Quantum Mechanics with Applications

David B Beard, George B Beard, George B Beard, Kevin B Beard

• Publisher: Dover Publications
• Year: 2014
• Print ISBN: 9780486779904
• eBook ISBN: 9780486798219
• Topic: Physical Sciences
• Subtopic: Quantum Theory

This introductory text emphasizes Feynman's development of path integrals and its application to wave theory for particles. Suitable for undergraduate and graduate students of physics, the well-written, clear, and rigorous text was written by two of the nation's leading authorities on quantum physics. A solid foundation in quantum mechanics and atomic physics is assumed.
Early chapters provide background in the mathematical treatment and particular properties of ordinary wave motion that also apply to particle motion. The close relation of quantum theory to physical optics is stressed. Subsequent sections emphasize the physical consequences of a wave theory of material properties, and they offer extensive applications in atomic physics, nuclear physics, solid state physics, and diatomic molecules. Four helpful Appendixes supplement the text.

XV

ELASTIC SCATTERING THEORY

Elastic scattering theory treats collisions between particles in which no internal excitation of the particles takes place. Inelastic scattering, on the other hand, occurs when the same kind of particle is present in the scattered beam as in the incident beam but the total translational kinetic energy of the system is less, the missing energy going into excitation of internal degrees of freedom of one or both particles. For example, if two atoms collide and part of their translational kinetic energy is lost to electronic excitation or ionization, the scattering is inelastic. Inelastic scattering theory is somewhat more complicated and is outside the scope of this book, so that the internal energy of the system or the complete eigenfunction of the colliding systems’ constituent parts will be ignored. A complex, many-component particle such as an atom will be regarded as a single particle.

A cross section for a given scattering event is defined as the fraction of the number of incident-particles-per-unit-area for which the scattering event happens. For example, suppose a group of boys threw 900 baseballs at random through an opening nine square feet in area, producing a beam of 100 balls/square foot. Suppose that out of every 100 balls/foot² 50 balls hit a target on the other side of the opening. The cross section would be given by

This is the fraction of the incident beam of 100 balls/ft² which hit the target on the average. We see that a cross section is given in units of area and gives the probability that a beam of one ball per square foot will hit the target.

Suppose that 15 balls cracked the target. The cross section for cracking the target would be

It is clear that there are different kinds of cross sections. We will be concerned solely with the probability of elastically scattering particles into a differential element of solid angle dω (the differential elastic scattering cross section) and the integral of this cross section over all angles (the total elastic scattering cross section).

Figure 15.1 illustrates the scattering of a beam of particles by a scattering center. The problem raised is: What is the probability that a particle will be deflected or absorbed and it or a like particle reemitted into a particular differential element of solid angle dω if the incident beam has unit intensity (1 particle per cm²-sec)? The dimension of this probability is, as we have just seen, the dimension of area used to express the incident beam intensity. The scattering cross section is normally measured in square centimeters, barns, or millibarns where 1 barn = 10⁻²⁴ cm². (The name “barn” is said to have been used first in the Manhattan Project during the Second World War, 10⁻²⁴ cm² being as huge as a barn compared with most nuclear scattering cross sections.)

To reiterate, in general the ratio of total number of scattered particles to the number of particles per unit area in the incident beam is called the total scattering cross section. Similarly, the ratio of the number of particles scattered into the differential element of solid angle dω to the number of particles per unit area in the incident beam is called the differential scattering cross section. The number of scattered particles having an energy between E and E + dE is also of interest, but this is for inelastic scattering and outside the scope of this chapter.

The subject of scattering theory is the relationship of scattering cross sections to the potential energy between the interacting particles. For example if the w...

Table of contents

• Cover
• Title Page
• Copyright Page
• Foreword
• Contents
• Preface
• I Black Body Radiation
• II Fourier Analysis
• III Diffraction And Interference of Photons And Particles
• IV The Feynman Path Integrals And The SchrÖDinger Equation
• V Use of The Wave Function To Obtain Physical Properties of A System
• VI One-Dimensional Problems
• VII The Wkb Approximation And The Bohr-Sommerfeld Quantum Conditions
• VIII Elementary Three-Dimensional Wave Functions In Spherical Coordinates
• IX Perturbation Theory
• X Time-Dependent Perturbation Theory
• XI The One-Electron Atom
• XII Multi-Electron Atoms
• XIII Molecular Structure
• XIV Free Electron And Band Theory of Metals
• XV Elastic Scattering Theory
• XVI Nuclear Forces And Models
• Appendix A The Vibrating String
• Appendix B Matrices
• Appendix C Reduction of Infinite Series In Higher Order Perturbation Theory
• Appendix D Parity
• Index