Quantum Mechanics I

Syllabus (under construction)

KKL - K. K. Likharev's Lectures on Quantum Mechanics
EM - E. Merzbacher, Quantum Mechanics
JJS - J.J. Sakurai's book
RRL - R. R. Liboff's book
MIT - MIT Open Courseware, Quantum Theory I
LL - L. D. Landau and E. M. Lifshitz, Quantum Mechanics
RS - R. Shankar, Principles of Quantum Mechanics

# Date Read Topic
1. Tue, Aug 31 KKL 1.1
JJS 1.1
Introduction. Physics at 1900. "Little problems" with classical physics. Experimental motivations for quantum mechanics. Mathematical preliminaries.
Thu, Sep 2 Class is cancelled (will be rescheduled later).
2. Tue, Sep 7 JJS 1.2-1.3
MIT part I
Classical vs. quantum mechanics. Postulates of quantum mechanics. Mathematical preliminaries. Vector spaces. Linear independence and bases. Dual spaces. Inner product spaces.
Thu, Sep 9 No classes, Rosh Hashanah.
3. Tue, Sep 14 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 16 JJS 1.5, 1.4
MIT part I
Projection operators. Eigenvalues and eigenvectors of Hermitian operators. Spectrum. Completeness relation. Matrix representation. Pauli matrices. Unitary transformations.
5. Tue, Sep 21 JJS 1.5, 1.4
MIT part I
Homework 2 is given. Diagonalization and simultaneous diagonalization of Hermitian operators. Back to the postulates of quantum mechanics. Measurements. Expectation value.
6. Thu, Sep 23 JJS 1.4
MIT part I
The example of spin 1/2 system (any two-level system). Hilbert space, operators. Algebra of Pauli matrices. Stern-Gerlach measurements in spin 1/2 system.
7. Tue, Sep 28 JJS 1.4
MIT part I
Homework 3 is given. Compatible and incompatible observables. A complete set of commuting observables. Quantum "behavior" and incompatibility of observables. The uncertainty relation. Tensor product of Hilbert spaces.
8. Thu, Sep 30 MIT part I Tensor product of Hilbert spaces. The Hilbert space of a composite system. Schrodinger equation - quantum dynamics. Example of spin precession in magnetic field.
9. Tue, Oct 5 JJS 1.6 Homework 4 is given. Quantum mechanics in one dimension. Operators with continuous spectra. Dirac's delta function. Position operator. Wave function in coordinate (position) representation. Translations. Infinitesimal translations.
10. Thu, Oct 7 JJS 1.7 Momentum as a generator of translations. Classical-quantum correspondence. Wave function in coordinate representation. Kernels of operators in coordinate representations. Position and momentum operators in coordinate representation.
11. Tue, Oct 12 KKL 2.1-2.2
LL 17-22
Homework 5 is given. Momentum-position uncertainty relation. Wave function in momentum representation. Gaussian wave packets. Time-dependent and time-independent Schrodinger equation. Time-independent Hamiltonians. Evolution operator.
12. Thu, Oct 14 LL 17-22
RS Ch.5
EM Ch.6
Particle moving in piecewise-constant potentials. Free particles. Evolution of Gaussian packets. Heavyside step function potential.
13. Tue, Oct 19 KKL 2.4
LL 25
Homework 6 is given. Transmission and reflection. Classically forbidden processes. Impenetrable wall limit. Boundary conditions. Particle in a box. Parity symmetry. Delta-functional potential. Boundary conditions.
14. Thu, Oct 21 KKL 2.4 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. Multiple scatterings. Transfer matrix method.
15. Tue, Oct 26 KKL 2.4
LL 23
Double delta-barrier. Resonances.
16. Thu, Oct 28 KKL 2.4, 4.9
LL 23
EM 10.6
Delta-lattice, band theory. Harmonic potential - quantum oscillator. Direct approach. Asymptotics, power series, energy eigenvalues. Ground state.
17. Tue, Nov 2 HW solutions,
All of the
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. ] Homework 7 is given.
Thu, Nov 4 Class is cancelled (will be rescheduled later).
18. Tue, Nov 9 KKL 4.9
EM 10.6
Hermite polynomials and excited states. Ladder operators. Raising and lowering operators.
19. Thu, Nov 11 LL 18 Homework 8 is given. Normalization of the states. Matrix elements of p and x. Ladder operators and relation to coordinate representations. Supersymmetry in quantum mechanics by example: -1/cosh(x) potential - ground state. Fundamental properties of Schroedinger equation. Discrete and continuous spectrum. Falling to the center.
20. Tue, Nov 16 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.
21. Tue, Nov 16 JJS 2.2 Special time and place: 5:25-6:45pm, B-131. Quantum dynamics. Schroedinger and Heisenberg representations. Evolution operator. The Heisenberg equation of motion.
22. Thu, Nov 18 JJS 2.2 Ehrenfest's theorem: relation between classical and quantum dynamics. Transition amplitudes. Equation for transition amplitudes: propagators.
23. Tue, Nov 23 EM 15.1-2
KKL 3.1
Homework 9 is given. 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
Thu, Nov 25 No classes, Thanksgiving.
24. Tue, Nov 30 your notes Homework 10 is given. b) vanishing boundary conditions. 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.
25. Thu, Dec 2 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.
26. Tue, Dec 7 JJS 2.6
KKL 3.2
Gauge invariance of Schrodinger equation. Gauge invariant current. Aharonov-Bohm effect. Motion in a uniform magnetic field. Landau gauge. 1d oscillator. Landau levels. Degeneracy of levels. Density of states. Shapes of orbits.
27. Tue, Dec 7 JJS 2.6
KKL 3.2
Special time and place: 5:30-6:50pm, B-131. Confining potential (1-dimensional). Radial gauge. Complex coordinates and wave functions on the lowest Landau level.
28. Thu, Dec 9 LL 111
EM 4.6
KKL 3.2
JJS 2.6
Dirac's magnetic monopole. Topological argument for magnetic charge quantization. Dirac's string. The Wu-Yang monopole.
Tue, Dec 14 FINAL EXAM: Room P-112, 2:15-4:45pm The final is "closed books and notes".