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Acta Physica Sinica

A DISCUSSION ABOUT THE PAPER “DETERMINISTIC PROPERTIES CONTAINER IN QUANTUM SYSTEMS WITH CONTINUOUS SPECTRUM”

LIANG FANG-HAO, 山东大学数学系,济南250100

Published Jan 1, 2001• 0 citations
Publisher

Abstract

For a single particle system with continuous energy spectrum, the paper “Deterministic Properties Contained in Quantum Systems with Continuous Spectrum” uses the “double wave functions” to describe the quantum behavior of each particle in the ensemble staying in an energy eigenstate. And the “equivalence theorem” in the paper points out that the double wave description will become the classical mechanics description in the classical limit. However, the statistical distribution of results of measurement of a physical quantity predicted by the double wave description for the ensemble is very different from that predicted by the usual quantum mechanics. And the proof given in the paper for the “equivalence theorem” is incorrect. Actually, this theorem is not true.

DOI
10.7498/aps.50.572

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For a single particle system with continuous energy spectrum, the paper “Deterministic Properties Contained in Quantum Systems with Continuous Spectrum” uses the “double wave functions” to describe the quantum behavior of each particle in the ensemble staying in an energy eigenstate. And the “equivalence theorem” in the paper points out that the double wave description will become the classical mechanics description in the classical limit. However, the statistical distribution of results of measurement of a physical quantity predicted by the double wave description for the ensemble is very different from that predicted by the usual quantum mechanics. And the proof given in the paper for the “equivalence theorem” is incorrect. Actually, this theorem is not true.
When a quantum system with continuous spectrum has a definite anergy, ordinary quantum mechanics uses a single continuous stationary state to describe its properties, the expectation values for arbitrary physical quantities are therefore either infinity or zero. In this paper we point out in the form of mathematical theorem that the system possesses new and deterministic results with evident physical meanings. Classical results for the system are the classical limit cases of these results, and the ordinarv expectation values are the statistical cases of our results.
Because of the use of different limiting procedures, there are two conflict conclusions on wave function. One is that a wave function can describe a single particle and the other is that it can only describe an ensemble in the classical limit. In this paper the limiting procedures have been compared. We put the synthesized limit n→∞, →0 but keep E(n, ) to be the actual measured value of energy. Under this limit condition, we can not only prove that a wave function can only describe an ensemble, but also can make clear the dynamical behavior of the particles of the ensemble. Calculations show that the reason why quantum mechanics cannot describe a particle is not because of the motion equation but because of the definition of state for the particle.
A deterministic quantum mechanics theory is presented. The proposed theory is shown to be consistent with the current mainstream statistical quantum theory as well as with classical physics. It produces solutions that demonstrate that causality, physical reality, and determinism are restored and can explain in simple form concerns raised by results from the current mainstream statistical quantum theory. The meaning of particle-wave duality and complementarity, the possibility of a particle, like an electron, crossing through the nucleus as it does when the angular momentum of the electron is zero at the ground state of the hydrogen atom, the possibility of a point-size particle to have an “intrinsic spin,” the possibility of “quantum jumps” as the electron transitions instantaneously from one stable orbital to another without passing through the space in between the orbitals and does that at irregular time intervals. The natural collapse of the wave function as part of the solution is a result that emerges from the proposed deterministic quantum mechanics theory. The phenomenon of entanglement is also discussed in the context that information transfer between entangled “particles” does not occur in a superluminal fashion and is not a “spooky action at a distance” but rather the local measurement of global property. A linear stability method presents actual analytical solutions consistent with current mainstream quantum theory and classical physics. The Bohr-Schrödinger energy levels leading to the experimentally confirmed spectral lines and the fine structure constant emerge from an approximate solution to these equations.
Quantum mechanics has been enormously successful in explaining the behavior of microsystems such as molecules, atoms, and nuclei. We have seen that quantum mechanics describes a quantum system by the wave function ψ, which is the solution of the Schro¨dinger equation. The wave function determines the probability amplitudes (whose modulus squared gives the probability) for all possible outcomes of a measurement corresponding to an observable on the system. In general, the wave function does not uniquely predict the result of a measurement; it only provides a probability distribution for all possible results of the measurement on a quantum system. Only if the system happens to be in an eigenstate corresponding to the observable being measured, is the result of a measurement of this observable completely predictable. On the other hand, the results of measurements corresponding to a conjugate or non-commuting observable, on the same system in the same state, are indeterminate. Quantum mechanical description of physical systems raises profound questions: How are the wave function and the probabilities for different outcomes of a measurement to be interpreted? Does the wave function represent a single system or an ensemble of systems? Does the physical system actually have the attribute corresponding to the measurement in question or does the measurement itself create the attribute? These questions led to different interpretations of quantum mechanics, which were vigorously debated during the years of its discovery as also in the following decades. The new idea of indeterminacy was especially difficult to reconcile with the long-held view that a physical theory should aim at predicting accurately, and without ambiguity, the result of any measurement on any system, big or small. These debates not only gave us a deeper understanding of quantum mechanics but also changed profoundly the way we view the micro-world.
The study of quantum statistical mechanics begins with a review of the basic principles of quantum mechanics. Schrödinger’s equation is introduced and Eigenstates (or stationary states) are defined. Model probability for quantum statistics is assumed to have a uniform distribution in phases. Wave functions for many-body systems are defined. The density matrix is introduced. The Planck entropy and the microcanonical ensemble are defined. The differences between classical and quantum statistical mechanics are all based on the differing concepts of a microscopic ‘state’. While the classical microscopic state (specified by a point in phase space) determines the exact position and momentum of every particle, the quantum mechanical state determines neither; quantum states can only provide probability distributions for observable quantities.
It is shown that the wave function describes the state of the statistical ensemble E[S] of individual particles, or the statistical average particle . This result follows from the fact that in the classical limit h=0 the Schroedinger equation turns to the dynamic equations for the statistical ensemble of classical particles. The statement that the wave function describes the state of an individual particle is incompatible with the quantum mechanics formalism. It is shown that the statistical interpretation of quantum mechanics is a corollary of the fact, that the QM formalism is the technique of the statistical ensemble description, restricted by constraints of the dynamic equation linearity.