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Quantum Mechanics I
Instructor: Dr. [Sabieh Anwar] Office hours: are Monday and Wednesday, 4 to 5 pm.
Teaching Fellows: [Aroosa Ijaz] Office hours: Monday (10:00 am to 12:00 noon), [Mustafa Afzal Saeed] Office hours: Thursday (10:00 am to 12:00 noon) and [Kaneez Amna]. Aroosa and Mustafa will be available in the Physics Department's lounge at the available times.
Textbooks: "Quantum Mechanics: Theory and Experiment" by Mark Beck. (Primary textbook). Three copies are available in the Course Reserve section in the LUMS Library. Copies of the supplementary textbook "Quantum Mechanics: Concepts and Applications" are also available in the Coourse Reserve Section.
Click here for the course outline. Homeworks, quizes and exams have been temporarily suppressed from the website.
Pre-mid term
Classical Description of Polarization (3 lectures)
- The Polarization Vector
- Linear, Circular, Elliptical Polarization
- Birefringent Materials
- Manipulating polarization through optical elements
Here is an interesting list of animations showing the polarization states of light.
Quantum States (2 lectures)
- Quantum Interference
- Inner product
- Basis states
- Experiments with beam splitters
Quantum Operators (3 lectures)
- Unit, Unitary, Hermitian operators
- Projection, Rotation operators
- Basis transformation
- Representation of states and operators
Measurement (2 lectures combined as one)
- Postulates of quantum mechanics
- Von Neumann measurements
- State preparation, measurement outcomes and their probabilities
- Indeterminacy relations between non-commuting observables
- Special Drill: (date=25 February 2014; 7:00 pm - 8:45 pm) Drill Session on quantum operators
Video recordings:
Unit, adjoint, rotation, projection operators. [1]
Representing states and operators, Similarity transformation.[2]
Change of Bases, Hermitian operators, real eigenvalues.[3]
Spectral decomposition of Hermitian operators, Measurement outcomes, Indeterminacy principle.[4]
Spin-1/2 (3 lectures)
- The Stern-Gerlach experiment
- Spin states
- Interference of spin states
- Commutation relations for spin angular momentum operators
- Bloch sphere, rotations on the Bloch sphere
- Single qubit logic gates (NOT and Hadamard gates)
Video recordings:
Force on a Magnetic Dipole, Interference of spin states. [5]
Commutation relations, Bloch sphere.[6]
Single spin-1/2 quantum gates. [7]
Post-mid term
Angular Momentum in Quantum Mechanics (3 lectures)
- Compatible observables
- Simultaneous eigenbasis
- Generalized angular momentum
- Eigenvalues and eigenstates of angular momentum
Video recordings:
Compatible observables, Simultaneous eigenbasis, Spin-1/2 angular momentum. [8]
Generalized angular momentum. [9]
Exemplifying the angular momentum. [10]
The Schrödinger Equation (4 lectures)
- Temporal evolution of quantum states
- Solutions to Schrödinger Equation
- Temporal evolution of expectation values
- Magnetic Resonance
- Muon spin rotation
- NMR Experiment
- Rabi Oscillations
Tutorial and reading suggestions:
- 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)
Video recordings:
Solution to time dependent Schrödinger Equation , temporal evolution of expectation values.[11]
Magnetic Resonance. [12]
NMR experiment, Rabi flipping. [13]
Muon spin rotation. [14]
The Wavefunction (3 lectures)
- Wavefunction
- Dirac delta function
- Wavefunction of a position eigenstate
- Position and momentum states
- Translation operator
- Position-momentum uncertainty
- Fourier transforms
Meaning of the wavefunction, wavefunction of a position eigenstate. [15]
Measuring momentum, position-momentum commutator. [16]
Momentum basis, momentum eigenstate in position basis, Fourier transforms. [17]
Wave Mechanical Formulation of Schrödinger Equation (3 lectures)
- The Schrödinger equation in the position basis
- The time-independent Schrödinger equation
- Potential step and barriers
- Infinite square well
- Tunneling and its applications
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.
- The world's smallest movie can also be seen here.
- Gallery of scanning tunneling micrographs shown in class.
- 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.
The Schrödinger equation in position basis, Potential step. [18]
Scattering and tunneling. [19]
Radioactivity, Wavepackets. [20]