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The position-momentum symmetry principle

de la Torre, Carlos Alberto

Published Nov 11, 2013• 0 citations
OpenAlex

Abstract

It is shown that the Fourier transformation that relates position and momentum representations of quantum mechanics can be understood as a consequence of a symmetry principle that establishes the equivalence of being and becoming in the description of reality. There are however other transformation compatible with the same principle that could lead to different formalisms of quantum mechanics.

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It is shown that the Fourier transformation that relates position and momentum representations of quantum mechanics can be understood as a consequence of a symmetry principle that establishes the equivalence of being and becoming in the description of reality. There are however other transformation compatible with the same principle that could lead to different formalisms of quantum mechanics.
From the hypotheses that the position-representation of a physical state is the Fourier transform of its momentum-representation and that the timerepresentation is the inverse Fourier transform of its energy-representation, we are able to obtain the Planck relation E = hν , the de Broglie relation p = h /λ , the Dirac fundamental commutation relation, the Schr?dinger equations, the Heisenberg uncertainty principle in quantum mechanics, and the annihilation/creation of a photon from excitation/de-excitation of an atom following Bohr.
The Heisenberg uncertainty principle is a consequence of the postulate that coordinate and momentum representations are related to each other by the Fourier transform. This postulate has been accepted from the beginning of quantum theory by analogy with classical electrodynamics. We argue that the postulate is based neither on strong theoretical arguments nor on experimental data. A position operator proposed in our recent publication resolves inconsis-tencies of standard approach and sheds a new light on important problems of quantum theory. We do not assume that the reader is an expert in the given field and the content of the paper can be understood by a wide audience of physicists.
Quantum states can be represented mathematically in different ways, depending on the chosen basis states. For example, position-based states have a definite position but an indefinite momentum, while momentum-based states have a definite momentum but an indefinite position.The Fourier transform is a mathematical operation that converts a quantum state from its position-space representation to its momentum-space representation, and vice versa. This transformation allows a function to be expressed in a different domain.This paper examines the concept and characteristics of the momentum-space representation of a quantum state and compares it to the position-space representation. Additionally, we investigate the applications and benefits of using the momentum-space representation to solve certain issues in quantum mechanics.
Based on the hypothesis that the position-representation of a physical state <x|α\alphaα> is the Fourier transform of its momentum-representation <P|α\alphaα> and that the time-representation <t|α\alphaα> is the inverse Fourier transform of its energy-representation <E|α\alphaα>, we are able to find the Planck relation E = hv, the de Broglie relation P = hλ\frac{h}{\lambda}λh​. Afterward, utilizing the Dirac delta function δ\deltaδ(x) and the property FTxf(x) = -iℏ\hbarℏ∂\partial∂, pFTxf(x) = -iℏ\hbarℏ∂\partial∂, pF(p) we obtain the links between operators in ordinary and Hilbert spaces leading to the Dirac fundamental commutation relation [p^,x^\hat{p}, \hat{x}p^​,x^] = -iℏ\hbarℏI^\hat{I}I^, the Schrödinger equations, the Heisenberg uncertainty principle in quantum mechanics, the annihilation & creation of a photon from excitation & de-excitation of an atom according to Bohr.