Contents

Cover

Half Title page

Title page

Copyright page

Dedication

Preface

Chapter 1: Classical Perspective

1.1 Importance of the Harmonic Oscillator

1.2 Newtonian Perspective

1.3 Restoring Force

1.4 Free Harmonic Oscillator

1.5 LC Resonator

1.6 Journey Through the Center of the Earth

1.7 Low Satellite Orbit

1.8 FREE-OSC, Free Oscillator Worksheet

1.9 Phase Space, Part 1

1.10 PHASE-SP, Phase Space Worksheet

1.11 Callisto, Galileo, and French

1.12 Searching for Planets

1.13 Franklin, Priestly, Verne, and Burroughs

What’s Next?

Chapter 2: Oscillator Energy

2.1 Conservative Forces

2.2 Potential Energy

2.3 Kinetic Energy

2.4 Conservation of Energy

2.5 Phase Space, Part 2

2.6 Energy Diagrams

2.7 Equipartition of Energy

2.8 General Potential Energy Function

2.9 Perturbed Satellite Orbit

2.10 High Power-Law Oscillators

2.11 Adiabatic Invariance

2.12 UNDAMPED, Free Oscillator Energy Worksheet

2.13 Bungee Jumpers Beware

2.14 Summary

Chapter 3: Damped Oscillators

3.1 Velocity-Dependent Resistive Force

3.2 FORCE-RV, Resistive Force Worksheet

3.3 Restoring Force and Resistive Force

3.4 Under-Damped Oscillator

3.5 Critically Damped Oscillator

3.6 Over-Damped Oscillator

3.7 Summary of Free, Damped Oscillators

What’s Next?

Chapter 4: Forced Oscillators

4.1 Forced Free Mass

4.2 Forced Damped Mass

4.3 General Solution for Forced Damped Mass

4.4 Power Transfer

4.5 Forced Damped Oscillator

4.6 Permittivity for Bound Charges

4.7 FORCEOSC, Forced Oscillator Worksheet

4.8 Summary

What’s Next?

Chapter 5: Q, R, X, Y, and Z

5.1 What Is Q?

5.2 Frequency Domain

5.3 Time Domain

5.4 Q of Various Oscillators

5.6 Resonant Detection, Mössbauer Effect

5.7 Q Multipliers and Dividers

5.8 Q Switches

5.9 Classical Uncertainty Principle, Part 1

5.10 ABC-OF-Q, Worksheet in Time and Frequency

5.11 Impedance and Admittance

5.12 Impedance and Admittance: Damped Mass

5.13 Impedance and Admittance: Harmonic Oscillator

5.14 Y-AND-Z, Admittance and Impedance Worksheet

5.15 How Henry Cavendish Weighed Planet Earth

What’s Next?

Chapter 6: Fourier Perspective

6.1 Signal Classification

6.2 Periodic Signals

6.3 Oscillator with Periodic Force

6.4 Bandwidth and Spectral Density

6.5 Laplace’s Equation

6.6 Non-Periodic Signals

6.7 Classical Uncertainty Principle, Part 2

6.8 Alternative Formats

6.9 Laplace Transform

6.10 Hilbert Transform

6.11 Parseval’s Theorem

6.12 Joint Time-Frequency Analysis

6.13 Inductive Leap

What’s Next?

Chapter 7: Elements of Linear Systems

7.1 What Is a Linear System?

7.2 Ode to the Sinusoid

7.3 Fast Fourier Transform

7.4 Test Signals

7.5 Step Function Response

7.6 Impulse Response

7.7 Frequency Domain Response Using Phasors

7.8 Impulse Response and Causality

7.9 System Function via Fourier Transform

7.10 Impulse Response via Fourier Transform

7.11 Convolution

7.12 Power Spectral Density

7.13 Wiener-Khintchine Theorem

7.14 Auto- and Cross-Correlation Functions

7.15 Coherence Function

7.16 System Identification

7.17 Deconvolution

7.18 Data Windows

7.19 Central Limit Theorem

What’s Next?

Chapter 8: Methods of Advanced Dynamics

8.1 Generalized Coordinates

8.2 Generalized Momenta

8.3 Lagrange’s Equations

8.4 Hamilton’s Equations

8.5 Poisson Brackets

8.6 Hamilton’s Principle

8.7 Hamilton-Jacobi Equation

8.8 Symmetry of the Hamiltonian Function

What’s Next?

