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Journal of Zunyi Normal College

To Calculate Density of Particle's State of Energy in the Statistical Physics

Youchang Yang

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

The article discusses how to calculate density of particle's state of energy in the statistical physics by classical physics and quantum mechanics, and analyzes results of two ways by relativistic quantum mechanics.

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SQM (Statistical QM) is a rigorously no-hidden-variables theory, based on the Schrödinger equation for a non-relativistic case where the eigenvalues of the Hamiltonian operator are possible outcomes of total energies, and wavefunctions are corresponding eigenvectors representing probability density (epistemic functions) that the outcome of measurement (in quantum ensemble) will have a particular eigenvalue. So, Schrödinger’s wavefunction can not represent a real individual particle and hence shows that SQM is an incomplete theory as noted by Einstein and de Broglie. Thus, it was necessary to provide the QM theory with missed individual particles theory IQM. The individual massive non-point-like particles in IQM are represented by the 3D rest-mass energy-density topology (like the stars), generating an autogravitational force, and with hidden variables represented by the energy-density flow inside the particle’s finite volume. So, a particle in QM is represented by an ontological complex wavefunction, based on its energy density and the principle of least action.In this paper, based on Noether’s continuity equation of energy-density flow, we take into consideration the ontological version of the Schrödinger equation, where the Hamiltonian operator is substituted by a fundamental classical ontological operator derived from a particle’s hidden variables. Then, we demonstrate how, by quantization of this ontological PDE of a massive particle, we derive the statistical Schrödinger equation with a physical explanation why, in this process, the hidden variables disappear. Finally, we define the classical ensemble of repeated measurements of a real particle with identically prepared measurement systems and derive the Born rule for it.In this way, not only are IQM and SQM two complementary parts of QM, giving the same results as demonstrated previously, but we show that the statistical QM can be derived from the more sophisticated IQM classical theory of individual particles. That is, we obtained the strong completion of Quantum Mechanics.
A consensus seems to have developed that the Gibbs paradox in classical thermodynamics (the discontinuous drop in the entropy of mixing when the mixed gases become equal to each other) is unmysterious: in any actual situation, two gases can be separated or not, and the associated harmless discontinuity from “yes” to “no” is responsible for the discontinuity. By contrast, the Gibbs paradox in statistical physics continues to attract attention. Here, the problem is that standard calculations in statistical mechanics predict a non-vanishing value of the entropy of mixing even when two gases of the same kind are mixed, in conflict with thermodynamic predictions. This version of the Gibbs paradox is often seen as a sign that there is something fundamentally wrong with either the traditional expression S=klnW or with the way W is calculated. It is the aim of this article to review the situation from the orthodox (as opposed to information theoretic) standpoint. We demonstrate how the standard formalism is not only fully capable of dealing with the paradox, but also provides an intuitively clear picture of the relevant physical mechanisms. In particular, we pay attention to the explanatory relevance of the existence of particle trajectories in the classical context. We also discuss how the paradox survives the transition to quantum mechanics, in spite of the symmetrization postulates.
Early experiments linked to bound state quantum mechanics noticed discrete energy E levels. (Ignoring the issue of radiation from accelerating charges), classical physics predicts continuous E levels. If one tries a statistical approach i.e. thinks in terms of a “perceived” density dt=C dx/v(x) where v(x) is velocity, one again has continuous energy levels. Thus, experimental results may suggest a non-classical condition which affects both acceptable densities and energy levels. Bypassing Newtonian mechanics which traces a particle through x(t), we note that even classically a potential interacts with p and not v i.e. pp/2m - popo/2m = -V(x)+V(xo). This equation holds for p= mo v = (.5mo) 2v etc. Thus, energy flow i.e. p is of interest and if one does not trace the particle as x(t) or consider acceleration (which is not considered statistically), one may view energy flux p as being “spread” over the x axis, it seems. The length 1/p then corresponds to one unit of energy flux for the case of p. In other words, each p is associated with a length 1/p. If this length may be physically resolved, one immediate trial result suggests a spatial probability which varies over 1/p. In other words one may have a periodic function such as cos(px). This physical scenario seems to arise only when the quantum particle interacts with a potential. Calling 2*3.14/p = wavelength, this length is not associated with the velocity of the particle, but with energy flow. Each particle has an identical type of cycling within its own wavelength i.e. cos(px) behaviour as each wavelength represents one unit of energy flux. The wavelength limits resolution and leads to a probability scheme. In a bound quantum system dx associated with V(x) is usually much smaller than the larger wavelengths and so classical ideas such as p and pp/2m still apply, but are modified i.e. multiplied by probabilities which may be positive or negative. Having a spatial probability linked to density which also creates a probability scheme used to calculate average kinetic energy at each x based on 1/p type wavelengths is independent of classical physics. Trying to force average kinetic energy at x in the wavelength scheme to match KE(x) classical leads to discrete E values.