Angular Momentum commutation angular Momentum
QuantumMechanics.angularMomentum_commutation_angularMomentum
Project documentation
The canonical commutation relations: [xᵢ, pⱼ] = iℏ δᵢⱼ𝟙. -/ lemma position_commutation_momentum : ⁅𝐱 i, 𝐩 j⁆ = (I * ℏ) • δ[i,j] • ContinuousLinearMap.id ℂ 𝓢(Space d, ℂ) := by ext ψ x show 𝐱 i (𝐩 j ψ) x - 𝐩 j (𝐱 i ψ) x = _ trans (I * ℏ) * (-x i * ∂[j] ψ x + ∂[j] ((fun x : Space d ↦ x i) • ⇑ψ) x) · simp only [positionCLM_apply, momentumCLM_apply,...
Exact Lean statement
lemma angularMomentum_commutation_angularMomentum : ⁅𝐋 i j, 𝐋 k l⁆ =
(I * ℏ) • (δ[i,k] • 𝐋 j l - δ[i,l] • 𝐋 j k - δ[j,k] • 𝐋 i l + δ[j,l] • 𝐋 i k)Formal artifact
Lean source
lemma angularMomentum_commutation_angularMomentum : ⁅𝐋 i j, 𝐋 k l⁆ = (I * ℏ) • (δ[i,k] • 𝐋 j l - δ[i,l] • 𝐋 j k - δ[j,k] • 𝐋 i l + δ[j,l] • 𝐋 i k) := by nth_rw 2 [angularMomentumOperator] simp only [angularMomentum_commutation_position, angularMomentum_commutation_momentum, lie_sub, lie_leibniz, comp_smul, smul_comp, comp_sub, sub_comp, ← smul_add, ← smul_sub] dsimp [angularMomentumOperator] ext simp only [nsmul_eq_mul, smul_apply, sub_apply, add_apply, mul_apply_eq_comp, comp_apply, _root_.natCast_apply, positionCLM_apply, momentumCLM_apply, neg_mul, mul_neg, smul_neg, sub_neg_eq_add, smul_eq_mul, smul_add] ring- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QuantumMechanics/Operators/Commutation.lean:302-312
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