Phys 401, Fall 2017

Topics Covered in Each Lecture

Lecture Number	Date	Content	Corresponding Reading material
1	Sep. 18	Review of Classical mechanics (CM): States, observables, and the measurement axiom in CM, Newtonian formulation of classical dynamics, conservative forces, one-dimensional gravitation fall, energy conservation; Lagrangian formulation of CM, action functional & the least action principle	Pages 75-77 of Shankar
2	Sep. 20	Derivation of the Euler-Lagrange equation from least action principle in one dimension, action function and its calculation for a simple harmonic oscillator, derivation of the equation of motion for a simple pendulum using Lagrangian mechanics, Euler-Lagrange equations in n dimensions, equations of motion for a central conservative force in two dimensions and angular momentum conservation	Pages 77-83 of Shankar
3	Sep. 25	Lagrangian coordinates and their associated conjugate momenta, symmetries in Lagrangian mechanics and Nöther’s theorem, translational symmetry and momentum consevation, rotational symmetry and angular momentum conservation, invariance under time translations and conservation of energy; Hamiltonian formulation of CM, the states and phase space of a classical system in Hamiltonian formulation, Hamilton’s equations of motion	Pages 86-90 of Shankar
4	Sep. 27	States, phase space, and 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; Canonical transformations: Definition, characterization, generating functions, examples	Pages 91-97 of Shankar
5	Oct. 02	Infinitesimal canonical transformation and the proof of the existence of their generator, Nöther’s theorem in Hamiltonians formulation of CM, consequences of symmetries, an example with scaling symmetry	Pages 98-104 of Shankar
6	Oct. 04	Hamilton-Jacobi formulation of CM: Canonical transformations that maps the Hamiltonian to zero, the Hamilton-Jacobi equation, the case of time-independent standard Hamiltonians in one dimension; Historical background leading to the development of Quantum Mechanics (QM).	Pages 104-113 of Shankar
7	Oct. 09	The clues for the mathematical structure of quantum mechanics: Discreteness of measured values for some observables and their being continuous for some others, the superposition/interference properties, the idea of using infinite matrices; 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
8	Oct. 11	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, dot product in R^3 and its relation to the notions of direction and distance, generalizing the dot product on R^n to C^n.	-
Exam 1	Oct. 13
9	Oct. 16	The Euclidean inner product on C^n, inner product and complex inner-product spaces, 6 different examples, the inner product on the space of square integrable functions, unit and orthogonal elements of an inner-product space, orthonormal subsets.	Pages 7-9 of Shankar
10	Oct. 18	Expansion of a vector in an orthonormal basis, projection operators along elements of an orthonormal basis and their orthogonality and completeness relations; Linear independence of subsets consisting of mutually orthogonal elements, Gram-Schmidt orthonormalization, and the existence of orthonormal bases in a finite-dimensional inner product space; 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.	Pages 9-16 of Shankar
11	Oct. 23	Operators that are linear combinations of elements of a complete orthonormal family of projection operators given by an orthonormal basis: The case of real and unimodular coefficients; Unitary operators mapping an inner product space onto another, inverse of a unitary operator, unitary equivalent inner product spaces, classification of finite-dimensional complex inner product spaces, characterization of unitary operators in finite dimensions in terms of their action on orthonormal bases; Adjoint of a linear operator acting in a finite-dimension of complex inner product space.	Pages 16-25 of Shankar
12	Oct. 25	Matrix representation of a linear operator and its adjoint in an orthonormal basis, adjoint of a unitary operator, unitary matrices, diagonalizable and Hermitian operators, the eigenvalues and eigenvectors of Hermitian and unitary operators; the spectral theorem for Hermitian and unitary operators in finite dimensions; the spectral resolution of Hermitian and unitary operators when some of the eigenvalues are not simple.	Pages 25-43 of Shankar
13	Oct. 30	Adjoint of linear combination and composition of operators, the eigenspaces (degeneracy subspaces) for the eigenvalues of a Hermitian operator, orthogonal direct sum decomposition in terms of eigenspaces, Statement and proof of the existence of an orthonormal basis consisting of eigenvectors of commuting Hermitian operators, functions of a Hermitian operator, Statement and proof of the characterization theorem for the unitary operators in terms of an associated Hermitian operators.	Pages 43-46 of Shankar
