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Quantum Mechanics 1 (BTech)

PYL122BTechEven 2022-23IIT Delhi

A first course in quantum mechanics.

Overview

A first course in quantum mechanics, developing the subject from the physics of quantum spin through the general formalism of states, operators and measurement, and on to wavefunctions, the harmonic oscillator, angular momentum, and the hydrogen atom.

Assessment

  • Minor 1 and Minor 2 (confidence-based assessments) and a Major exam
  • In-tutorial student presentations (in groups)
  • Tutorials covering the lecture content

Lecture outline

  1. General overview of quantum theory.
  2. Classical experiments with “classical spin”, quantum experiments with “quantum spin”; serial (series) vs ensemble (parallel) experiments, probabilities of outcomes, expectation values (averages). (Susskind §1.1–1.3)
  3. Quantum expectation values vs classical results, recovering classical mechanics: two limits, states and measurements in CM, breakdown of classical logic in QM, states in QM as rays in vector spaces. (Susskind §1.4–1.9)
  4. Measurements in CM & QM: preparation of states, classical and quantum spins as vectors in a vector space. (Susskind §1.9, Ch. 2)
  5. Dirac bra-ket notation, properties of vector spaces: inner products, orthogonality, quantum spins as vectors, notation and conventions. (Susskind §1.9, Ch. 2)
  6. Overview of the postulates of QM. (Susskind Ch. 2, §3.2)
  7. Measurements in QM: preparation of states, finite-dimensional vector spaces, operators and eigenvalue equations and their physical significance. (Susskind Ch. 2, §3.1–3.2)
  8. Representation of a spin along an arbitrary unit vector, Bloch sphere. (Susskind §3.1–3.2)
  9. Index notation (dummy and free indices in summations), linear operators and their explicit representation in terms of matrix elements. (Susskind §3.1–3.2)
  10. Expectation values, observables, measurement operators and their action on states, outer products, matrix elements of spin operators, Pauli matrices. (Susskind §3.1–3.7; Nielsen-Chuang §2.2)
  11. Observables and their measurement operators, complete set of commuting observables, first look at tensor products, uncertainty in measurements, uncertainty principle, commutation relations. (Susskind §3.8, Ch. 5, §6.1; Nielsen-Chuang §2.2)
  12. Standard deviation, generalized uncertainty relation, spectral decomposition theorem; composite systems, tensor product spaces, classical correlations. (Susskind Ch. 5, §6.1–6.2)
  13. Composite systems, tensor product spaces, classical correlations. (Susskind §6.1–6.2)
  14. Quantum composite systems, entanglement, tensor products. (Susskind §6.3–6.9, 7.1–7.2)
  15. Quantum composite systems, entanglement, tensor products (continued). (Susskind §6.3–6.9, 7.1–7.2)
  16. Quantum composite systems, entanglement, tensor products (continued). (Susskind §6.3–6.9, 7.1–7.2)
  17. The process of measurement from the point of view of entanglement. (Susskind §7.8)
  18. Einstein vs Bell: two meanings of locality in quantum mechanics, nonlocal behavior of QM; infinite-dimensional vector spaces / continuous spectrum, wavefunctions, position and momentum basis, Dirac delta function. (Susskind §7.9–7.11, §8.1–8.2)
  19. Schrödinger equation for states in a vector space and for wavefunctions, wave-particle duality.
  20. Origin of the Schrödinger equation, Hermitian operators in infinite-dimensional vector spaces, position operators and their commutation relations, translation operator. (Sakurai §1.6)
  21. Properties of the translation operator, unitarity, momentum operator as a generator of translation, commutation relations. (Sakurai §1.6–1.7)
  22. Momentum operator in the position basis; solution to the time-dependent Schrödinger equation (stationary states, time-independent Schrödinger equation). (Sakurai §1.6–1.7; Griffiths §2.1–2.2)
  23. Quantum harmonic oscillator. (Griffiths §2.3; Shankar Ch. 7)
  24. Rotationally invariant problems, spherical harmonics, angular momentum. (Shankar Ch. 12; Griffiths §4.1, 4.3)
  25. Rotationally invariant problems, spherical harmonics, angular momentum (continued). (Shankar Ch. 12; Griffiths §4.1, 4.3)
  26. Hydrogen atom. (Shankar Ch. 13; Griffiths §4.2)

References

  • L. Susskind and A. Friedman, Quantum Mechanics: The Theoretical Minimum
  • M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information
  • J. J. Sakurai, Modern Quantum Mechanics
  • David J. Griffiths, Introduction to Quantum Mechanics
  • R. Shankar, Principles of Quantum Mechanics