Lecture 69 of 115Part 2: Quantum Physics in One-dimensional Potentials16 min
Ground state wavefunction
Prof. Barton Zwiebach · Massachusetts Institute of Technology
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- Quantum Physics I, Massachusetts Institute of Technology (at MIT OpenCourseWare)
- Licence
- CC BY-NC-SA 4.0, as MIT OpenCourseWare states it. Non-commercial use only.
- Attribution
- Prof. Barton Zwiebach. Quantum Physics I. Spring 2016. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu/. License: Creative Commons BY-NC-SA.
- Who can watch
- Teens profiles and older. How courses are placed
All the lectures
- Quantum mechanics as a framework. Defining linearity
- Linearity and nonlinear theories. Schrödinger's equation
- Necessity of complex numbers
- Photons and the loss of determinism
- The nature of superposition. Mach-Zehnder interferometer
- More on superposition. General state of a photon and spin states
- Entanglement
- Mach-Zehnder interferometers and beam splitters
- Interferometer and interference
- Elitzur-Vaidman bombs
- The photoelectric effect
- Units of h and Compton wavelength of particles
- Compton Scattering
- de Broglie’s proposal
- de Broglie wavelength in different frames
- Galilean transformation of ordinary waves
- The frequency of a matter wave
- Group velocity and stationary phase approximation
- Motion of a wave-packet
- The wave for a free particle
- Momentum operator, energy operator, and a differential equation
- Free Schrödinger equation
- The general Schrödinger equation. x, p commutator
- Commutators, matrices, and 3-dimensional Schrödinger equation
- Interpretation of the wavefunction
- Normalizable wavefunctions and the question of time evolution
- Is probability conserved? Hermiticity of the Hamiltonian
- Probability current and current conservation
- Three dimensional current and conservation
- Wavepackets and Fourier representation
- Reality condition in Fourier transforms
- Widths and uncertainties
- Shape changes in a wave
- Time evolution of a free particle wavepacket
- Fourier transforms and delta functions
- Parseval identity
- Three-dimensional Fourier transforms
- Expectation values of operators
- Time dependence of expectation values
- Expectation value of Hermitian operators
- Eigenfunctions of a Hermitian operator
- Completeness of eigenvectors and measurement postulate
- Consistency condition. Particle on a circle
- Defining uncertainty
- Uncertainty and eigenstates
- Stationary states: key equations
- Expectation values on stationary states
- Comments on the spectrum and continuity conditions
- Solving particle on a circle
- Energy eigenstates for particle on a circle
- Infinite square well energy eigenstates
- Nodes and symmetries of the infinite square well eigenstates
- Finite square well. Setting up the problem
- Finite square well energy eigenstates
- Nondegeneracy of bound states in 1D. Real solutions
- Potentials that satisfy V(-x) = V(x)
- Qualitative insights: Local de Broglie wavelength
- Correspondence principle: amplitude as a function of position
- Local picture of the wavefunction
- Energy eigenstates on a generic symmetric potential. Shooting method
- Delta function potential I: Preliminaries
- Delta function potential I: Solving for the bound state
- Node Theorem
- Harmonic oscillator: Differential equation
- Behavior of the differential equation
- Recursion relation for the solution
- Quantization of the energy
- Algebraic solution of the harmonic oscillator
- Ground state wavefunction
- Number operator and commutators
- Excited states of the harmonic oscillator
- Creation and annihilation operators acting on energy eigenstates
- Scattering states and the step potential
- Step potential probability current
- Reflection and transmission coefficients
- Energy below the barrier and phase shift
- Wavepackets
- Wavepackets with energy below the barrier
- Particle on the forbidden region
- Waves on the finite square well
- Resonant transmission
- Ramsauer-Townsend phenomenology
- Scattering in 1D. Incoming and outgoing waves
- Scattered wave and phase shift
- Incident packet and delay for reflection
- Phase shift for a potential well
- Excursion of the phase shift
- Levinson's theorem, part 1
- Levinson's theorem, part 2
- Time delay and resonances
- Effects of resonance on phase shifts, wave amplitude and time delay
- Modelling a resonance
- Half-width and time delay
- Resonances in the complex k plane
- Translation operator. Central potentials
- Angular momentum operators and their algebra
- Commuting observables for angular momentum
- Simultaneous eigenstates and quantization of angular momentum
- Associated Legendre functions and spherical harmonics
- Orthonormality of spherical harmonics
- Effective potential and boundary conditions at r=0
- Hydrogen atom two-body problem
- Center of mass and relative motion wavefunctions
- Scales of the hydrogen atom
- Schrödinger equation for hydrogen
- Series solution and quantization of the energy
- Energy eigenstates of hydrogen
- Energy levels and diagram for hydrogen
- Degeneracy in the spectrum and features of the solution
- Rydberg atoms
- Orbits in the hydrogen atom
- More on the hydrogen atom degeneracies and orbits
- The simplest quantum system
- Hamiltonian and emerging spin angular momentum
- Eigenstates of the Hamiltonian