT part I KKL 4 |
Homework 3 is given. Tensor product of Hilbert spaces (the Hilbert space of composite system). Tensor product of Hilbert spaces. The Hilbert space of a composite system. Schrodinger equation - quantum dynamics. Example of spin precession in magnetic field. |
| 8. | Thu, Sep 24 | JJS 1.6 | Quantum mechanics in one dimension. Operators with continuous spectra. Dirac's delta function. Position operator. Translations. Momentum as a generator of translations. Momentum-position uncertainty relation. Classical-quantum correspondence. |
| Tue, Sep 29 | No classes, correction day (Monday). | ||
| 9. | Thu, Oct 1 | JJS 1.7 | Homework 4 is given. Wave function in coordinate representation. Wave function in momentum representation. Gaussian wave packets. |
| 10. | Tue, Oct 6 | KKL 2.1-2.2 LL 17-22 |
Time-dependent and time-independent Schrodinger equation. Time-independent Hamiltonians. Evolution operator. Particle moving in piecewise-constant potentials. Free particles. Evolution of Gaussian packets. |
| 11. | Thu, Oct 8 | LL 17-22 RS Ch.5 EM Ch.6 |
Evolution of Gaussian packets. Heavyside step function potential. Transmission and reflection. Classically forbidden processes. Impenetrable wall limit. Boundary conditions. Particle in a box. Parity symmetry. |
| 12. | Tue, Oct 13 | KKL 2.4 LL 25 |
Homework 5 is given. Delta-functional potential. Boundary conditions. Discrete spectrum of a delta-well - a single bound state. Scattering by a delta-scatterer. Scattering matrix, transfer matrix. Unitarity and symmetry of an S-matrix in general case. Transmission and reflection by delta-scatterer. |
| 13. | Thu, Oct 15 | KKL 2.4 | Multiple scatterings. Transfer matrix method. Double delta-barrier. Resonances. |
| 14. | Tue, Oct 20 | KKL 2.4 LL 23 |
Homework 6 is given. Delta-lattice, band theory. Harmonic potential - quantum oscillator. Direct approach. Asymptotics, power series, energy eigenvalues. |
| 15. | Thu, Oct 22 | KKL 2.4, 4.9 LL 23 EM 10.6 |
Ground state. Hermite polynomials and excited states. Ladder operators. Raising and lowering operators. Normalization of the states. Matrix elements of p and x. |
| 16. | Tue, Oct 27 | KKL 4.9 EM 10.6 |
Ladder operators and relation to coordinate representations. Supersymmetry in quantum mechanics. -1/cosh(x) potential - bound states. |
| 17. | Thu, Oct 29 | HW solutions, All of the above. |
Midterm - one-hour exam. [Math Foundations of QM. QM of finite dimensional systems: two-level system etc. Time-dependent Schroedinger equation. Uncertainty relation. QM in one dimension. Sectionally-constant potentials, scattering theory in 1d QM. Harmonic oscillator.] |
| 18. | Tue, Nov 3 | LL 18 | Homework 7 is given. -1/cosh(x) potential - the absence of reflection. Fundamental properties of Schroedinger equation. Discrete and continuous spectrum. Falling to the center. |
| 19. | Thu, Nov 5 | LL 21 | General properties of motion in 1d. No degeneracies for a discrete spectrum. Oscillation theorem. Bound state in a shallow potential well. Variational principle. Applications. Excited states. Rayleigh-Ritz trial wave functions. |
| 20. | Tue, Nov 10 | JJS 2.2 | Homework 8 is given. Quantum dynamics. Schroedinger and Heisenberg representations. Evolution operator. The Heisenberg equation of motion. Ehrenfest's theorem: relation between classical and quantum dynamics. Transition amplitudes. |
| Tue, Nov 10 | EXTRA RECITATION SECTION Solving midterm problems. | ||
| 21. | Thu, Nov 12 | EM 15.1-2 KKL 3.1 |
Transition amplitudes for free particle and for harmonic oscillator. The composition property of transition amplitudes. Quantum mechanics in two dimensions. Free particle in higher dimensions and separation of variables. Particle in a box: a) periodic boundary conditions b) vanishing boundary conditions. |
| 22. | Tue, Nov 17 | your notes | Homework 9 is given. 2D oscillator, separation of variables in Cartesian coordinates. Plane rotator. Ambiguity of quantization of the classical plane rotator. First encounter with Aharonov-Bohm effect. Central potential in 2D. Separation of variables in polar coordinates. Laplacian in polar coordinates. |
| 23. | Thu, Nov 19 | your notes EM 4.6 |
Angular momentum in 2D. Free particle inside a circular impenetrable wall. Bessel's equation. Zeros of Bessel's functions. Charged particle in e/m fields. QM of the charged particle in an external E/M field and gauge invariance. Velocity operator. |
| Fri, Nov 20 | EXTRA RECITATION SECTION by Tin Sulejmanpasic. Math foundations of QM. In B-131, 3pm. Recitation 1 | ||
| 24. | Tue, Nov 24 | JJS 2.6 KKL 3.2 |
Homework 10 is given. 20-minute Quiz. Gauge invariance of Schrodinger equation. Gauge invariant current. Aharonov-Bohm effect. |
| Thu, Nov 26 | No classes, Thanksgiving. | ||
| 25. | Tue, Dec 1 | LL 111 EM 4.6 KKL 3.2 |
Motion in a uniform magnetic field. Landau gauge. 1d oscillator. Landau levels. Degeneracy of levels. Density of states. Shapes of orbits. |
| 26. | Thu, Dec 3 | LL 111 EM 4.6 KKL 3.2 |
Homework 11 is given. Confining potential (1-dimensional). Radial gauge. Complex coordinates and wave functions on the lowest Landau level. |
| Fri, Dec 4 | EXTRA RECITATION SECTION by Tin Sulejmanpasic. 1d problems of QM. In B-131, 3pm. Recitation 2 | ||
| 27. | Tue, Dec 8 | JJS 2.6 | 20-minute Quiz. Dirac's magnetic monopole. Topological argument for magnetic charge quantization. Dirac's string. The Wu-Yang monopole. |
| 28. | Thu, Dec 10 | your notes | Arbitrary gauge and use of ladder operators. Magnetic translations and magnetic rotations. Canonic, kinematic, and "magnetic" momenta. |
| Fri, Dec 11 | EXTRA RECITATION SECTION by Tin Sulejmanpasic. In B-131, 3pm. | ||
| Thu, Dec 17 | FINAL EXAM: Room B-131, 2-4:30pm The final is "closed books and notes". |
