Time Evolution and the Pictures of Quantum Mechanics

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Quantum Mechanics A
Schrödinger Equation
The most fundamental equation of quantum mechanics; given a Hamiltonian \mathcal{H}, it describes how a state |\Psi\rangle evolves in time.
Basic Concepts and Theory of Motion
UV Catastrophe (Black-Body Radiation)
Photoelectric Effect
Stability of Matter
Double Slit Experiment
Stern-Gerlach Experiment
The Principle of Complementarity
The Correspondence Principle
The Philosophy of Quantum Theory
Brief Derivation of Schrödinger Equation
Relation Between the Wave Function and Probability Density
Stationary States
Heisenberg Uncertainty Principle
Some Consequences of the Uncertainty Principle
Linear Vector Spaces and Operators
Commutation Relations and Simultaneous Eigenvalues
The Schrödinger Equation in Dirac Notation
Transformations of Operators and Symmetry
Time Evolution of Expectation Values and Ehrenfest's Theorem
One-Dimensional Bound States
Oscillation Theorem
The Dirac Delta Function Potential
Scattering States, Transmission and Reflection
Motion in a Periodic Potential
Summary of One-Dimensional Systems
Harmonic Oscillator Spectrum and Eigenstates
Analytical Method for Solving the Simple Harmonic Oscillator
Coherent States
Charged Particles in an Electromagnetic Field
WKB Approximation
The Heisenberg Picture: Equations of Motion for Operators
The Interaction Picture
The Virial Theorem
Commutation Relations
Angular Momentum as a Generator of Rotations in 3D
Spherical Coordinates
Eigenvalue Quantization
Orbital Angular Momentum Eigenfunctions
General Formalism
Free Particle in Spherical Coordinates
Spherical Well
Isotropic Harmonic Oscillator
Hydrogen Atom
WKB in Spherical Coordinates
Feynman Path Integrals
The Free-Particle Propagator
Propagator for the Harmonic Oscillator
Differential Cross Section and the Green's Function Formulation of Scattering
Central Potential Scattering and Phase Shifts
Coulomb Potential Scattering

In this chapter, we discuss different "pictures" in which one may treat the dynamics of a quantum mechanical system. So far, we have been employing what is known as the Schrödinger picture, in which we treat operators as time-independent and place all of the time dependence of the system in the state vector. We will show that one may also consider the state vectors to be constant in time and the operators to be changing; this is known as the Heisenberg picture. We also consider the interaction, or Dirac, picture, in which the time dependence is split between the state vector and the operators. Both of these pictures are very useful. The Heisenberg picture allows one to make an even closer connection between quantum and classical mechanics via the equations of motion satisfied by observables, which resemble the classical equations of motion for the same system. The interaction picture is useful in describing the response of an exactly solvable system to external perturbations.

We also discuss Ehrenfest's theorem, which gives us yet another way to make a connection with classical mechanics, this time through the expectation values of operators, as well as the virial theorem, which gives us a relation between the expectation values of the kinetic and potential energies of a particle.

Chapter Contents

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