Quantum Mechanics 1 for Physicists, PHY-106.7

Course outline: 2022_QM1.pdf

Syllabus

1. Fundamental concepts: Stern-Gerlach experiment; State vectors and operators; Bra-Ket notation: Hilbert space, Inner products; Matrix representation: Eigenkets, Spin-1/2 system, Measurements: Observables, Compatible/Incompatible observables, Uncertainty relations; Change of basis: Transformation, Continuous representation: Position/Momentum representation, Dirac delta function, Gaussian Wavepackets.


2. Quantum dynamics: Time evolution and Schroedinger equation: Energy eigenkets, Stationary/nonstationary states, Spin precession; Schroedinger/Heisenberg picture: Ehrenfest theorem, Transition amplitude; Simple harmonic oscillator: Stationary states, Time-evolution; Wave mechanics: Probability density, Classical limit; Elementary solutions to Schroedinger wave equation: Free particles, Infinite-square well, Finite-square well, Transmission-Reflection problems, Simple harmonic oscillator, Linear potential.


3. Theory of angular momentum: Rotations: Finite/infinite rotations, Commutation; Spin-1/2 system; Pauli 2-component quantum mechanics; Continuous groups: SO(3), SU(3), Euler rotations; Density operators: Pure-vs-mixed ensembles, time-evolution of ensembles, Quantum statistical mechanics; Eigenvalues and eigenstates of angular momentum; Orbital angular momentum: Spherical harmonics; Central potential problems, Hydrogen atom; Angular momentum algebra: Angular momentum addition, Clebsh-Gordon coefficients; Oscillator model of angular momentum; Spin correlation measurements; Tensor operators: Wigner-Eckart theorem.


4. Approximation methods: Time-independent perturbation theory; Time-dependent perturbation theory; Application of perturbation theory to higher-order effects in Hydrogen atom; Degenerate and nondegenerate versions; Variational method; WKB method.

References

Modern Quantum Mechanics, J. J. Sakurai and J. J. Napolitano, Cambridge University Press (Edition-3, 2021).

Notes on selected topics for self-study

You can read these notes alongside the section in the reference textbook given on the right side. This section will be updated as the course progresses.

1. Application of Lambert-W function to derive Wien's displacement law , Preliminary topics
2. Paradoxes of a classical electron , 1.1: The Stern-Gerlach Experiment
3. Linear vector space and Hilbert space , 1.2: Kets, Bras and Operators
4. Canonical transformation , 1.6: Position, Momentum, and Translation
5. Degeneracy theorem and Wronskian , 2.4: Schroedinger's Wave Equation
6. Series solution for particle-in-a-box , 2.5: Elementary Solutions to Schroedinger's Wave Equation and Appendix B
7. Properties of a physically acceptable wavefunction , 2.5: Elementary Solutions to Schroedinger's Wave Equation

What are some good textbooks to consult for QM-1?

For the 2022 course, please follow Sakurai's book along with the additional notes provided above. In the next years, we may offer the QM-1 course based on a combination of the following texts.

1. Modern Quantum Mechanics, J. J. Sakurai, J. J. Napolitano, Cambridge University Press (Edition-3, 2021).
2. Principles of Quantum Mechanics, R. Shankar, Springer (Edition-2, Sixth Indian Reprint 2015).
3. Introduction to Quantum Mechanics, D. J. Griffiths, D. F. Schroeter, Cambridge University Press (Edition-3, 2018).
4. Quantum Physics, Michel Le Bellac, Cambridge University Press (Edition-1, 2006).
5. Quantum Mechanics: Fundamentals, Kurt Gottfried, Tung-Mow Yan, Springer (Edition-2, 2003).
6. Lectures on Quantum Mechanics, Steven Weinberg, Cambridge University Press (Edition-2, 2015).

There are several books that discuss certain topics remarkably well. Here is a short list.

1. Introductory Quantum Mechanics, Richard L. Liboff, Pearson (Edition-4, 2002).
2. A Modern Approach to Quantum Mechanics, John S. Townsend, Viva (First Indian Edition, 2010, Reprinted 2017).
3. Quantum Mechanics, David McIntyre, Corine A. Manogue, Janet Tate, Pearson (First Indian Edition, 2016).
4. Quantum Mechanics: Theory and Experiment, Mark Beck, Oxford University Press (Edition-1, 2012).

Further, there is a long list of classic texts that I will list some other time.

Optional reading on measurement, entanglement and related topics

We will not discuss these topics in this course, but for those interested in getting some idea of these topics, here is a list of references.

1. Quantum Physics: A First Encounter, Valerio Scarani, Oxford University Press (Edition-1, 2006).
2. A Short Introduction to Quantum Information and Quantum Computation, Michel Le Bellac, Cambridge University Press (Edition-1, 2006).

Optional reading of interesting articles and reviews

This list is maintained (and will be regularly updated) for the sake of collecting interesting articles that can be studied/discussed during the QM-1 course. Feel free to go through them. If you have any recommendations to this section, please send them to ramakrishnan@tifrh.res.in.

1. Against Measurement , John Bell, Physics World, Volume 3, Number 8 (1990), pages 33-40.
2. Ten theorems about quantum mechanical measurements , N. G. Van Kampen, Physica A, Volume 153, Issue 1 (1988), pages 97-113.
3. The Stern-Gerlach experiment revisited , Horst Schmidt-Böcking et al., The European Physical Journal H, Volume 41 (2016), pages 327-364. [arXiv link]
4. Albert Einstein's explanation of how science works from Physics: A Conceptual World View, Larry Kirkpatrick, Gregory E. Francis, Cengage Learning (2009).
5. Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics , Bretislav Friedrich and Dudley Herschbach, Physics Today, Volume 56, Number 12 (2003), pages 53-59.
6. One hundred years of Alfred Landé's g-factor , Bretislav Friedrich, Gerard Meijer, Horst Schmidt-Böcking, Gernot Gruber, Natural Sciences, Volume 1, Issue 2 (2021), pages 1-7.

Internet sources for images, videos, blogs, etc.

1. https://toutestquantique.fr/en/ — contains animations of experiments (such as the Stern-Gerlach experiment) that we will discuss in the course.