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Journal of Southwest University

An anomalous integral formula deduced from quantum mechanics

Luo Ling-xiao

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

Based on the normalizing of one dimensional infinite potential well, an anomalous integral formula is deduced.

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First, this article explains the formula of one - dimensional infinitely deep square potential well's energy levels from stationary state schrodinger equation, and then deduces the same formula from one - dimensional half infinitely high square potential well' s energy levels diagram of graphic. solution, and proves that one - dimensional leveles are limit of one - dimensional half infinitely high square potential well' s energy levels. Finally it discusses the energy levels' number of one - dimensional half infinitely high square potential well and gives out the differentiateating formula about it.
Quantum particle bound in an infinite, one-dimensional square potential well is one of the problems in Quantum Mechanics (QM) that most of the textbooks start from. There, calculating an allowed energy spectrum for an arbitrary wave function often involves Riemann zeta function resulting in a πππ series. In this work, two "πππ formulas" are derived when calculating a spectrum of possible outcomes of the momentum measurement for a particle confined in such a well, the series, π28=∑k=1k=∞1(2k−1)2\frac{π^2}{8} = \sum_{k=1}^{k=\infty} \frac{1}{(2k-1)^2}8π2​=∑k=1k=∞​(2k−1)21​, and the integral ∫−∞∞sin2xx2dx=π\int_{-\infty}^{\infty} \frac{sin^2 x}{x^2} dx =π∫−∞∞​x2sin2x​dx=π. The spectrum of the momentum operator appears to peak on classically allowed momentum values only for the states with even quantum number. The present article is inspired by another quantum mechanical derivation of πππ formula in \cite{wallys}.
Dirac's ket-bra formalism is the language of quantum mechanics and quantum field theory. In Refs.(Fan et al, Ann. Phys. 321 (2006) 480; 323 (2008) 500) we have reviewed how to apply Newton-Leibniz integration rules to Dirac's ket-bra projectors. In this work by alternately using the technique of integration within normal, antinormal, and Weyl ordering of operators we not only derive some new operator ordering identities, but also deduce some useful integration formulas regarding to Laguerre and Hermite polynomials. This opens a new route of deriving mathematical integration formulas by virtue of the quantum mechanical operator ordering technique.