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Contents |
Quantum Mechanics I (PHY 212) and Quantum Theroy I (PHY 513)
Instructor: Dr. [Sabieh Anwar]. Office hours are Tuesday, 10am to 12 noon.
Textbooks: "Quantum Mechanics: Theory and Experiment" by Mark Beck and "A Modern Approach to Quantum Mechanics" by John S. Townsend. (Primary textbooks). Three copies are available in the Course Reserve section in the LUMS Library.
Click here for the course outline.
Pre-mid term
Invoking quantum states and their transformations through the polarization concept (3 lectures)
- The Polarization Vector
- Linear, Circular, Elliptical Polarization
- Birefringent Materials
- Manipulating polarization through optical elements
- Polarizers, polarization analyzers and quantum projective measurements
- Thought experiments demonstrating the role of superpositions, and measurements creating states
- Inner products, probability amplitudes and quantum probabilities
Here is an interesting list of animations showing the polarization states of light.
Homework 1: (due date=10 September 2015; 5:00 pm) Homework on Polarization Description of Light in the Ket Notation. Solution.
Recitation on Quantum states exemplified by the polarization of light, 4 September 2015. Solution.
States and operators (5 lectures)
- Representing quantum states and operators
- Similarity transformation and bases change
- Unitary, projective, Hermitian operators and their properties
- Observables and their relation to Hermitian operators, expectation values
- Indeterminacy principle and its applications. Here is a clean derivation.
Homework 2: (due date=1 October 2015; 5:00 pm) Homework on Quantum Operators. Solution.
Quiz 1: Quiz 1 and its solution.
Reading suggestions for the first eight lectures: Beck Ch 1 to 5.
Spin-1/2 systems (3 lectures)
- Stern-Gerlach's experiments
- Interference of spin states
- Bloch sphere and Pauli operators
- Commutation relations between spin-1/2 operators
- Rotations on the Bloch sphere and quantum logic gates (Quantum NOT, Hadamard gates)
Reading suggestions Beck Ch. 6.
Time evolution of quantum states (4 lectures)
- Time-dependent Schrodinger equation
- Spin-1/2 inside a magnetic field
- Nuclear magnetic resonance, rotating wave approximation
- Ammonia molecule inside an electric field
- Muon spin rotation
Reading suggestions Beck Ch. 9 and Townsend Ch. 4.
- Slides on Muon Spin Rotation
- Review article for self-study: On the Spin-polarized muons in condensed matter physics (Blundell, S. J., 1999, J. Contemp Phys.: Condens. Matter, 40, 175)
Homework 3: (due date=15 October 2015; 10:00 am) Homework on Miscellaneous Concepts. Solution.
Recitation 2: (16 October 2015) Evolution on the Bloch sphere and its solution.
Midterm
Post-mid term
Angular momentum in quantum mechanics (3 lectures)
- Simultaneous eigenstates of commuting observables
- Non-commutativity of rtation operators and angular momentum operators
- Angular momentum operators are generators of rotations
- Eignestates and eigenvalues of angular momentum operators
Homework 4: (due date=4 November 2015; 5:00 pm) Homework on Generalized Angular Momentum. Solution.
Quiz 2: Quiz 2 and its solution.
Quiz 3: Quiz 3 and its solution.
Homework 5: (due date=20 November 2015; 10:00 am) Homework on Miscellaneous Topics. Solution.
Reading suggestions Beck Ch. 7 and Townsend Ch. 3.
Wavefunction (3 lectures)
- Concept of wavefunction and probabilistic interpretation
- Wavefunction of a position eigenstate
- Position and momentum states
- Translation operator
- Position-momentum uncertainty
- Fourier transforms
- Evolution of expectation values and correspondence with classical laws
Reading suggestions Beck Ch. 10 and Townsend Ch. 6.
Wave mechanical problems in 1D (4 lectures)
- Wavepacket and its time evolution
- Wave mechanical formulation of Shcrodinger equation
- Infinite and finite Well
- Scattering problems and probability currents
- Tunneling and its applications: here are some examples taken from Modern Physics by Taylor.
Reading suggestions Beck Ch. 11
Recitation 3: (26 October 2015) Recitation on wave mechanics.
Supplementary material and simulations:
- A nice java applet that simulates the behavior of plane waves and wave packets as they encounter potential energy discontinuities is shown in class and can be downloaded from here.
- For introducing and showing the time dependence of the wave functions (stationary as well as superpositions) I am relying on the wonderful java scripts written by W. Christian. You can download these, read up a little bit on them and enjoy in your comfort zone.
- An animation on the scanning tunneling microscope can be seen here.
- A collection of slides on single electron transistors, showing actual micrographs and Coulomb oscillations.
Homework 6: (due date=2 December 2015; 5 pm) Homework on wave mechanics in one dimension. Solution.
Quiz 4: Quiz 4 and its solution.
The Harmonic Oscillator (4 lectures)
- Eignestates and eigenvalues of the Hamiltonian
- Creation and annihilation operators, number states
- Hermite polynomials as wavefunctions and classical correspondence
- Coherent states: coherent states as displaced ground states, as eignestates of the annihilation operator, their significance
- Some applications: ground state of a mechanical oscillator, photons in a cavity, quantized oscillations in an LC circuit
Homework 7: (due date=15 December 2015; 5 pm) Homework on the harmonic oscillator. Solution.
Reading suggestions Beck Ch. 12 and Townsend Ch. 7
Supplementary material and simulations:
- A nice java applet that that shows the behavior of a single particle in bound states in one dimension. It solves the Schrödinger equation and allows you to visualize the solutions can be downloaded from here.
I will not assess you on three-dimensional wave mechanics.
Wave Mechanics in Three Dimensions (0 lectures)
- Central potential
- Solving the radial, polar and azimuthal wave functions under central potential
- Orbital angular momentum
- Rotational symmetry and conservation of angular momentum
Recitation, reading suggestions and simulations:
- Solving the Angular part of the Schrodinger Equation.
- Solving the radial Schrodinger equation under central potential
- List of some low order spherical harmonics, radial wavefunctions and their plots
- Some solved problems: dealing with wavefunctions of the hydrogen atom. Solution
- Now here is a link to a nice java applet that let's you visualize the wavefunctions of the hydrogen atom.
Video recordings:
Angular part of Schrodinger equation for a central potential. [1]
Orbital angular momentum. [2]
Radial momentum operator, Rigid rotor[3]