#### Lecture

##### SEARCH COURSES

#### L6.2 Is probability conserved? Hermiticity of the Hamiltonian. (M-I-T)

**Course : **
Quantum Physics I (M-I-T)

**Discipline : **
Basic and Health Sciences

**Institute : **
MIT

**Instructor(s) : **
Prof. Dr. Barton Zwiebach

**Course Number : **
MIT 8.04

**Level: **
--Undergraduate

#### Description

#### Quantum Physics I (M-I-T)

- L1.1 Quantum mechanics as a framework. Defining linearity. (M-I-T)
- L1.2 Linearity and nonlinear theories. Schrödinger’s equation. (M-I-T)
- L1.3 Necessity of complex numbers. (M-I-T)
- L1.4 Photons and the loss of determinism. (M-I-T)
- L1.5 The nature of superposition. Mach-Zehnder interferometer. (M-I-T)
- L2.1 More on superposition. General state of a photon and spin states. (M-I-T)
- L2.2 Entanglement. (M-I-T)
- L2.3 Mach-Zehnder interferometers and beam splitters. (M-I-T)
- L2.4 Interferometer and interference. (M-I-T)
- L2.5 Elitzur-Vaidman bombs. (M-I-T)
- L3.1 The photoelectric effect. (M-I-T)
- L3.2 Units of h and Compton wavelength of particles. (M-I-T)
- L3.3 Compton Scattering. (M-I-T)
- L3.4 de Broglie’s proposal. (M-I-T)
- L4.1 de Broglie wavelength in different frames. (M-I-T)
- L4.2 Galilean transformation of ordinary waves. (M-I-T)
- L4.3 The frequency of a matter wave. (M-I-T)
- L4.4 Group velocity and stationary phase approximation. (M-I-T)
- L4.5 Motion of a wave-packet. (M-I-T)
- L4.6 The wave for a free particle. (M-I-T)
- L5.1 Momentum operator, energy operator, and a differential equation. (M-I-T)
- L5.2 Free Schrödinger equation. (M-I-T)
- L5.3 The general Schrödinger equation. x, p commutator. (M-I-T)
- L5.4 Commutators, matrices, and 3-dimensional Schrödinger equation. (M-I-T)
- L5.5 Interpretation of the wavefunction. (M-I-T)
- L6.1 Normalizable wavefunctions and the question of time evolution. (M-I-T)
- L6.2 Is probability conserved? Hermiticity of the Hamiltonian. (M-I-T)
- L6.3 Probability current and current conservation. (M-I-T)
- L6.4 Three dimensional current and conservation. (M-I-T)
- L7.1 Wavepackets and Fourier representation. (M-I-T)
- L7.2 Reality condition in Fourier transforms. (M-I-T)
- L7.3 Widths and uncertainties. (M-I-T)
- L7.4 Shape changes in a wave. (M-I-T)
- L7.5 Time evolution of a free particle wavepacket. (M-I-T)
- L8.1 Fourier transforms and delta functions. (M-I-T)
- L8.2 Parseval identity. (M-I-T)
- L8.3 Three-dimensional Fourier transforms. (M-I-T)
- L8.4 Expectation values of operators. (M-I-T)
- L8.5 Time dependence of expectation values (M-I-T)
- L9.1 Expectation value of Hermitian operators. (M-I-T)
- L9.2 Eigenfunctions of a Hermitian operator. (M-I-T)
- L9.3 Completeness of eigenvectors and measurement postulate. (M-I-T)
- L9.4 Consistency condition. Particle on a circle. (M-I-T)
- L9.5 Defining uncertainty. (M-I-T)
- L10.1 Uncertainty and eigenstates. (M-I-T)
- L10.2 Stationary states: key equations. (M-I-T)
- L10.3 Expectation values on stationary states. (M-I-T)
- L10.4 Comments on the spectrum and continuity conditions. (M-I-T)
- L10.5 Solving particle on a circle. (M-I-T)
- L11.1 Energy eigenstates for particle on a circle. (M-I-T)
- L11.2 Infinite square well energy eigenstates. (M-I-T)
- L11.3 Nodes and symmetries of the infinite square well eigenstates. (M-I-T)
- L11.4 Finite square well. Setting up the problem. (M-I-T)
- L11.5 Finite square well energy eigenstates. (M-I-T)
