Modern quantum mechanics developed from wave mechanics by abstracting from its most fundamental features. These are the superposition principle and the linearity of the equation of motion. Any theory finds its justification from experiment, but all the aspects of a successful theory do not gain similar weights from such corroborations. The dynamical development of a theory, such as the transition from wave mechanics to the more general scheme of quantum mechanics, can thus be fully understood only if a few important experimental facts are kept in mind, namely, those whose description really requires that particular sets of principles be valid. The chance is good then that these principles will remain as guides in the development of the theory. Although the experiments that corroborate wave-mechanics are very numerous indeed, most of them are quite similar in their essence. The experiment of electron diffraction may be chosen, or, alternatively, the conceptually even simpler experiment of particle diffraction by two holes.
Modern quantum mechanics developed from wave mechanics by abstracting from its most fundamental features. These are the superposition principle and the linearity of the equation of motion. Any theory finds its justification from experiment, but all the aspects of a successful theory do not gain similar weights from such corroborations. The dynamical development of a theory, such as the transition from wave mechanics to the more general scheme of quantum mechanics, can thus be fully understood only if a few important experimental facts are kept in mind, namely, those whose description really requires that particular sets of principles be valid. The chance is good then that these principles will remain as guides in the development of the theory. Although the experiments that corroborate wave-mechanics are very numerous indeed, most of them are quite similar in their essence. The experiment of electron diffraction may be chosen, or, alternatively, the conceptually even simpler experiment of particle diffraction by two holes.
One of the earliest and most important tenets of quantum mechanics is the wave-particle duality: light behaves sometimes like a wave and at other times as particle and similarly an electron can also behave both like a particle and as a wave. When the formal laws of quantum mechanics are formulated, the central quantity that describes the particles is the wave function. This points to the need for a good understanding of the properties of the waves. This chapter introduces the concepts and most essential applications that are required to follow the discussion of quantum mechanical laws and systems. The basic characteristics of the waves, such as the superposition principle are presented, and the interference and the diffraction phenomena are discussed. The Young’s double slit experiment in analysed and the formation of interference pattern is explicitly shown. The Rayleigh criterion for the microscopic resolution is also derived.
Classical physics identified matter with particles and radiation with waves. Although these descriptions accounted for a large number of very important facts, it became apparent at the beginning of the twentieth century that other phenomena, such as blackbody radiation and photoelectric effect, are satisfactorily explained only if radiation is described in terms of particles. Louis de Broglie developed a theory according to which, conversely, matter has wave properties. As an outcome of this theory, he was able to write down a formula relating the corresponding wavelength with the momentum of the particle. Diffraction patterns characterize waves, and the basic fact that corroborated the validity of the notion of matter waves was that the Davisson and Germer experiment proved electrons can be diffracted. In classical optics the simplest experiment displaying this phenomenon is the well-known Young two-slit experiment, in which a beam of light is passed through a diaphragm with two slits.
This chapter shows that the laws of conventional optics are a consequence of the linearity of Maxwell's equations, which permit the superposition of different waves. Experiments on electron diffraction show that the interference phenomena associated with these "matter waves" obey the familiar laws of physical optics. The chapter provides the linearly independent solutions of the Schrodinger equation form a complete set of functions. The form of the Schrodinger equation is intimately connected with the requirement of gauge invariance . In classical transformation theory, the Hamiltonian H is the generator of displacements in time. In classical mechanics the total linear and angular momenta are constants of the motion if the Hamiltonian is invariant under translations and rotations, respectively. The chapter focuses on eigenfunctions, and in particular, on stationary states. It presents several illustrations of the fact that there frequently are more than one linearly independent energy eigenfunction belonging to an energy eigenvalue.
Abstract The present work is a discussion of a number of simple problems of the free motion of electrons and atoms from the point of view of the wave mechanics. The author has had the advantage of many conversations with Prof. N. Bohr on the subject, and it was composed under this inspiration. Bohr has recently published his views on the foundations of the theory, and it would be out of place to enter here deeply into the matter, but some review of principles is unavoidable if the technical processes are to be understood. Perhaps the chief point of the work is to show how a simple, even old-fashioned, technique is entirely adequate to deal with these very new problems. 1. General Principles. The matrix and the wave mechanics have both been already developed to great lengths as a calculus of stationary states, but they have not yet got so far in what we may call dynamics, a description of the progress of events. More and more complicated phenomena have been fitted into the same scheme, but not much has been done in making this scheme intuitively understandable. One of the most important contributions is a recent work by Heisenberg, who develops an “uncertainty relation” by showing how each observation of a system, say an electron, always itself introduces some disturbance, so that the exact state at the beginning of each new experiment is essentially unknowable. Heisenberg works from the matrix point of view, but Bohr has pointed out that the “uncertainty relation,” exhibited by Heisenberg simply as an experimental result, becomes quite natural on wave principles, and is indeed nothing more than an expression of those principles.
The principle of superposition of states requires that the states of a dynamical system form a linear vector space. This hypothesis of linearity has usually been regarded as a fundamental postulate in quantum mechanics, of a kind that cannot be explained by classical concepts. Indeed, Dirac [2, p. 14] comments that "the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory, as is shown by the fact that the quantum superposition principle demands indeterminacy in the results of observations in order to be capable of a sensible physical interpretation."
This chapter begins with a brief outline of some of the key motivations for considering a quantum theory: the early attempts to determine the spectral distribution of energy density of black bodies; stability of atoms and molecules; specific heats of solids; interference and diffraction of light beams; polarization of photons. The experimental foundations of wave mechanics are then presented in detail, but in a logical order quite different from its historical development: photo-emission of electrons by metallic surfaces, X- and γ-ray scattering from gases, liquids and solids, interference experiments, atomic spectra and the Bohr hypotheses, the experiment of Franck and Hertz, the Bragg experiment, diffraction of electrons by a crystal of nickel (Davisson and Germer), and measurements of position and velocity of an electron. The need for a quantum theory In the second half of the nineteenth century it seemed that the laws of classical mechanics, developed by the genius of Newton, Lagrange, Hamilton, Jacobi and Poincaré, the Maxwell theory of electromagnetic phenomena and the laws of classical statistical mechanics could account for all known physical phenomena. Still, it became gradually clear, after several decades of experimental and theoretical work, that one has to formulate a new kind of mechanics, which reduces to classical mechanics in a suitable limit, and makes it possible to obtain a consistent description of phenomena that cannot be understood within the classical framework. It is now appropriate to present a brief outline of this new class of phenomena, the systematic investigation of which is the object of the following sections and of chapters 4 and 14.