Motion in One Dimension

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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 will study the quantum mechaincs of one-dimensional systems. We study such problems for two reasons. Not only is it interesting to study the simplest cases to demonstrate some of the general properties of quantum mechanical systems, but it also turns out that there are two- and three-dimensional systems that can be reduced to effective one-dimensional problems, such as a particle in a central potential (also see the problem below).

We will discuss both bound and scattering states of one-dimensional potentials, thus illustrating the basic features of each, such as the discrete energy spectrum of the bound states, as opposed to the continuous scattering state spectrum. We also give an introduction to scattering from one-dimensional potentials, and show how to calculate the probabilities of transmission and reflection from such a potential. Finally, we treat two special cases, namely the Dirac delta function potential and periodic potentials, the latter of which will introduce the concept of energy bands, a topic of great importance in, for example, the study of the electronic properties of crystalline solids.

Chapter Contents

Problem

(Based on Problem 3.19 in Schaum's Theory and Problems of Quantum Mechanics)

Consider a particle of mass m\! in a three dimensional potential of the form, V(x,y,z) = X(x)+Y(y)+Z(z).\! Show that we can treat the problem as three independent one-dimensional problems. Relate the energy of the three-dimensional state to the effective energies of the one-dimensional problems.

Solution

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