# Introduction to Quantum Mechanics

Phys 451 — Fall 2014 |

Department of Physics & Astronomy |

Instructor | Dr. Kevin Beach |

Office: | 206 Lewis Hall |

Email: | kbeach@olemiss.edu |

Website: | https://www.phy.olemiss.edu/~kbeach |

More details provided in the syllabus.

# Assignments

- due Tuesday, September 9
- due Thursday, September 18
- (ungraded) preparation for the test on Thursday, September 25
- due Tuesday, October 14
- due Tuesday, October 28
- due Tuesday, November 18

# Lectures

- Review of the syllabus; differences between classical and quantum mechanics in the Lagrangian and Hamiltonian schemes; features of QM that break our classical intuition; Dirac notation and the operator formalism; linear vector spaces; orthonormality and completeness of states; time evolution of quantum states; expectation values and matrix elements; position representation; wave function of a freely propagating particle—see Robinett 1.1–1.5, 6.1–6.2, 12.1–12.2; Fitzpatrick 3.1–3.4
- Probabilistic interpretation of QM; position and momentum representations as Fourier transform pairs; momentum operator; particle in a nonzero potential; properties of the wave function; current operator and continuity equation for the local probability density—see Robinett 3.1–3.2, 4.1–4.5; Fitzpatrick 2.1–2.5, 3.14–3.16
- Probabilty current arising from a spatially varying phase angle; time evolution of expectation values; commutator relations; stationary states; eigenfunctions and eigenvalues; inner products and the expansion postulate—see Robinett 4.6–4.8, 12.4–12.5; Fitzpatrick 4.1–4.12
- Expansion in a complete basis of energy eigenstates; closure relation; normalization; expectation value of energy; measurement postulate; infinite square well; continuity and smoothness of the wave function; reflection symmetry and the parity of the wave function—see Robinett 5.1–5.3, 6.3–6.7; Fitzpatrick 5.1–5.2
- Finite square well; applying matching conditions for a piecewise-defined wave function; particle in a double well; two-level systems—see Robinett 8.2; Fitzpatrick 5.7
- Review of Assignment 1; image potential near a metallic surface—see Robinett 8.3; Fitzpatrick 7.1–7.3
- Exponential envelope in the asymptotic limit; power series solutions; barrier tunnelling problems; transmission and reflection coefficients—see Robinett 11.1–11.2; Fitzpatrick 5.8, 9.4, 5.3
- Transfer Matrix approach to barrier tunnelling
- Bras, kets, and the relationship to linear algebra; matrix mechanics using a finite basis set; review of Assignment 3—see Robinett Ch 10.4
- Review of the first in-class test; expansion postulate; basis of free-particle states and the relationship to Fourier transforms—see Robinett 2.2–2.3
- Time evolution of a wavefunction in a basis of energy eigenstates—see Fitzpatrick 4.9, 4.11–12
- Gaussian wave packets; probability current; dispersion; tunnelling—see Robinett Ch 2.5; Fitzpatrick 3.10–3.13
- Expectation values and their time dependence; numerical approaches based on discretization and iteration; cutoffs and resolution; spectral methods—see Robinett 10.1, Fitzpatrick 3.15, 4.1–4.5
- Quantum harmonic oscillator (QHO), factorization and second quantization—see Robinett 9.1–9.2, 13.1–13.3; Fitzpatrick 5.8
- QHO, manipulation of ladder operators, Hermite polynomials, vibrational modes of diatomic molecules—see Robinett E.3, 9.3
- Two-dimensional QHO—see Robinett 15.1
- Formalism for two interacting particles in three-dimensional space, hydrogen atom, transformation to centre-of-mass and relative coordinates, separability, components of angular momentum and their operator algebra—see Robinett 14.1–14.3, 16.1; Fitzpatrick 6.4
- Spherical coordinates, angular momentum eigenstates, vibrational and rotational modes of diatomic molecules—see Robinett 16.2–16.3; Fitzpatrick 8.1–8.7