Physlib.QuantumMechanics.Hydrogen.Basic

Hydrogen atom

This module introduces the `d`-dimensional hydrogen atom with `1/r` potential.

In addition to the dimension `d`, the quantum mechanical system is characterized by a mass `m > 0` and constant `k` appearing in the potential `V = -k/r`. The standard hydrogen atom has `d=3`, `m = mₑmₚ/(mₑ + mₚ) ≈ mₑ` and `k = e²/4πε₀`.

The potential `V = -k/r` is singular at the origin. To address this we define a regularized Hamiltonian in which the potential is replaced by `-k·r(ε)⁻¹`, where `r(ε)² = ‖x‖² + ε²`. This goes by several names including "soft-core" and "truncated" Coulomb potential. e.g. see https://doi.org/10.1103/PhysRevA.80.032507 and https://doi.org/10.1063/1.3290740.

A. Basic

B. Regularization

2 declarations

theorem

The potential function of a hydrogen atom is $V(x) = -k/\|x\|$

#potential_eq

For a $d$ -dimensional hydrogen atom $H$ with coupling constant $k$ , the potential energy function $V$ at a position $x$ is given by $V(x) = -\frac{k}{\|x\|}$ where $\|x\|$ denotes the Euclidean norm of the position vector $x$ .

theorem

The potential $V$ of a hydrogen atom is AE-strongly measurable

#potential_AESM

For any $d$ -dimensional hydrogen atom $H$ , the potential function $V: \text{Space } d \to \mathbb{R}$ is almost everywhere strongly measurable with respect to the Lebesgue measure on $\text{Space } d$ .