Phys 503, Fall 2012
Topics Covered in Each Lecture
|
Lecture Number |
Date |
Content |
Corresponding Reading material |
|
1 |
Sep. 20 |
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 |
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|
2 |
Sep. 26 |
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; |
Pages 7-12 of
textbook |
|
3 |
Sep. 27 |
Action functional, Legendre transformations and passage from Lagrangian
to Hamiltonian formulation of CM, canonical transformations, the
Hamilton-Jacobi formulation of CM; Introduction to Mathematical Structures
used in QM: Complex vector and inner-product spaces, norm and distance
defined by an inner-product, convergence of sequences in an inner-product
space, Cauchy sequences, Hilbert spaces. |
|
|
4 |
Oct. 03 |
Orthonormal bases of an inner product space, separable Hilbert
spaces, the space of square-summable sequences, the space of
square-integrable functions, C^0[0,1] with L^2-inner product as an example of
an inner-product space that is not complete. |
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|
5 |
Oct. 04 |
Closed and dense subsets of an inner-product space, subspace of
a vector space, closed subspaces of a Hilbert spaces, linear operators,
continuous functions relating inner-product spaces, bounded linear operators,
equivalence of continuity and boundedness for linear operators, X and D
operators as examples of unbounded operators acting in L^2(R), matrix
representation of linear operators relating finite-dimensional inner-product
spaces, the matrix representation of a linear operator in an orthonormal
basis, topological dual of an inner-product space, Dirac notation for dual
vectors for a separable Hilbert space. |
|
|
6 |
Oct. 10 |
Projection operators defined by an orthonormal basis, complete
and orthogonal sets of projection operators, symmetric operators,
orthogonality of the null space and range of a symmetric operator, orthogonal
projections, eigenvalue problem for linear operators acting in a vector
space, matrix eigenvalue problem |
|
|
7 |
Oct. 11 |
Matrix representation of the projection operators |a><a|
for a finite-dimensional inner-product space, spectral theorem for Hermitian
matrices, application to Pauli matrices, spectral theorem for symmetric
operators acting in a finite-dimensional inner-product space, densely defined
linear operators, adjoint of a linear operator acting in an
infinite-dimensional inner-product space, spectrum of a linear operator, spectrum
of a self-adjoint operator, spectrum of the X operator [(Xf)(x):=xf(x)]
acting in the Hilbert space L^2(R). |
|
|
8 |
Oct. 17 |
Spectrum of a self-adjoint operator (point and continuous spectrum), geometric multiplicity of eigenvalues, operators with a discrete spectrum, closed subspaces and the orthogonal complement of a subset, orthogonal 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. |
|
|
9 |
Oct. 18 |
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, 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. |
|
|
Kurban Bayramý |
Oct. 24-28 |
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|
10 |
Oct. 31 |
Review of Lagrangian and Hamiltonian formalism of classical mechanics, Legendre transformations, symmetry principles in hamiltonian formalism and canonical transformations |
|
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|
Nov 02 |
Canonical transformations and harmonic oscillator as an example of canonical transformations, Hamilton-Jacobi theory, construction of hamilton-jacobi equation and harmonic oscillator problem using hamilton-jacobi equation |
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|
12 |
Nov. 07 |
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.) |
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13 |
Nov. 08 |
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, Projection (measurement) axiom for observables with a continuous spectrum |
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|
14 |
Nov. 14 |
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; Fourier transformation as a basis transformation, the momentum eigenfunctions and their orthonormality and completeness relation, the position and momentum wave functions |
Pages 68-82 of
textbook |
|
15 |
Nov. 15 |
Various identities and relations involving the x-, k-, and p-bases; Schwarz Lemma and the general uncertainty relation, Heisenberg's Uncertainty principle, Quantum Dynamics: Schrödinger equation, time-evolution operator and its unitarity, Stone-von Neumann Theorem. |
Pages 82-87 & 100-102 of
textbook |
|
Exam 1 |
Nov. 21 |
|
|
|
16 |
Nov. 22 |
Stationary states, time-evolution operator for systems with a complete set of stationary states, Schrödinger equation in the energy and position representations. |
Pages 100-106 of textbook |
|
17 |
Nov. 28 |
Schrödinger equation in position representations and 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. | Pages 379-382 and 578-582 of textbook |
|
18 |
Nov. 29 |
Probability Current density as the density of the expectation value of the velocity observable, solution of the time-independent Schrödinger equation for a piecewise constant potential: The transfer matrix, bound states, and tunneling |
Pages of 141-146 textbook |
|
19 |
Dec. 05 |
Basic setup for one-dimensional scattering problem, Reflection and transmission coefficients and their expressions in terms of the transfer matrix M. Drivation of the formula for the determinant of M using Wronskian of scattering solutions from the left and right, Bound states as real and negative poles of the Reflection and transmission amplitudes. | Pages 146-148 of textbook |
|
20 |
Dec. 06 |
Manifestation of the conservation of the total probability in terms of the reflection and transmission coefficients; Scattering from a delta-function potential and its bound state; Formulation of dynamics in the Heisenberg picture, Heisenberg's equations of motion vs Hamiltonian equations, matrix elements of the Heisenberg's equations for a time-independent Hamiltonian with a discrete spectrum |
Pages 146-150, 125-127 & 155 of textbook |
|
21 |
Dec. 12 |
Definition and properties of the propagator for time-dependent Schrödinger operator, derivation of the propagator for a free particle in one dimension; Simple harmonic oscillator: the ladder operators, the coherent state wave functions, the minimum energy state |
Pages 109-112 of Sakurai & Pages 158-159 of textbook |
|
22 |
Dec. 13 |
Construction of the energy eigenvectors and the spectrum of a simple harmonic oscillator; Perturbation theory: Perturbative solution of transcendental equations; setup and recursion relation for time-independent perturbation theory |
Pages 159-164 & 357-359 of textbook |
|
Exam 2 |
Dec. 19 |
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23 |
Dec. 20 |
Class was cancelled because of unfavorable weather conditions. |
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24 |
Dec. 26 |
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, application of time-independent perturbation theory to quartic oscillator |
Pages 357-361 & 363-364 of textbook |
|
25 |
Dec. 27 |
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, perturbative computation of the transition probabilities in typical scattering processes and Fermi's Golden Rule. |
Pages 127-128 & 366-367 of textbook and Pages 325-333 of Sakurai |
Note: The pages from the textbook listed above may not include some of the subjects covered in the lectures.