The Physical Principles of the Quantum Theory

Werner Heisenberg

• Publisher: Dover Publications
• Year: 2013
• eBook ISBN: 9780486318417
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

The contributions of few contemporary scientists have been as far reaching in their effects as those of Nobel Laureate Werner Heisenberg. His matrix theory is one of the bases of modern quantum mechanics, while his "uncertainty principle" has altered our whole philosophy of science.
In this classic, based on lectures delivered at the University of Chicago, Heisenberg presents a complete physical picture of quantum theory. He covers not only his own contributions, but also those of Bohr, Dirac, Bose, de Broglie, Fermi, Einstein, Pauli, Schrodinger, Somerfield, Rupp, ·Wilson, Germer, and others in a text written for the physical scientist who is not a specialist in quantum theory or in modern mathematics.
Partial contents: introduction (theory and experiment, fundamental concepts); critique of physical concepts of the corpuscular theory (uncertainty relations and their illustration); critique of the physical concepts of the wave theory (uncertainty relations for waves, discussion of an actual measurement of the electromagnetic field); statistical interpretation of quantum theory (mathematical considerations, interference of probabilities, Bohr's complementarity); discussion of important experiments (C. T. R. Wilson, diffraction , Einstein-Rupp, emission, absorption and dispersion of radiation, interference and conservation laws, Compton effect, radiation fluctuation phenomena, relativistic formulation of the quantum theory).
An 80-page appendix on the mathematical apparatus of the quantum theory is provided for the specialist.

APPENDIX¹

THE MATHEMATICAL APPARATUS OF THE QUANTUM THEORY²

For the derivation of the mathematical scheme of the quantum theory, whether based on the wave or the particle picture, two sources are available: empirical facts and the correspondence principle. The correspondence principle, which is due to Bohr,³ postulates a detailed analogy between the quantum theory and the classical theory appropriate to the mental picture employed. This analogy does not merely serve as a guide to the discovery of formal laws; its special value is that it furnishes the interpretation of the laws that are found in terms of the mental picture used.

We commence with a derivation of the mathematical structure of quantum mechanics from the corpuscular analogy.⁴

§ 1. THE CORPUSCULAR CONCEPT OF MATTER

The fundamental equations of classical mechanics for a system of f-degrees of freedom may be written in the so-called “canonical” form,

where q₁ q₂ …., q_f are the generalized co-ordinates, p₁ p₂, …., p_f their conjugate momenta, and H the Hamiltonian function. When H does not depend explicitly on the time the energy equation

where W, the total energy, is a constant, follows at once. For simplicity it may be assumed that the system is multiply periodic, in which case any co-ordinate q_k as a function of the time may be written as a Fourier series, that is, as a sum of harmonic terms in the form

The

are amplitudes independent of the time and v₁ v₂ …., v_j are the fundamental frequencies of the motion. Similar expressions involving the same frequencies may be written for the p_k and in general for any function of the p_k and q_k.

By a canonical transformation – that is, one which leaves invariant the form of equations (1) – it is possible to introduce a new set of canonical conjugates J_k, w_k, known as “action-angle variables.” These are essentially defined by the following properties: The Hamiltonian H depends on the J_k only and the w_k are related to the fundamental frequencies of the motion by equations of the form

where the β_k are constants. In these variables the equations of motion therefore become

According to classical electrodynamics the frequencies of the spectral lines emitted by an atom will be the frequencies of the harmonic terms in equation (3) and the amplitudes will determine the corresponding intensities.

According to the correspondence principle there must exist a close relationship between the mechanics of classical particles as outlined above and the mechanics of the quantum theory. For the latter we must therefore seek a set of equations analogous in form to the equations of classical theory, but which also take account of certain well-established empirical facts of atomic physics. Primary among these are the following:

1. The Rydberg-Ritz combination principle. – The observed spectral frequencies of an atom possess a characteristic term structure. That is, all the spectral lines of an element may be represented as the differences of a relatively small number of terms. If these terms are arranged in a one-dimensional array T₁ T₂ …., the atomic frequencies form a two-dimensional array

from which follows at once the combination principle

2. The existence of discrete energy values. – The fundamental experiments of Franck and Hertz on electronic impacts show that the energy of an atom can take on only certain definite discrete⁵ values, W₁ , W₂, ….

3. The Bohr frequency relation. – The characteristic frequencies of an atom are related to its characteri...

Table of contents

• Cover
• Title Page
• Copyright Page
• Foreword
• Preface
• Table of Contents
• I. Introductory
• II. Critique of the Physical Concepts of the Corpuscular Theory
• III. Critique of the Physical Concepts of the Wave Theory
• IV. The Statistical Interpretation of Quantum Theory
• V. Discussion of Important Experiments
• Appendix: The Mathematical Apparatus of the Quantum Theory
• Index