Discrete Eigenvalues and Bound States - The Harmonic Oscillator and the WKB Approximation

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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

This chapter is devoted to a further discussion of systems with bound states; here we discuss two important problems. We first discuss the solution of the harmonic oscillator in quantum mechanics. This problem is of great interest because not only is it exactly solvable, but also because it emerges, at least as an approximation, in the study of systems of physical interest, such as molecular vibrations and phonons in solids. We also discuss the problem of a charged particle in a magnetic field, since it shares a number of features in common with the harmonic oscillator and many of the same techniques are applicable to it. Finally, we develop the WKB approximation, which is a semiclassical approximation to the solution of the Schrödinger equation, and show how to obtain approximate bound-state energies. We find that this approximation reduces to the Bohr-Sommerfeld quantization condition from the old quantum theory, thus showing that said theory is in fact a semiclassical approximation to the modern quantum theory that we have been studying.

Chapter Contents

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