8.04 | Spring 2016 | Undergraduate

# Quantum Physics I

## Video Lectures

The videos in this course are broadly divided into three parts:

Part 1: Basic Concepts

In Part 1, we introduce the basic concepts: Interpretation of the wavefunction, relation to probability, Schrödinger equation, Hermitian operators and inner products. We also discuss wave-packets, time evolution, Ehrenfest theorem and uncertainty.

Part 2: Quantum Physics in One-dimensional Potentials

Part 2 deals with solutions of the Schrödinger equation for one-dimensional potentials. We discuss stationary states and the key problems of a particle moving in: A circle, an infinite well, a finite square well, and a delta-function potential. We examine qualitative properties of the wavefunction. The harmonic oscillator is solved in two ways: Using the differential equation and using creation and annihilation operators. We study barrier penetration and the Ramsaur—Townsend effect.

Part 3: One-dimensional Scattering, Angular Momentum, and Central Potentials

Part 3 begins with the subject of scattering on the half-line. One can learn in this simpler context the basic concepts needed in 3-dimensional scattering theory: Scattered wave, phaseshifts, time delays, Levinson theorem, and resonances. We then turn to three-dimensional central potential problems. We introduce the angular momentum operators and derive their commutator algebra. The Schrödinger equation is reduced to a radial equation. We discuss the hydrogen atom in detail.

lec # topics
Lecture 1: An overview of quantum mechanics.
L1.1 Quantum mechanics as a framework. Defining linearity (17:46)
L1.2 Linearity and nonlinear theories. Schrödinger’s equation (10:01)
L1.3 Necessity of complex numbers (07:38)
L1.4 Photons and the loss of determinism (17:20)
L1.5 The nature of superposition. Mach-Zehnder interferometer (14:30)
Lecture 2: Overview of quantum mechanics (cont.). Interaction-free measurements.
L2.1 More on superposition. General state of a photon and spin states (17:10)
L2.2 Entanglement (13:07)
L2.3 Mach-Zehnder interferometers and beam splitters (15:32)
L2.4 Interferometer and interference (12:26)
L2.5 Elitzur-Vaidman bombs (10:29)
Lecture 3: Photoelectric effect, Compton scattering, and de Broglie wavelength.
L3.1 The photoelectric effect (22:54)
L3.2 Units of h and Compton wavelength of particles (12:39)
L3.3 Compton Scattering (22:34)
L3.4 de Broglie’s proposal (10:39)
Lecture 4: de Broglie matter waves. Group velocity and stationary phase. Wave for a free particle.
L4.1 de Broglie wavelength in different frames (14:53)
L4.2 Galilean transformation of ordinary waves (12:16)
L4.3 The frequency of a matter wave (10:23)
L4.4 Group velocity and stationary phase approximation (10:32)
L4.5 Motion of a wave-packet (08:58)
L4.6 The wave for a free particle (14:35)
Lecture 5: Momentum operator, Schrödinger equation, and interpretation of the wavefunction.
L5.1 Momentum operator, energy operator, and a differential equation (20:33)
L5.2 Free Schrödinger equation (09:56)
L5.3 The general Schrödinger equation. x, p commutator (17:58)
L5.4 Commutators, matrices, and 3-dimensional Schrödinger equation (16:12)
L5.5 Interpretation of the wavefunction (08:01)
Lecture 6: Probability density and current. Hermitian conjugation.
L6.1 Normalizable wavefunctions and the question of time evolution (16:50)
L6.2 Is probability conserved? Hermiticity of the Hamiltonian (20:42)
L6.3 Probability current and current conservation (15:14)
L6.4 Three dimensional current and conservation (18:13)
Lecture 7: Wavepackets and uncertainty. Time evolution and shape change time evolutions.
L7.1 Wavepackets and Fourier representation (12:23)
L7.2 Reality condition in Fourier transforms (09:11)
L7.3 Widths and uncertainties (19:13)
L7.4 Shape changes in a wave (16:56)
L7.5 Time evolution of a free particle wavepacket (09:44)
Lecture 8: Uncovering momentum space. Expectation values and their time dependence.
L8.1 Fourier transforms and delta functions (13:58)
L8.2 Parseval identity (15:50)
L8.3 Three-dimensional Fourier transforms (06:04)
L8.4 Expectation values of operators (28:15)
L8.5 Time dependence of expectation values (7:37)
Lecture 9: Observables, Hermitian operators, measurement and uncertainty. Particle on a circle.
L9.1 Expectation value of Hermitian operators (16:40)
L9.2 Eigenfunctions of a Hermitian operator (13:05)
L9.3 Completeness of eigenvectors and measurement postulate (16:56)
L9.4 Consistency condition. Particle on a circle (17:45)
L9.5 Defining uncertainty (10:31)

