PHYS 684: Quantum Mechanics I
Announcements
Dec. 6, Last lecture. Pick up your homework.
Dec. 20, Final exam, 7:30-10:15pm, in class, close-book.
Time and Location
Monday 7:20-10:00 pm, Aug 30 to Dec 21, Science and Technology I, Rm 310
Office hours
Monday and Friday 2:00-3:00 pm, Science and Technology I, Rm 363B
Grades
Homework (50%) + Midterm (25%) + Final exam (25%)
Homework is due each week before the start of each class.
Late homework will not be graded.
The solutions will be distributed electronically.
Both midterm and final exam will be in-class, closed book. The difficult level of exam problems is comparable
to homework.
Course Goal and Prerequisites
This course will cover the fundamentals of quantum mechanics (QM)
including its concepts and principles, its mathematical structure and applications.
It assumes that you had some previous exposure to quantum mechanics
(either in Modern Physics, Phys 308, or undergraduate QM, Phys 502). If QM is completely new to you,
you are strongly encouraged to work through Griffith, Introduction to Quantum Mechnics, or its equivalent
before coming into the class
or in parallel to the lectures.
Topics
- The leap from classical to quantum
- Review of classical mechanics: trajectory, phase space, Hamilton equation of motion.
- How quantum theory describes a system: state vector, Hilbert space, operator & eigenvalues,
measurement and probability, Schrodinger equation of motion.
- Double slit experiments with photons and electrons.
- Stern-Gerlach experiment.
- Linear vector space in Dirac notation
- Linear vector space, basis, inner product, Dirac ket and bra.
- Linear operators, matrix representation, adjoint, properties of Hermitian and Unitary operators.
- Eigenvalue problem, degeneracy and eigen-space, simultaneous
eigenkets of Hermitian operators.
- Change of basis
- Hilbert space of continuous functions
- From finite to infinite dimension, x basis, inner product, delta function normalization, resolution identity.
- Momentum operator, plane waves, position operator in momentum basis, commutator.
- Transformation between x and p basis.
- Proper and improper vectors, physical Hilbert space.
- Postulates of quantum mechanics
- Wave function/probability amplitude, superposition.
- Ideal measurement, probability (density) and quantum ensemble, collapse of state vector, state preparation.
- Expectation value and uncertainty, compatible v.s. incompatible operators.
- Heisenberg uncertainty relation.
- Gaussian wave packet.
- Quantum dynamics
- Schrodinger equation, examples of Hamiltonian, probability current.
- Dynamics as unitary transformation, propagator, application to 2-level systems.
- Propagator for free particle. Time evolution of a Gaussian wave packet.
- [optional] Propagator as path integral.
- Classical limit, Ehrenfest's theorem.
- QM in one dimension (2 lectures)
- Particle in an infinite potential well, origin of energy quantization, zero point energy.
- 1D harmonic oscillator, energy levels and wave functions.
- Algebraic solution to harmonic oscillator, ladder operator, energy basis.
- Scattering problem in 1D, transmission and reflection coefficients.
- Continuous symmetries: translation and rotation
- Symmetry and conservation law, active versus passive point of view
- Infinitesimal spatial translation, generator and finite translation, translational invariance
- Translation in time, Hamiltonian as the generator of time evolution operator
- Rotation in 2D, orbital angular momentum as rotation generator
- Rotation in 3D, commutation relation of angular momentum
- Spectrum of angular momentum
- Ladder operator, commutation relations, quantization of angular momentum
- Matrix elements of angular momentum operators
- [optional] Finite rotation, invariant irreducible subspace, example: spin 1/2 and spin 1
- Eigen wave function of orbital angular momentum, spherical harmonics
- Rotationally invariant potentials
- Radial equation and its boundary condition; degeneracy due to rotational symmetry
- Bound states in spherical quantum well
- Spherical harmonic oscillators, counting the degeneracy
- Energy levels and wave functions of hydrogen atom
- Discrete symmetries
- Parity operator and parity invariance, 1D examples, parity of spherical harmonics
- [optional] Time reversal, anti-unitary operators, Kramers degeneracy
- Identical particles, permutation operator and permutation symmetry
- QM of many particles
- Hilbert space of N particles, direct/tensor product states
- Symmetrization postulate, bosonic and fermionic statistics
- [optional] Occupation number representation, Fock space
- [optional] Second quantization for bosons and fermions
- [optional] Field operators, single and 2-particle operators
- Density matrix and entanglement [optional]
- Mixed versus pure states, density operator (density matrix).
- Qubits and Bloch sphere.
- Bipartite system, entangled states,
reduced density matrix.
- Revised postulates of quantum mechanics.
Textbook
Principles of Quantum Mechanics, by R. Shankar, 2nd Ed., Springer 1994
There are many excellent textbooks to supplement Shankar, e.g.,
Modern quantum mechanics, by J. J. Sakurai, revised Ed., Addison-Wesley, 1994
Quantum mechanics: non-relativistic theory, L. D. Landau and E. M. Lifshitz