Physlib.QuantumMechanics.Hydrogen.LaplaceRungeLenzVector
Laplace-Runge-Lenz vector
In this file we define - The (regularized) LRL vector operator for the quantum mechanical hydrogen atom, `𝐀(ε)ᵢ ≔ ½(𝐩ⱼ𝐋ᵢⱼ + 𝐋ᵢⱼ𝐩ⱼ) - mk·𝐫(ε)⁻¹𝐱ᵢ`.
The main results are - The commutators `⁅𝐋ᵢⱼ, 𝐀(ε)ₖ⁆ = iℏ(δᵢₖ𝐀(ε)ⱼ - δⱼₖ𝐀(ε)ᵢ)` in `angularMomentum_commutation_lrl` - The commutators `⁅𝐀(ε)ᵢ, 𝐀(ε)ⱼ⁆ = (-2iℏm·𝐇(ε) + iℏmkε²·𝐫(ε)⁻³))𝐋ᵢⱼ` in `lrl_commutation_lrl` - The commutators `⁅𝐇(ε), 𝐀(ε)ᵢ⁆ = iℏε²(⋯)` in `hamiltonianReg_commutation_lrl` - The relation `𝐀(ε)² = 2m 𝐇(ε)(𝐋² + ¼ℏ²(d-1)²) + m²k² + ε²(⋯)` in `lrlOperatorSqr_eq`
1 declaration
Commutator
In -dimensional quantum mechanics, for any indices , the commutator of the angular momentum operator component and the operator is given by: where is the angular momentum operator, is the -th component of the momentum operator, is the reduced Planck constant, is the imaginary unit, and is the Kronecker delta.
