On momentum states in quantum mechanics

Anna Lipniacka, Bertrand Martin dit Latour

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 $π$ series. In this work, two "$π$ formulas" are derived when calculating a spectrum of possible outcomes of the momentum measurement for a particle confined in such a well, the series, $\frac{π^2}{8} = \sum_{k=1}^{k=\infty} \frac{1}{(2k-1)^2}$, and the integral $\int_{-\infty}^{\infty} \frac{sin^2 x}{x^2} dx =π$. 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 $π$ formula in \cite{wallys}.

Will Raven

Abstract In this chapter, we explore angular momentum, a key concept in quantum mechanics and atomic physics. We cover the quantization of angular momentum, the role of quantum numbers, and their impact on atomic states. We will learn about the different types of angular momentum, the states representing these types of angular momentum, and how angular momenta are added together in quantum mechanics. The chapter also examines the compatibility of different observables, such as the magnitude and components of angular momentum, and the impact of measurements on quantum states.

E. U. Condon

Xin‐Yan Wu

Representation theory is the basic content of quantum mechanics.This paper discusses the form of mechanical quantity operator and state vector of quantum mechanics under momentum image and coordinate image.At last,it gives an example to explain the superiority by using the momentum image.

E. U. Condon