LEJ4Learning

Quantum Mechanics as a Framework. Defining Linearity

Linearity and Non-Linear Theories. Schrodinger’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.

L4.2 Galilean transformation of ordinary waves.

L4.3 The frequency of a matter wave.

L4.4 Group velocity and stationary phase approximation.

L4.5 Motion of a wave-packet.

L4.6 The wave for a free particle.

L5.1 Momentum operator, energy operator, and a differential equation.

L5.2 Free Schrödinger equation

L5.3 The general Schrödinger equation. x, p commutator.

L5.4 Commutators, matrices, and 3-dimensional Schrödinger equation.

L5.5 Interpretation of the wavefunction.

L6.1 Normalizable wavefunctions and the question of time evolution.

L6.2 Is probability conserved? Hermiticity of the Hamiltonian.

L6.3 Probability current and current conservation.

L6.4 Three dimensional current and conservation.

L7.1 Wavepackets and Fourier representation.

L7.2 Reality condition in Fourier transforms.

L7.3 Widths and uncertainties.

L7.4 Shape changes in a wave.

L7.5 Time evolution of a free particle wavepacket.

L8.1 Fourier transforms and delta functions.

L8.2 Parseval identity.

L8.3 Three-dimensional Fourier transforms.

L8.4 Expectation values of operators.

L8.5 Time dependence of expectation values

L9.1 Expectation value of Hermitian operators.

L9.2 Eigenfunctions of a Hermitian operator.

L9.3 Completeness of eigenvectors and measurement postulate.

L9.4 Consistency condition. Particle on a circle.

L9.5 Defining uncertainty

L10.1 Uncertainty and eigenstates.

L10.2 Stationary states: key equations.

L10.3 Expectation values on stationary states.

L10.4 Comments on the spectrum and continuity conditions.

L10.5 Solving particle on a circle.

L11.1 Energy eigenstates for particle on a circle.

L11.2 Infinite square well energy eigenstates.

L11.3 Nodes and symmetries of the infinite square well eigenstates.

L11.4 Finite square well. Setting up the problem.

L11.5 Finite square well energy eigenstates.

L12.1 Nondegeneracy of bound states in 1D. Real solutions.

L12.2 Potentials that satisfy V(-x) = V(x).

L12.3 Qualitative insights: Local de Broglie wavelength.

L12.4 Correspondence principle: amplitude as a function of position.

L12.5 Local picture of the wavefunction.

L12.6 Energy eigenstates on a generic symmetric potential. Shooting method.

L13.1 Delta function potential I: Preliminaries.

L13.2 Delta function potential I: Solving for the bound state.

L13.3 Node Theorem.

L13.4 Harmonic oscillator: Differential equation.

L13.5 Behavior of the differential equation.

L14.1 Recursion relation for the solution.

L14.2 Quantization of the energy.

L14.3 Algebraic solution of the harmonic oscillator.

L14.4 Ground state wavefunction.

L15.1 Number operator and commutators.

L15.2 Excited states of the harmonic oscillator.

L15.3 Creation and annihilation operators acting on energy eigenstates.

L15.4 Scattering states and the step potential.

L16.1 Step potential probability current.

L16.2 Reflection and transmission coefficients.

L16.3 Energy below the barrier and phase shift.

L16.4 Wavepackets.

L16.5 Wavepackets with energy below the barrier.

L16.6 Particle on the forbidden region.

L17.1 Waves on the finite square well.

L17.2 Resonant transmission.

L17.3 Ramsauer-Townsend phenomenology.

L17.4 Scattering in 1D. Incoming and outgoing waves.

L17.5 Scattered wave and phase shift.

L18.1 Incident packet and delay for reflection.

L18.2 Phase shift for a potential well.

L18.3 Excursion of the phase shift.

L18.4 Levinson’s theorem, part 1.

L18.5 Levinson’s theorem, part 2.

L19.1 Time delay and resonances.

L19.2 Effects of resonance on phase shifts, wave amplitude and time delay

L19.3 Modeling a resonance.

L19.4 Half-width and time delay.

L19.5 Resonances in the complex k plane.

L20.1 Translation operator. Central potentials.

L20.2 Angular momentum operators and their algebra.

L20.3 Commuting observables for angular momentum.

L20.4 Simultaneous eigenstates and quantization of angular momentum.

L21.1 Associated Legendre functions and spherical harmonics.

L21.2 Orthonormality of spherical harmonics.

L21.3 Effective potential and boundary conditions at r=0.

L21.4 Hydrogen atom two-body problem.

L22.1 Center of mass and relative motion wavefunctions.

L22.2 Scales of the hydrogen atom.

L22.3 Schrödinger equation for hydrogen.

L22.4 Series solution and quantization of the energy.

L22.5 Energy eigenstates of hydrogen.

L23.2 Degeneracy in the spectrum and features of the solution.

L23.3 Rydberg atoms.

L23.4 Orbits in the hydrogen atom.

L24.1 More on the hydrogen atom degeneracies and orbits.

L24.2 The simplest quantum system.

L24.3 Hamiltonian and emerging spin angular momentum.

L24.4 Eigenstates of the Hamiltonian.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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