14	Nov. 01	Application of the spectral theorem to examples in two and three dimensions	-
15	Nov. 06	Convergence of sequences in an inner-product space, Cauchy sequences and 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, projection axiom, expectation value of observables	Pages 116-128 of Shankar
16	Nov. 08	Application of the kinematic postulates of QM for a system with a 3-dimensional Hilbert space and a observable with two eigenvalues; Dynamical Aspects of QM, the Hamiltonian operator and formulation of dynamics	Page 143 of Shankar
Exam 2	Nov. 11
17	Nov. 13	Time-dependent Schrödinger equation and the time-evolution operator for a system with a finite-dimensional Hilbert space, the time-evolution operator for time-independent Hamiltonian operators, an example of two-level system with an observable measured at two different instants of time.	Page 145-147 of Shankar
18	Nov. 15	Uncertainty in a measurement, probability of measuring a value in a range of values, Heisenberg picture of quantum dynamics, the dynamical equation in the Heisenberg picture and its correspondence with the Hamilton’s equations, the common properties of the Poisson bracket and the commutator	Pages 128-129 of Shankar
19	Nov. 20	Infinite-dimensional Hilbert spaces, dense and complete subsets, separable Hilbert spaces, Schauder and orthonormal bases, the spaces of square integrable functions and square summable sequences, the operator X f(x)=xf(x) and its domain, symmetric operators, adjoint of an operator and its domain, Hermitian (self-adjoint) operators	-
20	Nov. 22	The operators P=hbar K and K f(x)=-i f’(x). The proof that K and P are symmetric operators and that they do not have any eigenvectors (belonging to L^2(R). Domain of K and P. The correspondence with classical position and momentum observables. The solution of the eigenvalue equation for K in the space of all differentiable functions. The kets |k> and their orthogonality and completeness relations, Dirac delta function and Fourier transform.	Pages 63-67 of Shankar
21	Nov. 27	The spectrum of a Hermitian operator acting in an infinite dimensional Hilbert space, the point and continuous spectra, the spectral theorem for K and functions of K. The operator P and the corresponding (generalized) eigenfunctions, their orthonormality and completeness relations, the spectral theorem for P and functions of P. The operator X and the evaluation functional, the (generalized) eigenfunctions of X, their orthonormality and completeness relations, the spectral theorem for X and functions of X.	Pages 58-61 of Shankar
22	Nov. 29	Spectral representation of a general Hermitian operator, the application of projection (measurement) axiom for an observable with a continuous spectrum, the position measurement in one dimension, the probability density of localization of a particle in space, the position and momentum representations, the kernel of operators in the position and momentum representations.	Pages 67-70 of Shankar
23	Dec. 04	Free particle in one dimension: Solution of the time-dependent Schrödinger equation in the momentum representation, the evolution of the eigenstates of the Hamiltonian, the position wave function for the stationary solutions and plane waves, the position wave function for a general evolving state: calculation of the propagator for the free particle.	Pages 151-154 of Shankar
24	Dec. 06	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 stationary states for time-independent vector and scalar potentials, the time-independent Schrödinger and the spectrum of the Hamiltonian operator, a classical particle in an infinite potential well, a quantum particle in an infinite potential well: The solution of the time-independent Schrödinger	Pages 154-160 of Shankar
Exam 3	Dec. 09
25	Dec. 11	Particle in an infinite potential well (box): The solution of the time-dependent Schrödinger, A particle in a semi-infinite rectangular barrier: The classical dynamics, the solution of the time-independent Schrödinger	-
26	Dec. 13	The energy spectrum and eigenfunctions for a particle interacting with a semi-infinite rectangular barrier potential: The bound and scattering states, and the calculation of scattering phase shift	-
27	Dec. 18	Finite rectangular barrier potential, reflection and transmission amplitudes, tunneling	-
28	Dec.  20	Cauchy-Schwartz inequality and the Heisenberg uncertainty principle, the minimum uncertainty (quasi-classical) states 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.