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About the Choice of Momentum Operator Under Coordinate Representation

Liu Han-kui

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

This article discusses in detail about the presentation of the momentum operator P under coordinate representation. The result indicates that it is not the only way to choose P=-ih (?) in common sense for simple-ness,And it has more general significance to define coordinate operator X and momentum operator P by using the canonical commutation rule of [X,P]=ih.

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This article discusses in detail about the presentation of the momentum operator P under coordinate representation. The result indicates that it is not the only way to choose P=-ih (?) in common sense for simple-ness,And it has more general significance to define coordinate operator X and momentum operator P by using the canonical commutation rule of [X,P]=ih.
Based on (i) the spectral resolution definition of the momentum operator, (ii) the linearity of correspondence between physical observables and quantum Hermitian operators, (iii) the definition of conjugate coordinate-momentum variables in classical mechanics, and (iv) the invariance of the classical Hamiltonian to canonical transformations (transformations that change from one set of conjugate variables to another), we show that the $[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x},\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}]$ coordinate-momentum commutator must have the value ihihih where hhh is a real nonzero number (which we can identify with the experimentally determined $\ensuremath{\hbar}$). The results are then extended to include all generalized coordinates and their conjugate momenta as well as the Cartesian special-relativistic case.
On the basis of physical principles having a very general nature, A. Landé [1] has demonstrated that the mathematical structure of quantum mechanics can be derived without having recourse to the introduction of special assumptions of an ad hoc type (such as commutation rules governing canonical observables) which are not immediately suggested by our knowledge of the physical world, but which have simply originated as rules which mathematical physicists have discovered by past experience to yield conclusions in conformity with experiment.