Patterns of Discovery: An Inquiry into the Conceptual Foundations of Science

By Norwood Russell Hanson | Go to book overview

APPENDIX II

Wave-packets, pulses of disturbance maxima, etc., have been dealt with in a qualitative way; this has helped in locating large features in the pattern of microphysical inquiry. The 'packets' or 'pulses' are not more than an analogy, instrumental to an exposition of Schrödinger's Ψ function. Let us pursue these points somewhat less informally. The mathematics which follows is meant to mark out the conceptual pattern of the foregoing in the actual symbolism of wave mechanics;1 it will focus attention on just those features of the theory about which there has been controversy concerning interpretation.

Consider light: the rays are the orthogonal trajectories of the wave surfaces, and, given screens with large apertures, we can ignore diffraction and compute the ray paths by Fermat's Principle

where u is the phase velocity.

Refractive index is here a function of position; this is a paradigm case of what we called 'W notation' in section D. (With very small objects (10 -8 cm.), geometrical optics fails us. Diffraction phenomena can no longer be ignored, nor can they be described in terms of light rays.)

Consider now classical particles. The equation

is the mathematical expression of Hamilton's Principle of Least Action. The natural motion of a system has an extreme value taken between two configurations of the system, when

(the time integral of the Lagrangian function). Here is an element of what we earlier called 'P notation'.

The analogy with optics is clear when comparison orbits are

-161-

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Patterns of Discovery: An Inquiry into the Conceptual Foundations of Science
Table of contents

Table of contents

  • Title Page iii
  • Contents vii
  • Preface ix
  • Introduction 1
  • Chapter I - Observation 4
  • Chapter II - Facts 31
  • Chapter III - Causality 50
  • Chapter IV - Theories 70
  • Chapter V - Classical Particle Physics 93
  • Chapter VI - Elementary Particle Physics 119
  • Appendix I 159
  • Appendix II 161
  • Notes 176
  • Index 235
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