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Finite energy transfer and better approximations of delta function in the position measurement.

Q. H. Liu, Sen Kuang, Z. Li, Shangzhe Dai, S. F. Xiao

Published Aug 14, 2018• 0 citations
OpenAlex

Abstract

It is well-accepted that the Dirac delta function can be defined by the limitation of the square function covering a unit area with decrease of the width of the square to zero, and the eigenvectors of the position operator take the form of the delta function. When discussing the position measurement in quantum mechanics, one is prompted by the mathematical convention that uses the square function of sufficiently narrow width to approximate the delta function in order to making the state of the position physical. We argue that such an approximation is improper in physics, because during the position measurement the energy transfer to the particle might be infinitely large. The continuous and square-integrable functions of both sharp peak and sufficiently narrow width can then be better approximations of the delta function to represent the physical states of position.

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It is well-accepted that the Dirac delta function can be defined by the limitation of the square function covering a unit area with decrease of the width of the square to zero, and the eigenvectors of the position operator take the form of the delta function. When discussing the position measurement in quantum mechanics, one is prompted by the mathematical convention that uses the square function of sufficiently narrow width to approximate the delta function in order to making the state of the position physical. We argue that such an approximation is improper in physics, because during the position measurement the energy transfer to the particle might be infinitely large. The continuous and square-integrable functions of both sharp peak and sufficiently narrow width can then be better approximations of the delta function to represent the physical states of position.
The Dirac delta function can be defined by the limitation of the rectangular function covering a unit area with decrease of the width of the rectangle to zero, and in quantum mechanics the eigenvectors of the position operator assume the form of the delta function. When discussing the position measurement in quantum mechanics, one is tempting to take the mathematical convention that uses the rectangular wave function of sufficiently narrow width to approximate the delta function in order to making the state of the position physical. We argue that such an approximation is improper in physics, because during the position measurement the energy transfer to the particle might be infinitely large. The continuous and square-integrable functions of both sharp peak and sufficiently narrow width can then be better approximations of the delta function to represent the physical states of the position. When the slit experiment is taken as an apparatus of position measurement, no matter what potential is used to model the slit, only the ground state of the slit-dependent wave function matters.
We review the postulates of quantum mechanics with respect to the representation of physical states and measurable quantities, their time evolution, and the interpretation of measurements. We first formulate the postulates in terms of wave functions and differential operators, and then reformulate them in the abstract Hilbert space of state vectors, using Dirac’s notations. Improper states subject to Dirac’s delta normalization are introduced, and the space of physical states is extended to include them. The postulates are rationalized by associating each Hermitian linear operator with a complete orthonormal system of its eigenvectors, where measurement probabilities depend on the projections of these eigenvectors on the system’s state vector. Particularly, wave functions are identified as projections of state vectors on the position operator eigenstates. State vectors representing multidimensional systems are formulated as tensor products of vectors in their subspaces. Finally, we address the general uncertainty relations in simultaneous measurements of different observables.
Classical mechanics usually employs two variables x,t to describe motion in the form of x(t). Derivatives of this function yield velocity, acceleration etc. For a bound system, velocity (momentum) changes from one x to the next, but one does not necessarily need time to describe the scenario as one could use the pair of variables p(x),x instead of x(t),t. An issue which arises is that p(x) has double values as it is positive when the bound particle is moving to the right and negative when it is moving to the left. In such a case, time distinguishes between the two motions which occur at the same spatial point. From the point of view of energy, however, such a degeneracy at x is irrelevant because kinetic energy is pp/2m. Thus, it seems one may remove time from a bound state problem as long as one uses an energy (i.e. energy conservation) picture. As a result, we propose using a p(xi), xi model to describe a bound state with p(xi) (momentum at xi) being different at each xi. In order to calculate average of the absolute value of p at xo we suggest using a Dirac delta function delta(p-p(xo)) i.e. related to {Integral dp p delta(p-p(xo))} . If one expands the delta function in terms of the basis exp(ipx), it seems that quantum mechanics for the bound state arises as a classical steady stream picture in terms of energy conservation, with exp(ipx) representing a steady stream picture of a particle with a constant momentum. We discuss informational probability properties of exp(ipx) which make it the basis of choice.
The electromagnetic scattering of a spin-0 charged particle off a fixed center is calculated in first-order quantum perturbation theory. This implies evaluating the square of a `Dirac delta-function,' an operation that is not defined in Schwartz distribution theory, and which in elementary text-books is dealt with according to `Fermi's golden rule.' In this paper these conventional calculations are carefully reviewed, and their crucial parts reformulated in a Colombeau algebra -- in which the product of distributions is mathematically well defined. The conclusions are: (1) The Dirac delta-function insuring energy conservation in first order perturbation theory belongs to a particular subset of representatives of the Schwartz distribution defined by the Dirac measure. These particular representatives have a well-defined square, and lead to a physically meaningful result in agreement with the data. (2) A truly consistent mathematical interpretation of these representatives is provided by their redefinition as Colombeau generalized functions. This implies that their square, and therefore the quantum mechanical rule leading from amplitudes to probabilities, is rigorously defined.