The famous exchange forces arise in quantum mechanics from the uncertainty at any given time as to which of the two potential wells a particle, which is capable of leaking back and forth between the two wells, is occupying. This entirely quantum-theoretical result can be given an energetic explanation, and in fact be estimated quantitatively, by finding the possible energy eigenfunctions for the particle, which correspond to nearly-equal or to nearly-opposite probability amplitudes for the wave function in each of the two wells. The variation of system energy with the separation of the wells can be interpreted as a force. The present paper is an effort to give a much more intuitive account of the origin of these important forces. The force is here interpreted purely kinematically as the result of the fact that the particle bombards the well edges both from within and from without. The net attractive or repulsive force between the wells then arises from the fact that for one type of wave function the bombardment is more frequent on the inner, facing edges, and so the wells are forced together, and conversely for the other type. The ideas are generalized to wells other than the idealized square potential well, and compared quantitatively with more familiar treatments of the problem.
The famous exchange forces arise in quantum mechanics from the uncertainty at any given time as to which of the two potential wells a particle, which is capable of leaking back and forth between the two wells, is occupying. This entirely quantum-theoretical result can be given an energetic explanation, and in fact be estimated quantitatively, by finding the possible energy eigenfunctions for the particle, which correspond to nearly-equal or to nearly-opposite probability amplitudes for the wave function in each of the two wells. The variation of system energy with the separation of the wells can be interpreted as a force. The present paper is an effort to give a much more intuitive account of the origin of these important forces. The force is here interpreted purely kinematically as the result of the fact that the particle bombards the well edges both from within and from without. The net attractive or repulsive force between the wells then arises from the fact that for one type of wave function the bombardment is more frequent on the inner, facing edges, and so the wells are forced together, and conversely for the other type. The ideas are generalized to wells other than the idealized square potential well, and compared quantitatively with more familiar treatments of the problem.
Abstract In the present work the wave mechanics are applied to the discussion of a collision problem of a more complicated type than has, I think, hitherto been done. Though there is nothing very new in the process it is rather interesting to see how a wave function behaves which involves two bodies, and is therefore outside our ordinary space intuitions. But the work had a much more general aim, as an attempt to see whether a sharper line of demarcation could be drawn between the particle-like and the wave-like properties of matter. Put briefly, I take a problem which would be regarded at first sight as irreconcilable with a pure wave theory, but thoroughly typical of the behaviour of particles, and show how in fact the correct result arises naturally from the consideration of wave alone. I have been led in this way to certain not very original speculation on the basis of the quantum theory and these are given at the end of the paper. These are not in the least intended to be a general formulation of the theory, but rather as an examination of a particular type of attempted formulation. I fear that the views expressed will be found partly obvious and perhaps partly inconsistent or even erroneous. My excuse for giving them is that the quantum theory seem already sufficiently developed for us to attempt something more than a mathematical formulation of it, and that a generally accepted physical understanding will be more quickly reached by agreement or disagreement with expressed opinions, than by waiting for the arrival of a full-grown theory. It has been an extraordinarily difficult task to express the ideas I intended, and I wish to thank Prof. D. R. Hartree for many valuable criticisms of my first attempts to do so, though he must be exonerated from all blame if the final result is unintelligible or erroneous. 1. The quantum mechanics have been expressed in a variety of ways, which place the emphasis differently, but it has been shown that they are all exactly equivalent, so what may choose which we will. I shall use the following description, limiting myself to the simpler cases which do not involve considerations of relativity. We take a partial differential equation of the wave type, set down according to certain rules depending on the particular system considered. We solve this subject to certain boundary conditions, expressive of all the limitations imposed by silts, mirrors, diffraction, gratings, moving shutters, etc., that may occur in the proposed system, and obtain as result the wave function Ψ, a function of all the co-ordinates of the system and of the time. From this we from a set of functions, of which the simplest is |Ψ|2; this process may be called taking the intensities. We next have to prepare for the particle aspect, by applying certain rules so as to normalise the intensities, a process entirely foreign to the idea of waves. Finally, we make the interpretation, in which we say that the normalised intensities measure the probability of the particles having such and such positions, momenta, energies, etc., and these are the results verified by observation.
The paper wishes to demonstrate that, in quantum systems with boundaries, different boundary conditions can lead to remarkably different physical behaviour. Our seemingly innocent setting is a one-dimensional potential well that is divided into two halves by a thin separating wall. The two half wells are populated by the same type and number of particles and are kept at the same temperature. The only difference is in the boundary condition imposed at the two sides of the separating wall, which is the Dirichlet condition from the left and the Neumann condition from the right. The resulting different energy spectra cause a difference in the quantum statistically emerging pressure on the two sides. The net force acting on the separating wall proves to be nonzero at any temperature and, after a weak decrease in the low-temperature domain, to increase and diverge with a square-root-of-temperature asymptotics for high temperatures. These observations hold for both bosonic and fermionic type particles, but with quantitative differences. We work out several analytic approximations to explain these differences and the various aspects of the found unexpectedly complex picture.
