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 + ½ε²r(ε)⁻³), where
r(ε)² = ‖x‖² + ε². The ε²/r³ term limits to the zero distribution for ε → 0
but is convenient to include for two reasons:
- It improves the convergence:
r(ε)⁻¹ + ½ε²r(ε)⁻³ = r⁻¹(1 + O(ε⁴/r⁴))with noO(ε²/r²)term. - It is what appears in the commutators of the (regularized) LRL vector components.
The hydrogen atom Hamiltonian regularized by ε ≠ 0 is defined to be
𝐇(ε) ≔ (2m)⁻¹𝐩² - k(𝐫(ε)⁻¹ + ½ε²𝐫(ε)⁻³).