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feat(LaplaceRungeLenzVector): prove angularMomentum_commutation_lrl #991

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feat(LaplaceRungeLenzVector): prove angularMomentum_commutation_lrl #991

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aef181c
feat(LaplaceRungeLenzVector): prove `angularMomentum_commutation_lrl`
pitmonticone Mar 14, 2026
9eff056
refactor(LaplaceRungeLenzVector): address review comments
pitmonticone Mar 15, 2026
971f6d3
fix(LaplaceRungeLenzVector): restore split simp calls (combining brea…
pitmonticone Mar 15, 2026
88d554e
refactor(LaplaceRungeLenzVector): remove unused simp args, drop unuse…
pitmonticone Mar 17, 2026
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38 changes: 34 additions & 4 deletions PhysLean/QuantumMechanics/DDimensions/Hydrogen/LaplaceRungeLenzVector.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -136,14 +136,45 @@ private def constRadiusRegInvCompPosition (ε : ℝˣ) (i : Fin H.d) :=
-/

/-- `⁅𝐋ᵢⱼ, 𝐀(ε)ₖ⁆ = iℏ(δᵢₖ𝐀(ε)ⱼ - δⱼₖ𝐀(ε)ᵢ)` -/
@[sorryful]
lemma angularMomentum_commutation_lrl (ε : ℝˣ) (i j k : Fin H.d) :
⁅𝐋[i,j], H.lrlOperator ε k⁆ = (Complex.I * ℏ * δ[i,k]) • H.lrlOperator ε j
- (Complex.I * ℏ * δ[j,k]) • H.lrlOperator ε i := by
sorry
simp only [lrlOperator_eq' H ε]
simp only [lie_sub, lie_add, lie_sum, lie_smul, lie_leibniz]
simp only [angularMomentum_commutation_angularMomentum,
angularMomentum_commutation_momentum,
angularMomentum_commutation_radiusRegPow,
angularMomentum_commutation_position]
dsimp only [kroneckerDelta]
simp only [comp_sub, comp_smul, zero_comp, add_zero,
smul_sub, smul_add, smul_smul]
simp only [Finset.sum_add_distrib, Finset.sum_sub_distrib]
simp only [Nat.cast_ite, Nat.cast_one, CharP.cast_eq_zero, mul_ite, mul_one, mul_zero,
ite_smul, zero_smul, smul_ite, smul_zero,
Finset.sum_ite_eq, Finset.mem_univ, ↓reduceIte]
rw [angularMomentumOperator_antisymm k i, angularMomentumOperator_antisymm k j]
simp only [neg_comp, smul_neg]
have ite_comp_right : ∀ (p : Prop) [Decidable p]
(A B : 𝓢(Space H.d, ℂ) →L[ℂ] 𝓢(Space H.d, ℂ)),
(if p then A else 0).comp B = if p then A.comp B else 0 :=
fun p _ A B ↦ by split_ifs <;> simp
simp only [ite_comp_right,
sub_comp, add_comp, smul_comp,
Finset.sum_add_distrib, Finset.sum_sub_distrib,
Finset.sum_ite_eq, Finset.mem_univ, ↓reduceIte]
rw [angularMomentumOperator_antisymm i k, angularMomentumOperator_antisymm j k]
simp only [neg_comp, smul_neg, neg_neg]
split_ifs with hik hjk <;>
simp_all only [smul_zero, sub_zero, zero_sub, zero_smul, zero_mul,
Finset.sum_const_zero, add_zero, sub_self, neg_zero, neg_neg,
← Finset.smul_sum, angularMomentumOperator_eq_zero, zero_comp] <;>
(try simp only [smul_comm (H.m * H.k) (Complex.I * ↑↑ℏ)]) <;>
(try conv_lhs => rw [show
2⁻¹ * Complex.I * ↑↑ℏ * (↑H.d - 1) * (Complex.I * ↑↑ℏ) =
Complex.I * ↑↑ℏ * (2⁻¹ * Complex.I * ↑↑ℏ * (↑H.d - 1)) from by ring]) <;>
abel

/-- `⁅𝐋ᵢⱼ, 𝐀(ε)²⁆ = 0` -/
@[sorryful]
lemma angularMomentum_commutation_lrlSqr (ε : ℝˣ) (i j : Fin H.d) :
⁅𝐋[i,j], H.lrlOperatorSqr ε⁆ = 0 := by
unfold lrlOperatorSqr
Expand All @@ -153,7 +184,6 @@ lemma angularMomentum_commutation_lrlSqr (ε : ℝˣ) (i j : Fin H.d) :
simp [Finset.sum_add_distrib, Finset.sum_sub_distrib]

/-- `⁅𝐋², 𝐀(ε)²⁆ = 0` -/
@[sorryful]
lemma angularMomentumSqr_commutation_lrlSqr (ε : ℝˣ) :
⁅angularMomentumOperatorSqr (d := H.d), H.lrlOperatorSqr ε⁆ = 0 := by
unfold angularMomentumOperatorSqr
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