The ground-state energies and wave functions of a particle moving in one-dimensional potentials of the form V(x)=x2p, p=1,2,..., are obtained nearly exactly by iterative solution of the Schrödinger equation. These solutions are used as the ‘‘experimental benchmark’’ against which approximate solutions for the general case are tested. Both finite depth wells and infinitely deep wells are studied, as is the three-dimensional version (spherical symmetry, angular momentum quantum number l=0). The approach is aimed at student exploration of the physics of the family of potential wells, with emphasis upon connecting the familiar extreme cases, V(x)=x2 (simple harmonic oscillator) and the limiting case xk, k→∞ (square well).
Part 1: the need for the quantum mechanics quantization of radiation the special theory of relativity wave-particle duality the nature of matter waves the uncertainty principle the Bohr atom. Part 2 Schrodinger's equation: the postulates of quantum mechanics the wave equation the wave equation in three dimensions the time-independent Schrodinger equation stationary states normalisation probability flux physical constraints on the wave-function wave packets Gaussian wave-packets and the uncertainty principle. Part 3 One-dimensional potential wells: potential wells the one-dimensional infinite square well discussion of infinite square-well results eigenvalue equations the finite square well finite square well eigenfunctions the control of the wave-function by the wave equation numerical integration of Schrodinger's equation continuum solutions of the square-well potential interpretation of unbound-particle wave functions the double potential well the quantum bouncer electric fields and the tilted well the harmonic oscillator - introduction the quantum oscillator discussion of harmonic oscillator eigenvalues and eigenfunctions the classical probability density more potential wells. Part 4 Schrodinger's equation in three dimensions: particle in a three dimensional box degeneracy Schrodingers equation in spherical polar coordinates - the hydrogen atom normalisation. Part 5 Formal development of quantum mechanics: eigenvalues equations, operators and measurements the momentum operator, eigenvalues and eigenfunctions the position operator and its eigenfunctions Hermitian operators orthogonality of eigenfunctions the expansion postulate example - the b-decay of tritium expectation values compatibility of observables - commutators non-commuting operators time-dependence of expectation values - Ehrenfest's theorem abstract operators - the parity operator, matrix mechanics. Part 6 Angular momentum: angular momentum operators angular momentum commutators angular momentum eigenvalues and eigenfunctions interpretation of angular momentum eigenvalues interpretation of angular momentum eigenfunctions the Stern-Gerlach apparatus and electron spin spin operators - the Pauli Spin matrices the Zeeman effect measurements in quantum mechanics. Part 7 Spherically symmetric potential wells: the rigid rotator the radial equation the hydrogen atom and hydrogen like ions the isotropic harmonic oscillator molecular vibrations - the Morse potential the spherical square well general properties of spherically symmetric potential wells. (Part contents)
Exchange forces lie at the base of some of the most striking explanatory successes of modern physics.An exchange interaction, for instance, is held to be responsible in some key sense for the chemical bond.An exchange force is the origin of ferromagnetism, providing the force that lines up individual spins into an overall pattern.In nuclear physics an exchange force is characteristic for the description of nucleon-nucleon forces.And, in perhaps the most far-reaching example, an exchange interaction is taken as definitional for the quantum field-theoretic description of nature, in which not only the fundamental forces (gravity, electromagnetism, the weak and strong nuclear forces), but also such non-fundamental ones as that operative between pairs of superconducting electrons in a lattice, are traced to the exchange of field quanta.In all these cases, moreover, exchange forces are a peculiarly quantum mechanical phenomenon, something characteristic of the new era that began in 1925 and continues into the present.And as a peculiarly quantum mechanical phenomenon, they are central to the new understanding of matter and force that was ushered in by the theory.For such a central concept, the historical origins of exchange forces are distinctly unclear.What was the path by which these diverse manifestations of exchange-in atomic physics, in condensed matter, in nuclear physics, in quantum field theory-came to go by the same name?Indeed, what is the common thread running through all these uses?The very notion of exchange is nebulous.Consider the following definition from a technical dictionary: exchange [QUANT MECH] (1) Operation of exchanging the space and spin coordinates in a Schrodinger--Pauli wave function representing two identical particles; this operation must leave the wave function unchanged, except possibly for a sign.(2) Process of exchanging a real or virtual particle between two other particles.'Here two distinct meanings of the term 'exchange' are on offer.The first notion