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Massachusetts Institute of Technology

Quantum Physics I

Massachusetts Institute of Technology via MIT OpenCourseWare

Overview

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This is the first course in the undergraduate Quantum Physics sequence. It introduces the basic features of quantum mechanics. It covers the experimental basis of quantum physics, introduces wave mechanics, Schrödinger's equation in a single dimension, and Schrödinger's equation in three dimensions. The lectures and lecture notes for this course form the basis of Zwiebach’s textbook *Mastering Quantum Mechanics* published by {{% resource_link "0ae4d211-b767-459e-b759-e3094452392f" "MIT Press" %}} in April 2022. This presentation of 8.04 by Barton Zwiebach (2016) differs somewhat and complements nicely the presentation of [Allan Adams (2013)](/courses/8-04-quantum-physics-i-spring-2013/). Adams covers a larger set of ideas; Zwiebach tends to go deeper into a smaller set of ideas, offering a systematic and detailed treatment. Adams begins with the subtleties of superpostion, while Zwiebach discusses the surprises of interaction-free measurements. While both courses overlap over a sizable amount of standard material, Adams discussed applications to condensed matter physics, while Zwiebach focused on scattering and resonances. The different perspectives of the instructors make the problem sets in the two courses rather different.

Syllabus

  • Algebraic solution of the harmonic oscillator
  • Angular momentum operators and their algebra
  • Associated Legendre functions and spherical harmonics
  • Behavior of the differential equation
  • Center of mass and relative motion wavefunctions
  • Comments on the spectrum and continuity conditions
  • Commuting observables for angular momentum
  • Compton Scattering
  • Consistency condition. Particle on a circle
  • Correspondence principle: amplitude as a function of position
  • Creation and annihilation operators acting on energy eigenstates
  • de Broglie wavelength in different frames
  • de Broglie’s proposal
  • Defining uncertainty
  • Degeneracy in the spectrum and features of the solution
  • Delta function potential I: Preliminaries
  • Delta function potential I: Solving for the bound state
  • Effects of resonance on phase shifts, wave amplitude and time delay
  • Eigenfunctions of a Hermitian operator
  • Eigenstates of the Hamiltonian
  • Elitzur-Vaidman bombs
  • Energy below the barrier and phase shift
  • Energy eigenstates for particle on a circle
  • Energy eigenstates of hydrogen
  • Energy eigenstates on a generic symmetric potential. Shooting method
  • Energy levels and diagram for hydrogen
  • Entanglement
  • Excited states of the harmonic oscillator
  • Excursion of the phase shift
  • Expectation value of Hermitian operators
  • Expectation values of operators
  • Expectation values on stationary states
  • Finite square well energy eigenstates
  • Finite square well. Setting up the problem
  • Fourier transforms and delta functions
  • Free Schrödinger equation
  • Galilean transformation of ordinary waves
  • Ground state wavefunction
  • Group velocity and stationary phase approximation
  • Half-width and time delay
  • Hamiltonian and emerging spin angular momentum
  • Harmonic oscillator: Differential equation
  • Hydrogen atom two-body problem
  • Incident packet and delay for reflection
  • Infinite square well energy eigenstates
  • Interferometer and interference
  • Interpretation of the wavefunction
  • Is probability conserved? Hermiticity of the Hamiltonian
  • Linearity and nonlinear theories. Schrödinger's equation
  • Local picture of the wavefunction
  • Mach-Zehnder interferometers and beam splitters
  • Modelling a resonance
  • Momentum operator, energy operator, and a differential equation
  • More on superposition. General state of a photon and spin states
  • More on the hydrogen atom degeneracies and orbits
  • Motion of a wave-packet
  • Necessity of complex numbers
  • Node Theorem
  • Nodes and symmetries of the infinite square well eigenstates
  • Normalizable wavefunctions and the question of time evolution
  • Number operator and commutators
  • Orbits in the hydrogen atom
  • Orthonormality of spherical harmonics
  • Parseval identity
  • Particle on the forbidden region
  • Phase shift for a potential well
  • Photons and the loss of determinism
  • Potentials that satisfy V(-x) = V(x)
  • Probability current and current conservation
  • Qualitative insights: Local de Broglie wavelength
  • Quantization of the energy
  • Quantum mechanics as a framework. Defining linearity
  • Ramsauer-Townsend phenomenology
  • Reality condition in Fourier transforms
  • Recursion relation for the solution
  • Reflection and transmission coefficients
  • Resonances in the complex k plane
  • Resonant transmission
  • Rydberg atoms
  • Scales of the hydrogen atom
  • Scattered wave and phase shift
  • Scattering states and the step potential
  • Schrödinger equation for hydrogen
  • Series solution and quantization of the energy
  • Shape changes in a wave
  • Simultaneous eigenstates and quantization of angular momentum
  • Solving particle on a circle
  • Stationary states: key equations
  • Step potential probability current
  • The frequency of a matter wave
  • The general Schrödinger equation. x, p commutator
  • The nature of superposition. Mach-Zehnder interferometer
  • The photoelectric effect
  • The simplest quantum system
  • The wave for a free particle
  • Three dimensional current and conservation
  • Three-dimensional Fourier transforms
  • Time delay and resonances
  • Time dependence of expectation values
  • Time evolution of a free particle wavepacket
  • Translation operator. Central potentials
  • Uncertainty and eigenstates
  • Units of h and Compton wavelength of particles
  • Wavepackets
  • Wavepackets and Fourier representation
  • Wavepackets with energy below the barrier
  • Waves on the finite square well
  • Widths and uncertainties
  • Effective potential and boundary conditions at r=0
  • Levinson's theorem, part 1
  • Nondegeneracy of bound states in 1D. Real solutions
  • Scattering in 1D. Incoming and outgoing waves
  • Levinson's theorem, part 2
  • Commutators, matrices, and 3-dimensional Schrödinger equation

Taught by

Prof. Barton Zwiebach

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