Phys 401, Fall 2018

Topics Covered in Each Lecture

Lecture Number	Date	Content	Corresponding Reading material
1	Sep. 18	Review of Classical mechanics (CM): States, observables, and measurement in CM, Newtonian formulation of classical dynamics, conservative forces, one-dimensional gravitation fall, energy conservation	Pages 75-76 of Shankar
2	Sep. 20	Lagrangian formulation of CM, action functional & the least action principle, Derivation of the Euler-Lagrange equation from least action principle in one dimension, Application to standard Lagrangian	Pages 77-83 of Shankar
3	Sep. 25	Euler-Lagrange equations in n dimensions, from Cartesian to generalized coordinates, application to simple pendulum, the central forces in two dimension, conservation of angular momentum; Symmetry in CM and Nöther’s theorem, translational symmetry and momentum conservation, rotational symmetry and angular momentum conservation, time-translation symmetry and conservation of energy	-
4	Sep. 27	Hamiltonian formulation of CM, the states and phase space of a classical system in Hamiltonian formulation, Hamilton’s equations of motion, observables in Hamiltonians formulation of CM, rate of change of the value of an observable, Poisson bracket and its properties, fixing states and making the observables dynamical	Pages 86-92 of Shankar
5	Oct. 02	Canonical transformations: Definition, characterization, an example, infinitesimal canonical transformation and their generator, symmetries and Nöther’s theorem in Hamiltonians formulation of CM	Pages 92-104 of Shankar
6	Oct. 04	Review of linear algebra: Complex vector spaces, subspaces, span, finite-dimensional vector spaces, linear independence, (algebraic or Hamel) basis, dimension of a finite-dimensional vector space, linear operators, their domain and range, vector-space isomorphisms	Pages 1-7 of Shankar
7	Oct. 09	The unique representation theorem, classification of finite-dimensional complex vector spaces using their dimension, the dual space of a vector space and the dual basis, matrix representation of elements of a finite-dimensional vector space and its dual	-
8	Oct. 11	The Euclidean inner product on C^n, inner product and complex inner-product spaces, examples of inner-product spaces, L^2-inner product on the space of continuous functions defined in an interval, unit and orthogonal elements and subsets of an inner-product space, orthonormal subsets and bases, the matrix representation of vectors and linear operators in an orthonormal basis, the existence of orthonormal bases for a finite-dimensional vector spaces (Gram-Schmidt process)	Pages 7-16 of Shankar
-	Oct. 13	Exam 1
9	Oct. 16	Riesz lemma (Representation Theorem) in finite-dimensions and Dirac’s bra-ket notation, matrix representation of linear operators acting in a finite-dimensional inner product space, matrix representation in an orthonormal basis; Unitary operators mapping an inner product space onto another, inverse of a unitary operator, unitary equivalent inner product spaces, characterization of unitary operators as isomorphisms mapping orthonormal bases to orthonormal bases.	Pages 16-29 of Shankar
10	Oct. 18	The matrix representation of unitary operators in an orthonormal basis, the unitary matrices and unitary groups U(n), Symmetric operators, Hermitian (self-adjoint) operators in finite-dimensions, the matrix representation of Hermitian operators in an orthonormal basis and Hermitian matrices, the adjoint of a linear operator mapping a finite-dimensional inner-product space V to V, properties of the adjoint.	Pages 16-29 of Shankar
11	Oct. 23	Hermitian and unitary operators represented by diagonal matrices in an orthonormal basis, properties of the eigenvalues and eigenvectors of Hermitian and unitary operators, normal operators, statement of the spectral theorem for normal operators in finite-dimensions, spectral representation of a normal operator, Hermitian (unitary) operators as normal operators with real (unimodular) eigenvalues, change of bases and orthogonalization of the matrix representation of a normal operator.	Pages 30-43 of Shankar
12	Oct. 25	Application of the spectral theorem for a Hermitian operator acting in C^2, Degenerate eigenvalues, degeneracy subspaces, the associated projection operators and the orthogonal direct-sum decomposition of the inner-product space	Pages 30-43 of Shankar
13	Oct. 30	The existence of a basis consisting of eigenvectors of a pair of commuting Hermitian operators in finite dimensions, functions of a normal operator acting in a finite-dimensional inner-product space, unitary operators as functions of Hermitian operators	Pages 43-46 of Shankar
14	Nov. 01	Convergence of sequences in an inner-product space V, closure of a subset of V, limit of a function f mapping V to an inner-product space W, continuity of f at a point of its domain, continuous functions f:V->W, derivative of a function g:R->V, Cauchy sequences and the definition of a Hilbert space; Postulates of QM for systems with a finite-dimensional state space: Basic postulate: Representation of quantum systems by a Hilbert space and a Hermitian operator, Kinematical aspects: (Pure) States, state space, observables, and projection axiom	Pages 116-127 of Shankar
15	Nov. 06	Review of the projection axiom, outcomes of a measurement and their probabilities, the expectation value and uncertainty in measuring an observable, the relation between reality of the spectrum of a linear operator and its Hermiticity, the proof that zero uncertainty is achieved only and only by the eigenstates, Instantaneous repetition of a measurement, Application for a system with a 3-dimensional Hilbert space and an observable with two eigenvalues.	Pages 127-129 of Shankar
16	Nov. 08	Compatible observables; Dynamical Aspects of QM: The time-dependent Schrödinger equation, the invariance of the inner product of two evolving states, the time-evolution operator and its unitarity.	Page 129-131 & 145-146 of Shankar
