Introducing the γ function in quantum theory,International Journal of Quantum Chemistry

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Introducing the γ function in quantum theory
International Journal of Quantum Chemistry ( IF 2.3 ) Pub Date : 2020-04-08 , DOI: 10.1002/qua.26221
Péter R. Surján ₁

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Through a series of postulates, we define a function γ(x) whose square acts as Dirac's δ(x) and exhibits several unusual properties. Though the square root of δ cannot be defined among distributions, it appears in quantum theory if one wants to associate a wave function to a (quasi)classical particle having charge distribution δ(x). The newly defined function γ(x) serves to describe quasi‐classical particles using part of the quantum formalism (eg, wave functions, operators, expectation values) but exhibiting classical properties. The function γ(x) appears to be useful to define model wave functions for simple (quasi)quantum systems. In a spherical coordinate system, γ(r − r₀) leads to a quasi‐classical “bubble” model of the hydrogen atom, where the electron is distributed on the surface of a sphere with radius r₀, and it provides exact quantum mechanical energies of its total symmetric levels. For other simple quantum systems, it provides approximate but meaningful energies. In particular, exact energy differences for harmonic oscillator levels are obtained, with the zero‐point energy missing.

中文翻译：

在量子理论中介绍γ函数

通过一系列假设，我们定义一个函数γ（x），其平方充当狄拉克的δ（x）并展现出一些不同寻常的性质。尽管不能在分布之间定义δ的平方根，但是在量子理论中似乎有人希望将波函数与具有电荷分布δ（x）的（准）经典粒子相关联。新定义的函数γ（x）用于使用部分量子形式主义（例如，波动函数，算符，期望值）来描述准经典粒子，但具有经典性质。功能 γ（x）似乎对于定义简单（准）量子系统的模型波函数很有用。在球坐标系中， γ（r - r ₀）导致氢原子的准经典“气泡”模型，其中电子分布在半径为r ₀的球体表面上，并提供了精确的量子力学总对称能级的能量。对于其他简单的量子系统，它提供了近似但有意义的能量。特别是，可以获得谐波振荡器电平的精确能量差，而零点能量却丢失了。

更新日期：2020-04-08

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