Quantum Mechanics

Principles and Formalism

Roy McWeeny

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486143804
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

Concerned strictly with the principles and formalism of quantum mechanics, this graduate student-oriented volume develops the subject as a fundamental discipline. Opening chapters review the origins of Schrödinger's equations and the nature of the solutions in certain simple and well-known cases, advancing to the ideas associated with vector spaces. Having provided students with the appropriate mathematical language, the author proceeds to the formulation of the main principles of quantum mechanics and their immediate consequences. The book concludes with final generalizations in which alternative "languages" or representations are discussed — each with reference to its specific advantages and applications — and the Dirac transformation theory is developed and explained.

CHAPTER 1

PHYSICAL BASIS OF QUANTUM THEORY

1.1. Particles and waves

There is little doubt that the most concise and elegant exposition of the principles of quantum mechanics consists of a set of basic propositions, from which the whole theory may be derived without further appeal to experiment. The experimental basis of the subject is in this way absorbed into a set of postulates which, although by no means self evident, lead to a network of conclusions which may be tested and verified. The postulates consequently provide a very succinct expression of the results of a wide range of observations. From these postulates, it is possible in principle, though often difficult in practice, to follow the ramifications of the theory into many branches of physics and chemistry.

In spite of the many attractions of the axiomatic approach, to which we return in Chapter 4, it is useful first to recall some of the basic observations concerning wave and particle behaviour of light and electrons. These led to the generalizations on which the more formal theory is based. To this end, some familiarity with the wave-particle “dualism” will be assumed. We recall two of its main features:

(A) There is evidence (e.g. from the photo-electric effect and the Compton effect) that radiation exhibits particle properties. It appears to be transmitted in localized “packets” with energy E and momentum p related to frequency v in the following way:

(1.1)

(1.2)

Here c is the velocity of light, h is Planck’s constant;

A “light particle” with energy given by (1.1) and momentum by (1.2) is called a photon.

(B) There is evidence (e.g. from electron-diffraction experiments) that material particles exhibit wave properties. The relative frequency with which particles are found in a given region of space (measured, for example, by the intensity of darkening of a photographic plate on which a beam of particles falls) is found to be correctly predicted as the squared amplitude of a wave-like disturbance, propagated according to laws formally similar to those of physical optics. For particles travelling in a beam, with a constant velocity, the associated wave is plane and has its normal in the direction of motion. It was suggested by de Broglie, on the basis of relativistic considerations, that the wave length should be related to the particle momentum p = mv by

(1.3)

which agrees exactly with (1.2) since λ = c/v, and this conjecture was subsequently verified experimentally.

Wave mechanics, the particular formulation of quantum mechanics due to Schrödinger, arose in the attempt to reconcile the apparent coexistence of wave-like and particle-like properties in both material particles and photons. Here we indicate the argument, in a rudimentary form, by considering a harmonic wave travelling in the positive x direction:

(1.4)

where ψ measures the magnitude of the disturbance at point x and time t and u is the velocity of propagation.

We remember the interpretation of k. If x increases by 1/k, the values of ψ and its derivatives are unchanged: the disturbance is therefore periodic in space, at any given time, with period λ = 1/k. λ is the wave length and k is the wave number. Also if t increases by 1/(ku), ψ and its derivatives are again unchanged: the disturbance is therefore periodic in time at any given point in space, with period T = 1/(ku). T is the period of a complete oscillation, its reciprocal v = ku being the frequency of oscillation. The definitions are thus

k = wave number,	λ = 1/k = wave length,
v = ku = frequency,	T = 1/v = period.

Also, the two functions (1.4) (either choice of sign) clearly oscillate only between maximum and minimum values ± A; A is the amplitude. A disturbance which is everywhere real can of course be regarded as the real part of either function, or as the sum of the two since e^iθ = cos θ + i sin θ.

Wave packets

According to Fourier’s theorem, any disturbance, travelling to the right with velocity u, can be represented by combining waves such as (1.4) with variable k values and suitably chosen amplitudes:

(1.5)

which is the limit-of a sum of terms

A₁ exp {2πik₁(x−ut)} + A₂ exp {2πik₂(x − ut)} + ...

as the values of k₁, k₂, ... get closer together until they cover the whole range (− ∞, + ∞),¹ the amplitude then becoming a continuous function of k. An arbitrary disturbance which is everywhere real can always be represented in this form by suitably choosing the amplitudes; thus by taking just two terms with k₁ = − k₂ = k and...

Table of contents

• Title Page
• Copyright Page
• Table of Contents
• PREFACE
• CHAPTER 1 - PHYSICAL BASIS OF QUANTUM THEORY
• CHAPTER 2 - SOME SIMPLE SOLUTIONS OF SCHRÖDINGER’S EQUATION
• CHAPTER 3 - MATHEMATICAL DIGRESSION
• CHAPTER 4 - GENERAL FORMULATION OF QUANTUM MECHANICS
• CHAPTER 5 - GENERAL THEORY OF REPRESENTATIONS
• APPENDIX 1 - THE SCHRÖDINGER EQUATION IN GENERALIZED COORDINATES
• APPENDIX 2 - SEPARATION OF PARTIAL DIFFERENTIAL EQUATIONS
• APPENDIX 3 - SERIES SOLUTION OF SECOND-ORDER DIFFERENTIAL EQUATIONS
• APPENDIX 4 - PROJECTION OPERATORS AND NORMAL FORMS
• INDEX