-	Nov. 12	Exam 2
17	Nov. 13	Basic properties of time-evolution operator for time-independent Hamiltonian operators, time-evolution operator for a time-independent Hamiltonian operator, stationary states, computation of expectation values of spin observables for a two-level system.	Page 145-147 of Shankar
18	Nov. 15	The Dyson series for the evolution operator of time-dependent Hamiltonians, time-ordering operation, quantum dynamics in the Heisenberg picture, Heisenberg-picture observables, the Heisenberg equation and its correspondence with Hamilton’s equations of motion, Dirac’s classical to quantum correspondence, the antisymmetry, linearity, Jacobi identity, and Leibnitz rule property of the commutator.	Pages 71-75 & 80-84 of Sakurai’s “Modern Quantum Mechanics”
19	Nov. 20	Dirac’s canonical quantization prescription, the canonical commutation relations; The vector space of classical observables with the Poisson bracket and the vector space of anti-Hermitian operators with (iħ)^-1 time the commutator as Lie algebras, The definition of a Lie algebra, generators of finite-dimensional Lie algebras and its structure constants, representations of a Lie algebra, faithful and unitary representations, the Heisenberg Algebra hd. The proof that hd does not have a finite-dimensional faithful unitary representations.	-
20	Nov. 22	Schauder bases of an infinite-dimensional inner-product space, separable inner-product spaces, orthonormal Schauder bases of an infinite-dimensional separable inner product space, the Hilbert space of square-summable sequences (l^2) as the principal example of an infinite dimensional separable Hilbert space, the uniqueness theorem for infinite dimensional separable Hilbert spaces, Representing the Heisenberg Algebra h1 in l^2 , space of square-integrable functions L^2(D) where D is an interval of real numbers, Representing the Heisenberg Algebra h1 in L^2(R): The position and momentum operators.	-
21	Nov. 27	The domains of the operators X, K, and P, he proof that they are are symmetric operators; Adjoint of a linear operator in infinite dimensions, the distinction between the notions of symmetric operator and Hermitian (self-adjoint) operator; spectrum of Hermitian and unitary operators as the set of approximate eigenvalues, the point and continuous spectrum of a Hermitian or unitary operator, the evaluation map and the spectral problem for X, the Dirac delta function as “generalized eigefunctions” of X, their completeness and orthonormality relations, position basis, spectral representation of X and functions of X; The spectral problem for K, the orthonormality and completeness relation for the “generalized eigrnfunctions” of K, spectral representation of K and functions of K.	Pages 57-70 of Shankar
22	Nov. 29	Spectral theorem for Hermitian and unitary operators acting in an infinite-dimensional Hilbert space, the resolution of identity and the spectral representation of Hermitian and unitary operators, the measurement axiom for quantum system with an infinite-dimensional Hilbert space, the probability of the outcome of a measurement of an observable, its expectation value and uncertainty, the probability density; the measurement of position and momentum operators, the position and momentum wave functions, probability density of localization in space.	Pages 67-70 of Shankar
23	Dec. 04	Superposition principle, the position and momentum representation of observables, quantization of a free particle moving on a straight line, the factor-ordering problem, the system defined by the simplest factor-ordering for the Hamiltonian of a free particle moving on a straight line, the spectrum and (generalized) eigenfunctions of the Hamiltonian operator H=P^2/2m.	Pages 151-152 of Shankar
24	Dec. 06	Time evolution of the eigenstates of the Hamiltonian for a free particle, their position wave function and plane waves, the position wave function for a general evolving state: calculation of the propagator for the free particle. Evolution of a free particle with an initial Gaussian position wave function, the time-dependent Schrödinger equation in position representation for a particle interacting with possibly time-dependent vector and scalar potentials in one dimension, the time-independent Schrödinger and the spectrum of the Hamiltonian operator	Pages 152-157 of Shankar
-	Dec. 08	Exam 3
25	Dec. 11	Interpreting superposition of definite momentum states of the free particle using the standard Bohr/Copenhagen interpretation and Everett’s many world interpretation; Particle in an infinite potential well in one dimension: Classical dynamics, quantization and the Hamiltonian operator, solution of the time-independent Schrödinger equation, the Hilbert space, energy spectrum, energy eigenfunctions, and propagator; Differences between the classical and quantum behavior: Existence of a discrete set of possible outcomes of energy measurements, resistance of the quantum particle to be squeezed into a narrow infinite potential well	Pages 157-164 of Shankar
26	Dec. 13	Particle in a half infinite – half finite square potential barrier: Determination of the energy spectrum (point and continuous spectra) and energy eigenstates, physical interpretation of the generalized eigenfunctions associated with points of the continuous spectrum as scatting states, calculation of the reflection amplitude and the scattering phase shift	-
27	Dec. 18	Finite rectangular barrier potential, scattering solutions, reflection and transmission amplitudes, transfer matrix, tunneling	-
28	Dec. 20	Cauchy-Schwartz inequality and the Heisenberg uncertainty principle, the minimum uncertainty (quasi-classical) states as eigenstates of a non-Hermitian operator, states with minimal product of position and momentum uncertainties in one dimension	Pages 16-17 & 237-241 of Shankar

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