Announcements

Lecture No	Date	Content	Corresponding Reading material
1	Sep. 19	Review of classical mechanics (CM): States and observables in CM, kitenmatic and dynamical aspects of CM; Formulation based of the choice of dynamical eq (e.g. Newton's eqs of motion.); Conservative forces and conservation energy, integrability in dimension 1; Hamiltonian formulation of CM, Hamiltonian dynamical systems, Poisson bracket	--
2	Sep. 21	Lagrangian Formulation of CM: Variational principle and Euler-Lagrange eqs; Postulates of CM concerning measurement of observables: Principles of perfect determination, indefinite reducibility of measurement errors, and continuity of observables; Basics of Statistical CM: Probability density and mean value of observables, derivation of the continuity and Liouville eqs; Review of Linear Algebra: Complex vector and inner-product spaces, norm, convergence of sequences in an inner-product space, Cauchy sequences, Definition of a complex Hilbert space	Pages 7-12 of textbook
3	Sep. 28	Supplementary Lecture on Classical Mechanics I: Legendre transformations and Hamilton equations of motion, cyclic coordinates and conservation theorems; Canonical transformations, example: Harmonic oscillator.
4	Oct. 03	Complex Euclidean spaces and their orthonormal bases, Space of square summable sequences l^2 and its standard orthonormal (Schauder) basis, the space of square integrable functions L^2(R), the notions of orthonormal subsets and bases of a Hilbert space, separable Hilbert spaces.
5	Oct. 05	Subspaces of a vector space, Linear operators, Matrix representation of a linear operator mapping between finite-dimensional vector spaces, the matrix representation using orthonormal bases, Continuous functions and bounded operators, (topological) dual of an inner product space, the Dirac bra-ket notation, complete sets of orthogonal projection operators associated with an orthonormal basis of a separable Hilbert space, symmetric operators.
6	Oct. 07	Hamilton-Jacobi Equation, Hamilton's Principal and Characteristic functions, Harmonic Oscillator Problem as an example of Hamilton-Jacobi Method, Hamilton-Jacobi Equation for Hamilton's Characteristic Function
7	Oct. 10	Algebra of linear operators, connection of matrix algebra, linear operators in a finite-dim. Hilbert space: symmetric operators and Hermitian matrices, eigenvalue problem, spectral theorem for Hermitian matrices, Pauli matrices and their spectral representation
8	Oct. 12	Spectral theorem for symmetric operators acting in a finite-dim Hilbert space; Linear operators acting in an infinite-dimensional (separable) Hilbert space: Dense subsets, adjoint of a linear densely-defined operator, self-adjoint (Hermitian) operators and their difference with general symmetric operators; the point spectrum and regular values of a linear operator, The operator (Xf) (x)=xf(x) acting in L^2(R): The proof that it has no eigenvalues (eigenfunctions) but that real numbers are not regular values; Spectrum of a self-adjoint operator (point and continuous spectrum), the geometric multiplicity and degenerate eigenvalues, Example the parity operator; Linear operators with a discrete spectrum. Example (Hf)(x)=(-f''(x)+x^2f(x))/2.
9	Oct. 14	Projection Operators: Closed subspaces and orthogonal complement of a subset, projection operator onto a closed subspace and their spectral representation, Orthogonal direct sums, spectral theorem for self-adjoint operators with a discrete spectrum, functions of self-adjoint operators with a discrete spectrum.
10	Oct. 17	Isometric and unitary Operators: characterization of unitary operators as basis transformations; unitary operators acting in a single separable Hilbert space, adjoint of a unitary operator, unitary groups of the Hilbert space, the U(N) groups
11	Oct. 19	Spectral theorem for unitary operators with a discrete spectrum; Basic Structure of Classical Mechanics: Mathematical description of classical systems in terms of phase space+Hamilton function pairs, kinematics (states, phase space, observables, dynamics (Hamilton equations); equivalent phase space+Hamilton function pairs describing the same classical system (symplectomorohisms as generalizes canonical transformations), precise mathematical definition of a classical system; Basic Structure of Quantum Mechanics: Mathematical description of quantum system in terms of Hilbert space+Hamiltonian operator pairs, kinematics (pure states as rays in a Hilbert space, description in terms of projection operators, projective Hilbert space, complex projective spaces
12	Oct. 24	Observables in QM and the Projection Axiom for observables with a discrete spectrum: probabilistic nature of making observations in QM and its difference with the probabilistic description of classical systems, the formula for computing the probability of measuring each possible outcome and the expectation values of an observable. Probabilities and expection values as functions mapping the projective Hilbert space into [0,1] and R; Theorem: A linear operator with a discrete spectrum is Hermitian iff its expectation value in every state is real. Corollary observables must be Hermitian (once the Hilbert space is fixed.)
