Discrete wave mechanics is pursued further by extending the one-dimensional treatment to two (or more) dimensions in the light of explicit momentum considerations. Cognizance is taken of the effect of particle motion on mass and hence on the interactions between components of motion in different directions. The overall energy parameter turns out to be a product instead of a sum of parameters identified with each of several orthogonal axes. Accordingly, the separation of variables is most directly accomplished by factoring the principal energy parameter in conjunction with factoring the wave vector expression itself. Wave vector energies, on the other hand, remain additive. Finally, group velocity components are discussed for higher-dimensional systems.
Discrete wave mechanics is pursued further by extending the one-dimensional treatment to two (or more) dimensions in the light of explicit momentum considerations. Cognizance is taken of the effect of particle motion on mass and hence on the interactions between components of motion in different directions. The overall energy parameter turns out to be a product instead of a sum of parameters identified with each of several orthogonal axes. Accordingly, the separation of variables is most directly accomplished by factoring the principal energy parameter in conjunction with factoring the wave vector expression itself. Wave vector energies, on the other hand, remain additive. Finally, group velocity components are discussed for higher-dimensional systems.
Discrete wave mechanics is formulated for particles in one-dimensional systems by use of a simple finite difference equation. The solutions involve wave vectors (instead of wave functions) as well as a newly defined "wave vector energy." In the limit, as c --> infinity, the treatment reduces to that of Schrödinger's wave mechanics. Specific calculations are made for completely free particles as well as for particles confined to a one-dimensional box. The results exhibit a striking compatibility with relativistic considerations. The wave vectors show properties that can be identified with particles and anti-particles-each possess identical probability distributions with energies that add up to zero.
Reasons are given for believing that the boundary conditions at a discontinuity in the potential (V) in wave mechanics may be ${\ensuremath{\varphi}}_{1}={\ensuremath{\varphi}}_{2}{\ensuremath{\mu}}^{\frac{1}{2}}$ and $\frac{{\ensuremath{\varphi}}_{1}d{\ensuremath{\varphi}}_{1}}{\mathrm{dx}}=\frac{{\ensuremath{\varphi}}_{2}d{\ensuremath{\varphi}}_{2}}{\mathrm{dx}}$. Here $\ensuremath{\varphi}$ is the wave function and $\ensuremath{\mu}$ the ratio of the group velocities. With these boundary conditions the waves are either completely transmitted or completely reflected according as the energy E is greater or less than V. The particles therefore need not be supposed to enter the region in which V>E where their velocities would be imaginary and their kinetic energies negative. It is suggested that the group of waves associated with a particle should be identified with the particle and that the energy and momentum of the particle are the energy and momentum of the group of waves. When the group is stopped or absorbed it contracts to small dimensions and so becomes a particle.
Statistical mechanics seems to be based on the idea of distributing a fixed amount of energy among a number of particles. (Usually this number is very large.) Alternatively, one may think of a single particle changing its state (e.g. momentum) during many time intervals. In both cases, one takes an average of the energy, but this means one needs weights for each single particle state. These weights seem to be based on the idea that each ‘vector’ of states has the same probability of occurrence. Thus, if one has N particles and an energy eN, one vector involves one particle having eN and the rest 0. Many vectors, however, may be created with N particles having energy near e. Thus, the weight for a single particle to have energy e is much higher than for it to have energy E. The weight is ultimately taken to be the Boltzmann factor. In quantum mechanics, it seems a very different assumption is made. Unlike classical physics where one follows a particle in space and time, knowing its acceleration, position and velocity at each x, in quantum mechanics, collisions with the potential seem to be granular, and the main assumption seems to be that a quantum particle may be in any momentum plane wave state at a given time. Like statistical mechanics, there is a set of weights, but one no longer uses the scenario of distributing energy among different states to determine these weights. Instead, the weights are obtained by noting that a quantum average at a point x is taken using fp exp(ipx)/ (Sum over fp exp(ipx)). The weights must ensure that: -1/2m (Sum over p p2/2m fp exp(ipx)) / (Sum over fp exp(ipx)) = E-V(x) at each x point. Thus, the plane wave behavior of quantum particles becomes a dominant factor. For the case of V(x)=0, the plane wave nature of the possible single particle momentum states leads to an overall (Sum over fp exp(ipx)) which has the appearance to two plane waves. This is very different from the classical statistical mechanical case. In this note, we attempt to examine the difference of these two statistical theories, and understand why in one case, the ground state of an oscillator, the two become the same.
Introduction Under this name at present two theories are being carried on, which are indeed closely related but not identical. The first, which follows on directly from the famous doctoral thesis by L. de Broglie, concerns waves in three-dimensional space. Because of the strictly relativistic treatment that is adopted in this version from the outset, we shall refer to it as the four-dimensional wave mechanics. The other theory is more remote from Mr de Broglie's original ideas, insofar as it is based on a wave-like process in the space of position coordinates ( q -space) of an arbitrary mechanical system. We shall therefore call it the multi-dimensional wave mechanics. Of course this use of the q -space is to be seen only as a mathematical tool, as it is often applied also in the old mechanics; ultimately, in this version also, the process to be described is one in space and time. In truth, however, a complete unification of the two conceptions has not yet been achieved. Anything over and above the motion of a single electron could be treated so far only in the multi -dimensional version; also, this is the one that provides the mathematical solution to the problems posed by the Heisenberg–Born matrix mechanics. For these reasons I shall place it first, hoping in this way also to illustrate better the characteristic difficulties of the as such more beautiful four-dimensional version.
The problem of emission by an atom is treated according to Schroedinger’s rules. The translational motion of the atom as a whole is taken into account. The charge-current density vector is shown to be propagating with the velocity of light in the direction of the emitted quantum. Only unidirectional quanta can be emitted according to the theory. The reason is that the charge-current wave radiates infinitely more intensely if it travels with light velocity than otherwise.It is shown that Gordon’s treatment of the Compton effect involves an analogous phenomenon. It is suggested that Schroedinger’s equations designed for the conservation of energy and momentum work out better when the translational motion of the atom as a whole is considered.
According to classical physics (i.e., physics prior to the 20th century), particles and waves are distinct classes of physical entity that possess markedly different properties. For instance, particles are discrete, which means that they cannot be arbitrarily divided. In other words, it makes sense to talk about one electron, or two electrons, but not about a third of an electron. Waves, on the other hand, are continuous, which means that they can be arbitrarily divided. In other words, given a wave whose amplitude has a certain value, it makes sense to talk about a similar wave whose amplitude is one third, or any other fraction whatsoever, of this value. Particles are also highly localized in space. For example, atomic nuclei have very small radii of order 10−15 m, whereas electrons act like point particles (i.e., they have no discernible spatial extent). Waves, on the other hand, are nonlocalized in space. In fact, a wave is defined as a disturbance that is periodic in space, with some finite periodicity length (i.e., wavelength). Hence, it is fairly meaningless to talk about a disturbance being a wave unless it extends over a region of space that is at least a few wavelengths in size.