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Journal of Physics A Mathematical and General

Expectation values in the pilot wave interpretation of quantum mechanics

M C Robinson

Published Jan 1, 1982• 1 citation
Publisher

Abstract

It is shown that the expectation values of position, linear and angular momentum and potential and total energy, are the same in the pilot wave and usual interpretations of quantum mechanics, but are different for kinetic energy and the square of the linear and angular momentum. Averaging the potential energy over an ensemble in the pilot wave interpretation is shown to be mathematically equivalent to treating e mod Psi mod 2 as a charge distribution as in the usual interpretation.

DOI
10.1088/0305-4470/15/1/020

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It is shown that the expectation values of position, linear and angular momentum and potential and total energy, are the same in the pilot wave and usual interpretations of quantum mechanics, but are different for kinetic energy and the square of the linear and angular momentum. Averaging the potential energy over an ensemble in the pilot wave interpretation is shown to be mathematically equivalent to treating e mod Psi mod 2 as a charge distribution as in the usual interpretation.
According to quantum mechanics, the information with respect to any measurement on a physical system is contained in a mathematical object, the wave function. In this chapter we become familiar with the mathematical objects that represent the measured properties themselves, namely the quantum mechanical operators. We start from a brief introduction into operators and their properties, emphasizing linear operators, and noncommuting operators. Then we introduce the canonical position and momentum operators. Defining functions of operators, we derive different quantum mechanical operators that correspond to different physical observables, including angular momentum, kinetic energy, and the scalar potential energy. Finally, we introduce the quantum mechanical total energy operator (the Hamiltonian) and demonstrate its explicit generic form for nanoscale building blocks such as atoms and molecules.
A quantum mechanics representation based on position (r⃗\vec{r}r), linear momentum(p⃗\vec{p}p​) and energy(EEE) eigenvalues is presented here. A set of equations, explicitly independent on wave function, was derived relating these observables. In this view, a particle has a known trajectory and at any point on space there is a linear momentum associated. Trajectory here, can be viewed as if a measurement were taken continuously. This picture does not change current quantum mechanics interpretation, rather it comes as a new route of calculation. Also, wave function can be retrieved performing an integration in all space for linear momentum. Equations derived in this work, present a potential dependent on linear momentum originating what we call quantum force. A particle experiences a total force whose resultant is composed by classic and quantum force, evolving on time according to it. In terms of evaluation, a single particle could be described by a set of auxiliary particles named as wave particles(WP). They will start with a range of initial conditions and will collectively describe probabilistic aspects of quantum mechanics. We expect this route could be applied to many problems, such as multi-body systems.
A quantum mechanics representation based on position (r⃗\vec{r}r), linear momentum(p⃗\vec{p}p​) and energy(EEE) eigenvalues is presented here. A set of equations, explicitly independent on wave function, was derived relating these observables. In this view, a particle has a known trajectory and at any point on space there is a linear momentum associated. Trajectory here, can be viewed as if a measurement were taken continuously. This picture does not change current quantum mechanics interpretation, rather it comes as a new route of calculation. Also, wave function can be retrieved performing an integration in all space for linear momentum. Equations derived in this work, present a potential dependent on linear momentum originating what we call quantum force. A particle experiences a total force whose resultant is composed by classic and quantum force, evolving on time according to it. In terms of evaluation, a single particle could be described by a set of auxiliary particles named as wave particles(WP). They will start with a range of initial conditions and will collectively describe probabilistic aspects of quantum mechanics. We expect this route could be applied to many problems, such as multi-body systems.
Quantum theory and the de Broglie – Bohm pilot-wave theory are empirically equivalent. In addition to other objections to the pilot-wave theory, many physicists (and some philosophers) take this to be enough to dismiss the pilot-wave theory, as they say it adds nothing to the standard theory. In this short paper I review some objections and replies to the pilot-wave theory. In particular I respond to the empirical equivalence challenge arguing that, given their mutual relationship, there is no reason to expect the two theories to make different predictions, even if they actually might.
An overview of main operators in quantum mechanics is presented. Operators for position, linear momentum, kinetic energy, potential energy, total energy, and angular momentum are introduced. Properties of linear operators are described. Commutative property of operators has profound implications because a pair of non-commutative operators cannot be measured accurately at the same time. Eigenfunctions and eigenvalues of an operator are defined, and operator form of Schrodinger’s equation is given. Most operators in quantum mechanics are of the Hermitian category because they always have real eigenvalues. The condition to test Hermitian nature of operators is specified. Postulates of quantum mechanics are stated to serve as a convenient starting point.