- L12.1 Nondegeneracy of bound states in 1D. Real solutions. (M-I-T)
- L12.2 Potentials that satisfy V(-x) = V(x). (M-I-T)
- L12.3 Qualitative insights: Local de Broglie wavelength. (M-I-T)
- L12.4 Correspondence principle: amplitude as a function of position. (M-I-T)
- L12.5 Local picture of the wavefunction. (M-I-T)
- L12.6 Energy eigenstates on a generic symmetric potential. Shooting method. (M-I-T)
- L13.1 Delta function potential I: Preliminaries. (M-I-T)
- L13.2 Delta function potential I: Solving for the bound state. (M-I-T)
- L13.3 Node Theorem. (M-I-T)
- L13.4 Harmonic oscillator: Differential equation. (M-I-T)
- L13.5 Behavior of the differential equation. (M-I-T)
- L14.1 Recursion relation for the solution. (M-I-T)
- L14.2 Quantization of the energy. (M-I-T)
- L14.3 Algebraic solution of the harmonic oscillator. (M-I-T)
- L14.4 Ground state wavefunction. (M-I-T)
- L15.1 Number operator and commutators. (M-I-T)
- L15.2 Excited states of the harmonic oscillator. (M-I-T)
- L15.3 Creation and annihilation operators acting on energy eigenstates. (M-I-T)
- L15.4 Scattering states and the step potential. (M-I-T)
- L16.1 Step potential probability current. (M-I-T)
- L16.2 Reflection and transmission coefficients. (M-I-T)
- L16.3 Energy below the barrier and phase shift. (M-I-T)
- L16.4 Wavepackets. (M-I-T)
- L16.5 Wavepackets with energy below the barrier. (M-I-T)
- L16.6 Particle on the forbidden region. (M-I-T)
- L17.1 Waves on the finite square well. (M-I-T)
- L17.2 Resonant transmission. (M-I-T)
- L17.3 Ramsauer-Townsend phenomenology. (M-I-T)
- L17.4 Scattering in 1D. Incoming and outgoing waves. (M-I-T)
- L17.5 Scattered wave and phase shift. (M-I-T)
- L18.1 Incident packet and delay for reflection. (M-I-T)
- L18.2 Phase shift for a potential well. (M-I-T)
- L18.3 Excursion of the phase shift. (M-I-T)
- L18.4 Levinson's theorem, part 1. (M-I-T)
- L18.5 Levinson's theorem, part 2. (M-I-T)
- L19.1 Time delay and resonances. (M-I-T)
- L19.2 Effects of resonance on phase shifts, wave amplitude and time delay. (M-I-T)
- L19.3 Modeling a resonance. (M-I-T)
- L19.4 Half-width and time delay. (M-I-T)
- L19.5 Resonances in the complex k plane. (M-I-T)
- L20.1 Translation operator. Central potentials. (M-I-T)
- L20.2 Angular momentum operators and their algebra. (M-I-T)
- L20.3 Commuting observables for angular momentum. (M-I-T)
- L20.4 Simultaneous eigenstates and quantization of angular momentum. (M-I-T)
- L21.1 Associated Legendre functions and spherical harmonics. (M-I-T)
- L21.2 Orthonormality of spherical harmonics. (M-I-T)
- L21.3 Effective potential and boundary conditions at r=0. (M-I-T)
- L21.4 Hydrogen atom two-body problem. (M-I-T)
- L22.1 Center of mass and relative motion wavefunctions. (M-I-T)
- L22.2 Scales of the hydrogen atom. (M-I-T)
- L22.3 Schrödinger equation for hydrogen. (M-I-T)
- L22.4 Series solution and quantization of the energy. (M-I-T)
- L22.5 Energy eigenstates of hydrogen. (M-I-T)
- L23.1 Energy levels and diagram for hydrogen. (M-I-T)
- L23.2 Degeneracy in the spectrum and features of the solution. (M-I-T)
- L23.3 Rydberg atoms. (M-I-T)
- L23.4 Orbits in the hydrogen atom. (M-I-T)
- L24.1 More on the hydrogen atom degeneracies and orbits. (M-I-T)
- L24.2 The simplest quantum system. (M-I-T)
- L24.3 Hamiltonian and emerging spin angular momentum. (M-I-T)
- L24.4 Eigenstates of the Hamiltonian. (M-I-T)