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.

structure QuantumMechanics.HydrogenAtom : Type

A hydrogen atom is characterized by the number of spatial dimensions d, the mass m and the coefficient k for the 1/r potential.

• d : ℕ

  Number of spatial dimensions

• m : ℝ

  Mass (positive)

• hm : 0 < self.m
• k : ℝ

  Coefficient in potential (positive for attractive)

@[simp]

theorem QuantumMechanics.HydrogenAtom.m_ne_zero (H : HydrogenAtom) : H.m ≠ 0

def QuantumMechanics.HydrogenAtom.hamiltonianReg (H : HydrogenAtom) (ε : ℝˣ) : SchwartzMap (Space H.d) ℂ →L[ℂ] SchwartzMap (Space H.d) ℂ

The hydrogen atom Hamiltonian regularized by ε ≠ 0 is defined to be 𝐇(ε) ≔ (2m)⁻¹𝐩² - k·𝐫(ε)⁻¹.

Equations

• H.hamiltonianReg ε = (2 * H.m)⁻¹ • 𝐩² - H.k • 𝐫[ε,-1]