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Dali Xueyuan xuebao

Derivation of Position and Momentum Uncertainty Relation with Cauchy-Schwarz Inequality

Xie Yong

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

This paper introduces the derivation of position and momentum uncertainty relation from Cauchy-Schwarz inequality and the properties of Hermitian operator,and discusses this uncertainty relation.For a Gaussian wave function,the inequality of uncertainty relation will hold equality sign.

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The famous Cauchy-Schwarz inequality is deduced from the fundamental unequal relationship in different spaces in the nature.Considering the linear Hermitian property of the state physical operators on the Hilbert space,the uncertainty relation is derived by applying the inequality to the quantum system of microcosm.The significance of the uncertainty relation which is the objective report of the wave-particle duality of the microscopic particle and the basis for keeping the stability of atoms and even the whole world is discussed.
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We present several variations of a proof of the position-momentum uncertainty principle that are based on the calculus of variations and does not rely on the Cauchy–Schwartz inequality. We show that the stationary uncertainty wave functions are the Hermite–Gaussian solutions to the quantum harmonic oscillator problem, that the minimum uncertainty wave function is the Gaussian, and that stationary uncertainty wave functions must be their own Fourier transforms. We also provide a calculus of variations proof of the Cauchy–Schwartz inequality. Finally, we discuss the properties of wave functions that are their own Fourier transforms and provide examples of such functions that may be of interest to teachers of undergraduate and graduate level quantum mechanics courses.