lec # topics
Lecture 10: Uncertainty (cont.). Stationary states. Particle on a circle.
L10.1 Uncertainty and eigenstates (15:53)
L10.2 Stationary states: key equations (18:43)
L﻿10.3 Expectation values on stationary states (09:00)
L﻿10.4 Comments on the spectrum and continuity conditions (13:09)
L﻿10.5 Solving particle on a circle (11:05)
Lecture 11: Uncertainty (cont.). Stationary states. Particle on a circle.
L11.1 Energy eigenstates for particle on a circle (16:12)
L11.2 Infinite square well energy eigenstates (13:15)
L﻿11.3 Nodes and symmetries of the infinite square well eigenstates. (09:43)
L﻿11.4 Finite square well. Setting up the problem. (22:30)
L﻿11.5 Finite square well energy eigenstates (10:39)
Lecture 12: Properties of 1D energy eigenstates. Qualitative properties of wavefunctions. Shooting method.
L12.1 Nondegeneracy of bound states in 1D. Real solutions (12:36)
L12.2 Potentials that satisfy V(-x) = V(x) (14:18)
L12.3 Qualitative insights: Local de Broglie wavelength (15:52)
L12.4 Correspondence principle: amplitude as a function of position (05:53)
L12.5 Local picture of the wavefunction (12:52)
L12.6 Energy eigenstates on a generic symmetric potential. Shooting method (15:26)
Lecture 13: Delta function potential. Justifying the node theorem. Simple harmonic oscillator.
L13.1 Delta function potential I: Preliminaries (16:14)
L13.2 Delta function potential I: Solving for the bound state (15:21)
L13.3 Node Theorem (13:01)
L13.4 Harmonic oscillator: Differential equation (16:45)
L13.5 Behavior of the differential equation (10:31)
Lecture 14: Simple harmonic oscillator II. Creation and annihilation operators.
L14.1 Recursion relation for the solution (12:25)
L14.2 Quantization of the energy (23:23)
L14.3 Algebraic solution of the harmonic oscillator (16:50)
L14.4 Ground state wavefunction (15:58)
Lecture 15: Simple harmonic oscillator III. Scattering states and step potential.
L15.1 Number operator and commutators (15:49)
L15.2 Excited states of the harmonic oscillator (18:19)
L15.3 Creation and annihilation operators acting on energy eigenstates (21:03)
L15.4 Scattering states and the step potential (10:34)
Lecture 16: Step potential reflection and transmission coefficients. Phase shift, wavepackets and time delay.
L16.1 Step potential probability current (14:59)
L16.2 Reflection and transmission coefficients (08:12)
L16.3 Energy below the barrier and phase shift (18:40)
L16.4 Wavepackets (20:51)
L16.5 Wavepackets with energy below the barrier (05:54)
L16.6 Particle on the forbidden region (06:48)
Lecture 17: Ramsauer-Townsend effect. Scattering in 1D.
L17.1 Waves on the finite square well (15:44)
L17.2 Resonant transmission (17:49)
L17.3 Ramsauer-Townsend phenomenology (10:16)
L17.4 Scattering in 1D. Incoming and outgoing waves (18:05)
L17.5 Scattered wave and phase shift (08:40)
Lecture 18: Scattering in 1D (cont.). Example. Levinson’s theorem.
L18.1 Incident packet and delay for reflection (18:52)
L18.2 Phase shift for a potential well (09:13)
L18.3 Excursion of the phase shift (15:16)
L18.4 Levinson’s theorem, part 1 (14:46)
L18.5 Levinson’s theorem, part 2 (09:30)

lec # topics
Lecture 19: Resonances and Breit-Wigner distribution. The complex k-plane.
L19.1 Time delay and resonances (18:18)
L19.2 Effects of resonance on phase shifts, wave amplitude and time delay (14:53)
L19.3 Modelling a resonance (15:37)
L19.4 Half-width and time delay (08:17)
L19.5 Resonances in the complex k plane (15:14)
Lecture 20: Central potentials and angular momentum.
L20.1 Translation operator. Central potentials (19:13)
L20.2 Angular momentum operators and their algebra (14:28)
L20.3 Commuting observables for angular momentum (17:17)
L20.4 Simultaneous eigenstates and quantization of angular momentum (24:35)
Lecture 21: Legendre equation. Radial equation. Hydrogen atom 2-body problem.
L21.1 Associated Legendre functions and spherical harmonics (18:51)
L21.2 Orthonormality of spherical harmonics (17:57)
L21.3 Effective potential and boundary conditions at r=0 (14:28)
L21.4 Hydrogen atom two-body problem (25:04)
Lecture 22: Hydrogen atom (cont.). Differential equation, series solution and quantum numbers
L22.1 Center of mass and relative motion wavefunctions (14:22)
L22.2 Scales of the hydrogen atom (09:56)
L22.3 Schrödinger equation for hydrogen (20:59)
L22.4 Series solution and quantization of the energy (14:22)
L22.5 Energy eigenstates of hydrogen (12:24)
Lecture 23: Spectrum for hydrogen. Virial theorem, circular orbits and eccentricity.
L23.1 Energy levels and diagram for hydrogen (13:41)
L23.2 Degeneracy in the spectrum and features of the solution (14:20)
L23.3 Rydberg atoms (26:22)
L23.4 Orbits in the hydrogen atom (10:45)
Lecture 24: Hydrogen atom (conclusion). The simplest quantum system and emergent angular momentum.
L24.1 More on the hydrogen atom degeneracies and orbits (23:21)
L24.2 The simplest quantum system (13:55)
L24.3 Hamiltonian and emerging spin angular momentum (15:42)
L24.4 Eigenstates of the Hamiltonian (14:03)

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