Chapter 9: Coupled Oscillators

9.1 Two Coupled Oscillators

9.2 Magnetically Coupled Resonators

9.3 COUPLOSC, Worksheet for Coupled Oscillators

9.4 Time Evolution, Weak Coupling

9.5 COUPLOS2, Worksheet #2 for Coupled Oscillators

9.6 Normal Coordinates

9.7 Oscillating Galaxies

9.8 Synchronized Oscillators

What’s Next?

Chapter 10: Statistical Perspective

10.1 Distributions and Moments

10.2 Accuracy and Precision

10.3 Parable of the Bunny Rabbits

10.4 Displacement Probability Density

10.5 Uncertainty Due to Thermal Motion

10.6 More Distribution Functions

10.7 Natural Sources of Randomness

10.8 Poisson, Exponential, and Gaussian Distributions

10.9 Random Excitation

10.10 Entropy

10.11 Virial Theorem

10.12 Liouville’s Theorem

10.13 Information and Noise

10.14 Stochastic Resonance

10.15 Phase Space Correlation and Covariance

10.16 Predictability

10.17 Time Evolution and Predictions

10.18 Expectation Values: Caveat Emptor

10.19 Expert Opinions on Predictability and Causality

What’s Next?

Chapter 11: Quantum Perspective

11.1 Classical Crisis

11.2 A Collection of Oscillators

11.3 PLANCK, Worksheet for Blackbody Radiation

11.4 Harmonic Oscillators and Radiation

11.5 Wilson-Sommerfeld-Ishiwara Quantization Rule

11.6 Summary of Energy Level Spacing

11.7 Quantization of Angular Momentum

What’s Next?

Chapter 12: Quantum Oscillators

12.1 Wave-Like Property of Momentum

12.2 Schrödinger Equation

12.3 Dynamic Variables and Their Operators

12.4 Free Particle

12.5 A Particle in a Box

12.6 Superposition of States

12.7 Electric Dipole Radiation

12.8 Dipole Transitions, Particle in a Box

12.10 Harmonic Oscillator

12.11 Harmonic Oscillator Dipole Transitions

12.12 QUANTOSC, Quantum Oscillator Worksheet

12.13 Revival of Wavefunctions

12.14 Discovery of Classical Oscillators

12.15 Meanwhile, Back in Heligoland …

12.16 Single-Electron Oscillator

12.17 Trouble in Paradise

What’s Next?

Appendix 1: Worksheets

Installing the Files from the Diskette

Useful Spreadsheet Functions

Appendix 2: The Hilbert Transform

A2.1 Derivation via Unit Step

A2.2 Constructing an Analytic Function

A2.3 Fourier Transform of Unit Step

A2.4 Application of the Hilbert Transform

A2.6 References for the Hilbert Transform

Bibliography

Books

Software

Selected Papers from American Journal of Physics

Selected Papers from Other Journals

Index

INTRODUCTION TO
Classical and Quantum Harmonic Oscillators

Copyright © 1997 by John Wiley & Sons, Inc.

All rights reserved. Published simultaneously in Canada.

Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012

Library of Congress Cataloging in Publication Data:

Bloch, S. C. (Sylvan Charles)
Introduction to classical and quantum harmonic oscillators / Sylvan C. Bloch.
p. cm.
“A Wiley-Interscience publication.”
Includes bibliographical references and index.
ISBN 0-471-14744-3 (cloth: alk. paper)
1. Harmonic oscillators. I. Title.
QC125.2.B56 1997
530.4’16—dc21
96-44728
CIP

To my grandfathers
Charles David Bloch, Major, U.S. Army
Sylvan Myers, M.D.,

and my son
Lawrence Edward Bloch, J.D., M.D.

Preface

If calculus is the most pervasive concept in our description of the physical world, perhaps the harmonic oscillator is a close second. Harmonic oscillators appear in mechanics, electromagnetics, electronics, optics, acoustics, and quantum mechanics, to name a few. The simple harmonic oscillator is a fundamental building block in our system of knowledge of the physical universe.

People have studied the harmonic oscillator analytically for almost 300 years, but it was recognized as a useful device for musical instruments and weapons for perhaps thousands of years. There is speculation that stringed instruments originated when soldiers and hunters discovered that some of their bows emitted pleasing sounds when plucked. Reeds, sea shells, rams’ horns, and bamboo were employed as musical instruments by blowing into, or across, an opening. Stretched skins provided the first drum heads. Advances in metallurgy eventually produced bells and chimes. The historical connection between weapons and oscillators continues; high-power microwave oscillators have knocked down helicopters, the French army has experimented with acoustic anti-personnel weapons, and lasers are commonplace for aiming, guidance, and remote energy deposition.

