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Frame of Reference Moving Along a Line in Quantum Mechanics

Karl De Paepe

Published Jul 1, 2020• 0 citations
OpenAlex

Abstract

We consider a frame of reference moving along a line with respect to an inertial frame of reference. We expect the rate of total energy expectation value not to depend on the order of transformations of a transformation that is a composition of a translation and a transformation that has a frame of reference accelerating with a time dependent acceleration with respect to an inertial frame of reference.

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•Jun 30, 2008•2 citations
We consider a frame of reference moving along a line with respect to an inertial frame of reference. We expect the rate of total energy expectation value not to depend on the order of transformations of a transformation that is a composition of a translation and a transformation that has a frame of reference accelerating with a time dependent acceleration with respect to an inertial frame of reference.
The general framework of a time-dependent scale transformation as applied to non-relativistic time-dependent Schrödinger equations is described. The nature of the transformation is clarified by comparing it with a time-dependent translation and rotation. Particular attention is paid to the associated gauge transformation and the resulting harmonic inertial force. Corresponding classical results are also displayed. This work intends to pave the way for various applications to be presented in the subsequent papers of this series.
In physics, every observation is made with respect to a frame of reference. Although reference frames are usually not considered as degrees of freedom, in all practical situations it is a physical system which constitutes a reference frame. Can a quantum system be considered as a reference frame and, if so, which description would it give of the world? Here, we introduce a general method to quantise reference frame transformations, which generalises the usual reference frame transformation to a "superposition of coordinate transformations". We describe states, measurement, and dynamical evolution in different quantum reference frames, without appealing to an external, absolute reference frame, and find that entanglement and superposition are frame-dependent features. The transformation also leads to a generalisation of the notion of covariance of dynamical physical laws, to an extension of the weak equivalence principle, and to the possibility of defining the rest frame of a quantum system.
The general framework of a time-dependent scale transformation as applied to non-relativistic time-dependent Schrödinger equations is described. The nature of the transformation is clarified by comparing it with a time-dependent translation and rotation. Particular attention is paid to the associated gauge transformation and the resulting harmonic inertial force. Corresponding classical results are also displayed. This work intends to pave the way for various applications to be presented in the subsequent papers of this series.
The spatially-temporal structure which we are going to introduce is a distinctive feature in the description of mechanical objects. In classical mechanics the motion of point masses is described in a frame of reference which consists of a spatial Cartesian coordinate system and a clock. The distinguished class is formed by the so called inertial frames which are characterized by the property that a free point mass performs uniform rectilinear motion with respect to such a frame. It follows that the coordinates (ξ, τ) and (ξ′, τ′) of a point mass in any two inertial frames of reference are related by the Galilei transformation 3.1.1 ξ′=Rξ+x+vτ,τ′+τ+t, \xi ' = R\xi + x + v\tau , \tau ' + \tau + t, ξ′=Rξ+x+vτ,τ′+τ+t, where R is an orthogonal matrix (rotation) which describes the orientation of the new spatial axes with respect to the old ones, x is the vector of the spatial shift of the origin, ν is the relative velocity, and τ is the time shift showing the difference between the readings of the two clocks.
Abstract In quantum mechanics courses, the concept of a reference frame is frequently hidden. In this article, we present a basic perspective to introduce the transformations of a reference frame based on unitary transformations. These unitary transformations are inspired in canonical transformation for reference frame from the perspective of classical mechanics. The manner we deal with this change in reference frame allows us to discuss some fundamental concepts in Schrödinger and Heisenberg pictures, as well as the ideas of passive and active description in quantum mechanics. These unitary transformations are inspired in a reference-frame canonical transformation from the perspective of classical mechanics. The approach we take for this reference-frame change allows for a discussion on fundamental concepts in Schrödinger and Heisenberg pictures, as well as the ideas of passive and active description in quantum mechanics. We apply this unitary transformation in five examples.
The standard formalism of quantum mechanics is extended to describe a total system including the reference system (RS), with respect to which the total system is described. The RS is assumed to be able to act as a measuring apparatus, with measurement records given by the values of some reference properties of the RS. In order to describe the total system, we define a frame of reference (FR) as a set of states that can be used to express all other states of the total system. The theory is based on four basic postulates, which have, loosely speaking, the following contents. (i) A reference property of a RS has a definite value and is sufficiently stable in the FR directly related to the reference property. (ii) States of the total system are associated with vectors in the Hilbert space. (iii) Schrodinger equation is the dynamical law in each valid FR. (iv) Under certain condition a property of a system can be regarded as a reference property; vector descriptions of the total system given in different FRs of the same RS may have a probabilistic relationship like in Born's rule.