13	Oct. 26	Spectral theorem for self-adjoint operators having a continuous spectrum: Lack of eigenvectors, Gelfand triplets and generalized eigenfunctons, orthonormality and completeness relations, spectral representation of the operator and its functions, Dirac delta-function as a generalized function, representation of Dirac delta-function using a sequence of functions
14 Exam 1	Oct. 31	See Midterm Exam 1 for details.
Kurban Bayramı	Nov. 07-11
15	Nov. 14	Projection (measurement) axiom for observables with a continuous spectrum, the correspondence principle and Dirac's canonical quantization scheme, orthonormality and completeness relation associated with the position and wave number operators: the x- and k-basis
16	Nov. 16	Fourier transformation as a basis transformation, the momentum eigenfunctions and their orthonormality and completeness relation, the position and momentum wave functions, various identities and relations involving the x-, k-, and p-bases; Schwarz Lemma and the general uncertainty relation, Heisenberg's Uncertainty principle	Pages 68-87 of textbook
17	Nov. 21	Quantum Dynamics: Schrödinger equation, time-evolution operator and its unitarity, Stone-von Neumann Theorem, Liouville-von Neumann equation, Stationary states, time-evolution operator for systems with a complete set of stationary states (derivation of the necessary and sufficient condition of the latter, i.e., the existence of a complete set of constant eigenvectors of the Hamiltonian.), Propagator and its calculation for a free particle moving on R.	Pages 100-106, 121-123 of textbook
18	Nov. 23	Schrödinger equation in the energy and position representations, probability density for localization in space, polar form of the Schrödinger equation and the derivation of the quantum Hamilton-Jacobi and continuity equations, David Bohm's Causal Interpretation of QM and semiclassical approximation.	-
19	Nov. 28	Semiclassical (WKB) wave functions in 1-dim, application to a piecewise constant potential. The transfer matrix and bound states.	Pages 379-381, 578-582 of textbook
20	Nov. 30	Class was cancelled because of the Fire Department practice.
Exam 2	Dec. 02
21	Dec. 05	Probability Current density as the density of the expectation value of the velocity observable, basic setup for one-dimensional scattering problem, reduction of the problem from a wave packet to its Fourier modes.	Pages 141-146 of textbook
22	Dec. 07	Reflection and transmission coefficients R^2 and T^2 and their expressions in terms of the transfer matrix M. The proof that det(M)=1 and that probability conservation implies that R^2+T^2=1; Bound state and scattering states of a delta-function potential.	Pages 146-150 of textbook
23	Dec. 12	Formulation of dynamics in the Heisenberg picture, Heisenberg's equations of motion vs Hamiltonian equations, Treating the eigenvalue problem for a simple harmonic oscillator (SHO) in the Heisenberg's picture (Heisenberg's matrix approach.)	Pages 154-156 of textbook
24	Dec. 14	Computation of the energy spectrum of SHO using the Heisenberg' s approach; algebraic treatment of SHO, creation and annihilation operators and coherent states	Pages 156-164 of textbook
25	Dec. 19	Time-independent Perturbation Theory: General methodology of perturbation theory, perturbative solution of a transcendental equation, derivation of the perturbation series for the eigenvalues and eigenvectors of a self-adjoint operator with a discrete and nondegenerate spectrum.	Pages 357-361 of textbook
26	Dec. 21	Application of time-independent perturbation theory to quartic oscillator; time-independent degenerate perturbation theory.	Pages 362-364 of textbook
Exam 3	Dec. 23
27	Dec. 26	Time-dependent unitary transformations as the quantum analogs of time-dependent canonical transformations, Dynamics in Dirac's interaction picture, basics of time-dependent perturbation theory, Dyson series.	Pages 127-128 & 366-367 of textbook
28	Dec. 28	Perturbative computation of the transition probabilities in typical scattering processes and Fermi's Golden Rule.	Pages 325-333 of Sakurai

Note: The pages from the textbook listed above may not include some of the subjects covered in the lectures.