Galileo Galilei, a talented musician, painter, craftsman, and medical student, probably performed the first quantitative measurements of a harmonic oscillator while observing the periodic motion of a great pendulum chandelier during a long sermon in the cathedral of Pisa. This sort of thing would get him in trouble later on.

Today harmonic oscillators are ubiquitous. Nobel prizes await the person who builds (or finds) the most nearly perfect oscillator, like the laser and the Mössbauer effect. Atomic oscillators enable us to make our most accurate measurements. And where would rock bands be without music synthesizers?

Simple harmonic oscillators are used to measure physical quantities over an enormous range, from the cosmic implications of the universal gravitational constant G, to the equivalence of inertial mass and gravitational mass, to the density variations deep inside planet Earth, to the sub-atomic domain of electron charge and spin.

The harmonic oscillator is easily visualized as a mechanical system. It is just a bit more abstract as an electronic device, but still easily understood. Deeper, at atomic, nuclear, and sub-nuclear levels we find phenomena that seem to behave as if they were harmonic oscillators, even though they cannot yet be visualized or because we do not really understand what is oscillating.

And yet, despite the simplicity of the oscillator, there is a richness of analytical techniques that can be used to describe it. The harmonic oscillator is an excellent system to learn these methods so you can apply them to more complex systems.

This book is an elementary introduction to harmonic oscillators with practically no steps omitted. A diskette is provided so you can experiment with various aspects of harmonic oscillators. The diskette contains worksheets that are ready to run in the Microsoft Excel spreadsheet. You can also import these files into Quattro Pro and Lotus 1-2-3. Some people regard spreadsheets as “power tools” that enable the user to crunch large amounts of numbers, and that is true. But spreadsheets are more than that. The newer versions contain the fast Fourier transform (FFT) and eighteen complex functions that not only free you from the tedium of calculation but they also enable you to carry out numerical procedures that, in many cases, cannot be done in closed form or with elementary functions. Science and technology require insight, imagination, reasoning, and calculation. These mutually reinforce each other, so if you can get help with calculation you have a 25 percent advantage at the start. The other three are up to you.

You can ask “what if …?” and see the answer. You have instant visualization of new results and the ability to interface with data acquisition boards to import real-world information. Your spreadsheet and your computer become an extension of your mind. I urge you to use your computer and the diskette to explore and gain an intuitive as well as a quantitative understanding. Most of the graphics in this book were done on a spreadsheet; for better graphic clarity you can see them in color on your monitor and you can see new results as you change parameters and initial conditions.

Several threads run through the fabric of this book. They include adiabatic invariance, phase space and its discrete, incompressible, and fuzzy nature, classical and quantum uncertainty principles, and the almost inevitable Aristotelian Incline from the classical domain through the semi-classical to the quantum domain.

Some years ago Professor P. A. M. Dirac and his wife Margit came to my home for dinner. Robert and Edie Dressier stopped by for a while. In the course of our conversation I remarked that I had read that Einstein said it would be enough if we could understand the electron. Dirac replied that it would be enough if students could understand the harmonic oscillator.

This book is based on the works of some of the best minds of the last three hundred years, including Galileo Galilei, Sir Isaac Newton, Louis de Lagrange, Sir William Rowan Hamilton, Leonhard Euler, the Bernoullis (Jacob, John, and Daniel), Baron Jean Baptiste Joseph Fourier, Joseph Liouville, Pierre-Simon Marquis de Laplace, Siméon Denis Poisson, C. G. J. Jacobi, Karl Friedrich Gauss, Lord Kelvin, Lord Rayleigh, Max Planck, Albert Einstein, Niels Bohr, Arnold Sommerfeld, Paul Ehrenfest, Prince Louis Victor de Broglie, David Hilbert, Erwin Schrödinger, Werner Heisenberg, Wolfgang Pauli, Max Born, Paul Adrien Maurice Dirac, and Richard Phillips Feynman. There are many others, and this work is continuing.

It may be difficult to imagine that anything new can be said now about harmonic oscillators, yet hundreds of papers are published every year on harmonic oscillators and their multitude of applications in engineering, science, and technology. We would not have the Global Positioning System without atomic clocks. Nobel prizes are still being awarded for harmonic oscillators; one of the latest involves an oscillator made with a single electron.

