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Quantum Mechanics (QM), probability, initial conditions of the wave function, and interpretation

Michael J. Caola

Published Apr 8, 2021• 0 citations
OpenAlex

Abstract

We suggest that the experimentally-observed probabilistic nature of QM is associated with our (Nature's) inability to know the initial conditions c n (t = 0) of the wave function Ψ(t) = ∞ n c n (t)ψ n (r, t) describing the experiment.

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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.
This paper introduces the Particle Existence Probability Spectrum (PEPS), a novel theoretical framework that challenges the conventional understanding of wave function collapse in quantum mechanics. The core concept posits that the probability distribution of a particle's existence itself possesses a structured spectrum, denoted as *Sp(ω)*. This spectrum represents a dynamic and inherently probabilistic organization of potential locations rather than a sudden, deterministic collapse of a pre-existing wave function. The theoretical underpinnings are developed through a rigorous mathematical treatment of quantum mechanics, emphasizing the continuous nature of probability. The experimental validation of PEPS relies on high-precision interference experiments, designed to reveal the underlying structure of the probability distribution. The implications of PEPS extend to a re-evaluation of wave function interpretation and could potentially unlock new insights into the fundamental nature of reality. The central equation for the PEPS is: *Sp(ω) = ∫ |ψ(x, ω)|^2 dx* Where: * *Sp(ω)* is the Particle Existence Probability Spectrum. * *|ψ(x, ω)|^2* is the probability density of finding the particle at position *x* with frequency *ω*. * *∫* denotes the integral over all possible positions *x*.
For systems which obey quantum mechanics, the formulation of the problem of treating large numbers of particles is, of course, somewhat different than it is for classical systems. The microscopic description of the system is provided by a wave function which (in the absence of spin) is a function of the classical coordinates { q i }. The mathematical model is provided by a Hamiltonian operator H which is often obtained from the corresponding classical Hamiltonian by the replacement p i → ( ħ /i)(∂/∂ q i ). In other cases the form of the Hamiltonian operator is simply postulated. The microscopic dynamics are provided by the Schrödinger equation i ħ (∂Ψ/∂ t ) = H Ψ which requires as initial condition the knowledge of the wave function Ψ({ q i }, t ) at some initial time t 0 . (Boundary conditions on Ψ({ q i }, t ) must be specified as part of the description of the model as well.) The results of experiments in quantum mechanics are characterized by operators, usually obtained, like the Hamiltonian, from their classical forms and termed observables. Operators associated with observables must be Hermitian. In general, the various operators corresponding to observables do not commute with one another. It is possible to find sets of commuting operators whose mutual eigenstates span the Hilbert space in which the wave function is confined by the Schrödinger equation and the boundary conditions. A set of such (time independent) eigenstates, termed ψ ν ( q ), is a basis for the Hilbert space.