Physlib

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

theorem

Commutator [Lij,Lkp]=i(δikLjpδjkLip)[L_{ij}, \mathbf{L}_k \cdot \mathbf{p}] = i\hbar (\delta_{ik} \mathbf{L}_j \cdot \mathbf{p} - \delta_{jk} \mathbf{L}_i \cdot \mathbf{p})

In dd-dimensional quantum mechanics, for any indices i,j,k{1,,d}i, j, k \in \{1, \dots, d\}, the commutator of the angular momentum operator component LijL_{ij} and the operator (Lkp)=m=1dLkmpm(\mathbf{L}_k \cdot \mathbf{p}) = \sum_{m=1}^d L_{km} p_m is given by: [Lij,Lkp]=i(δikLjpδjkLip) [L_{ij}, \mathbf{L}_k \cdot \mathbf{p}] = i\hbar (\delta_{ik} \mathbf{L}_j \cdot \mathbf{p} - \delta_{jk} \mathbf{L}_i \cdot \mathbf{p}) where Lij=xipjxjpiL_{ij} = x_i p_j - x_j p_i is the angular momentum operator, pmp_m is the mm-th component of the momentum operator, \hbar is the reduced Planck constant, ii is the imaginary unit, and δij\delta_{ij} is the Kronecker delta.