I have attempted to start at the beginning and follow the yellow brick road slowly and without heavy mathematical baggage. This is possible because things are easier in retrospect. Everything is obvious, once you’ve seen it (to paraphrase the mathematician Felix Klein). Some things that were worthy of Nobel prizes can now be done in a page or two. I have tried to stay with the fundamentals, but I have also included some interesting applications.

I would like to express my appreciation to Dr. Robert Dressier for his helpful suggestions and insights, too numerous to list. I would also like to thank Prof. Robert W. Flynn and Prof. John A. Llewellyn for their comments and assistance, and Prof. Hans G. Dehmelt for permission to reproduce his results of the single-electron oscillator. For their ever-cheerful help, I am indebted to Marilyn Cureton, Jill Lones, Art Matheny, Debbie O’Hearn, and Shauna Schullo.

Special thanks go to Jack Drucker, Robb Frederick, Perry King, executive editor George J. Telecki, and managing editor Lisa Van Horn for their assistance and their patience. I prepared this book in camera-ready form, so any mistakes are my own.

Feynman said that research is seeing what everyone else has seen, but thinking what no one else has thought. Pasteur said fortune favors the prepared mind. I hope this book will be helpful or (in accord with the oath of Hippocrates) do no harm.

Sylvan Charles Bloch

Tampa, Florida
January 1997

… I have had to unteach myself the difficulties, and now beg to present … the parts that are not hard. Master these thoroughly, and the rest will follow.

Professor Silvanus Phillips Thompson, F.R.S.

Chapter 1

Classical Perspective

Si non oscillas noli tintinnare.
Aristotle

The harmonic oscillator is a pervasive concept in science and engineering; it has been applied from sub-nuclear quarks to cosmic systems. In electronics it is used from sub-audio frequencies to the gamma-ray region. The harmonic oscillator is a simple system that can be solved exactly in both classical and quantum descriptions. That, by itself, is a major claim to fame and places the harmonic oscillator in a special category but there is much more.

An examination of current databases reveals that several hundred articles are published every year on various aspects of the harmonic oscillator. The bibliography (pages 333–358) shows some of the broad applications.

1.1 Importance of the Harmonic Oscillator

From both a practical and theoretical viewpoint the harmonic oscillator occupies a central position as the first approximation for all vibrating systems. As you will soon see, when any realistic system vibrates with small amplitude about its equilibrium position the system can be described as a simple harmonic oscillator. Chapter 2 shows why this is so. In learning about the harmonic oscillator you will also be learning something about methods used to analyze a variety of systems.

The analogies between classical mechanical, electrical, and acoustical systems are well known. When we solve one system we are also solving its analogs. Some quantities which play similar roles in systems are shown in Table 1-1.

Table 1-1. Analogous Quantities in Physical Systems

Mechanical | Electrical | Acoustical
F force | V emf | Pressure
x displacement | q charge | Volume displacement
v velocity | l current | Volume current
a acceleration | dl/dt rate of change of current | Rate of change of volume current
m mass | L inductance | Inertance
κ restoring force constant | 1/C reciprocal of capacitance | Reciprocal of acoustic capacitance
R resistive force constant | R resistance | Acoustic resistance

Although the mechanical, acoustical, and electrical quantities are analogous, there are two major differences:

1. Displacement x is considered to be a continuous variable but electric charge q is discrete, quantized in units of the electron charge. This has a profound effect on randomness and noise in measurements and on the structure of the universe.

2. Mass m is quantized in a hiearchy of discrete masses. Electrons may be the smallest (5.11 × 10⁵ electron-Volts), although in 1996 neutrino oscillations seemed to indicate that neutrinos may have a mass of about 0.2 eV. (One eV is the energy equivalent to a mass of 1.78 × 10⁻³⁶ kg.)

Oscillators constructed with electrons trapped in quantum wells have important applications in electronics. For one thing, they make excellent little lasers. Single electrons have been confined in three-dimensional electromagnetic traps for months, and have been observed to behave as quantized harmonic oscillators; Hans Dehmelt was awarded the Nobel Prize in 1989 for this, and related, work.

We will begin our journey with mechanical oscillators because simple mechanical devices can be easily visualized. Classical mechanics has three foundations. If you start with any one you can derive the other two so it does not appear that any particular one is more fundamental than any other. For historical reasons you learn Newton’s formulation of mechanics first; also with Newton’s laws simple problems can be solved with easy mathematics. In Chapter 8 we will see that Lagrange’s formulation is useful for complicated systems that would be very difficult to solve with Newton’s laws. Hamilton’s formulation (also Chapter 8) provides deeper insight into fundamental questions and a bridge to quantum theory.

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!