Motion in One Dimension

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.

Physical Basis of Quantum Mechanics

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

Schrödinger Equation

Brief Derivation of Schrödinger Equation Relation Between the Wave Function and Probability Density Stationary States Heisenberg Uncertainty Principle Some Consequences of the Uncertainty Principle

Operators, Eigenfunctions, and Symmetry

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

Motion in One Dimension

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

Discrete Eigenvalues and Bound States

Harmonic Oscillator Spectrum and Eigenstates Analytical Method for Solving the Simple Harmonic Oscillator Coherent States Charged Particles in an Electromagnetic Field WKB Approximation

Time Evolution and the Pictures of Quantum Mechanics

The Heisenberg Picture: Equations of Motion for Operators The Interaction Picture The Virial Theorem

Angular Momentum

Commutation Relations Angular Momentum as a Generator of Rotations in 3D Spherical Coordinates Eigenvalue Quantization Orbital Angular Momentum Eigenfunctions

Central Forces

General Formalism Free Particle in Spherical Coordinates Spherical Well Isotropic Harmonic Oscillator Hydrogen Atom WKB in Spherical Coordinates

The Path Integral Formulation of Quantum Mechanics

Feynman Path Integrals The Free-Particle Propagator Propagator for the Harmonic Oscillator

Continuous Eigenvalues and Collision Theory

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

• One-Dimensional Bound States
• The Dirac Delta Function Potential
• Oscillation Theorem
• Scattering States, Transmission and Reflection
• Motion in a Periodic Potential
• Summary of One-Dimensional Systems

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