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Journal of Mathematical Chemistry

A postulate involving quantum mechanical momentum in position space, density function expression of the kinetic energy and Heisenberg’s uncertainty relation

Ramon Carbó‐Dorca, Emili Besalú

Published May 24, 2014• 3 citations
Publisher
DOI
10.1007/s10910-014-0358-5

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Heisenberg guessed, after he established the matrix quantum mechanics, that the non-commutativity of the matrices of position and momentum implied that the position and momentum of a particle could not be precisely simultaneously determined. He consequently conjectured that time and energy should also have a similar relationship. Soon after, Robertson derived an inequality concerning the space coordinate and momentum, which was thought to be the mathematical expression of the uncertainty relation guessed by Heisenberg. Since then, people have tried various devices to prove the correctness of these two relations. However, no one conducted a careful analysis of Heisenberg’s primary paper. In this work, we point out some serious problems in Heisenberg’s paper and the literature talking about the uncertainty relationships: the physical concepts involved in the uncertainty relations are not clear; one physical concept had more than one explanation, i.e., switching concepts; there has never been measurement experiment to support the relations. The conclusions are that the so-called coordinate–momentum uncertainty relation has never been related to actual measurement and there does not exist a time–energy uncertainty relation.
By interpolating between Fisher information and mechanical kinetic energy, we introduce a general notion of kinetic energy with respect to a parameter of Schrödinger wavefunctions from a statistical inference perspective. Kinetic energy is the sum of Fisher information and an integral of a parametrized analogue of quantum mechanical current density related to phase. A family of integral inequalities concerning kinetic energy and moments are established, among which the Cramér–Rao inequality and the Weyl–Heisenberg inequality, are special cases. In particular, the integral inequalities involving the negative order moments are relevant to the study of electron systems. Moreover, by specifying the parameter to a scale, we obtain a family of inequalities of uncertainty relation type which incorporate the position and momentum observables symmetrically in a single quantity.
Due to the requirements of the principle of causality in the theory of relativity, one cannot make a device for the simultaneous measuring of the canonical conjugate variables in the conjugate Fourier spaces. Instead of admitting that a particle’s position and its conjugate momentum cannot be accurately measured at the same time, we consider the only probabilities which can be determined when working at subatomic level to be valid. On the other hand, based on Schwinger's action principle and using the quadridimensional form of the unitary transformation generator function of the quantum operators in the paper, the general form of the evolution equation for these operators is established. In the nonrelativistic case one obtains the Heisenberg's type evolution equations which can be particularized to derive Heisenberg's uncertainty relations. The analysis of the uncertainty relations as implicit evolution equations allows us to put into evidence the intrinsic nature of the correlation expressed by these equations in straight relations with the measuring process. The independence of the quantisation postulate from the causal evolution postulate of quantum mechanics is also put into discussion.