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feat(Momentum): prove momentumOperatorSchwartz_isSymmetric #990

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7af0703
feat(Momentum): prove `momentumOperatorSchwartz_isSymmetric`
pitmonticone Mar 14, 2026
175438b
refactor(Momentum): remove private def derivSchwartz, use local let
pitmonticone Mar 16, 2026
bfcb7e6
golf(Momentum): shorten momentumOperatorSchwartz_isSymmetric proof
pitmonticone Mar 19, 2026
747d1d6
golf(Momentum): fold unfold and integral_neg into simp only
pitmonticone Mar 19, 2026
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66 changes: 49 additions & 17 deletions PhysLean/QuantumMechanics/DDimensions/Operators/Momentum.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -9,6 +9,7 @@ public import PhysLean.QuantumMechanics.DDimensions.Operators.Unbounded
public import PhysLean.QuantumMechanics.DDimensions.SpaceDHilbertSpace.SchwartzSubmodule
public import PhysLean.QuantumMechanics.PlanckConstant
public import PhysLean.SpaceAndTime.Space.Derivatives.Basic
import Mathlib.Analysis.Calculus.FDeriv.Star
/-!

# Momentum operators
Expand Down Expand Up @@ -97,31 +98,62 @@ lemma momentumOperatorSqr_apply (ψ : 𝓢(Space d, ℂ)) (x : Space d) :

-/

open SpaceDHilbertSpace
open SpaceDHilbertSpace MeasureTheory

/-- The momentum operators defined on the Schwartz submodule. -/
def momentumOperatorSchwartz : schwartzSubmodule d →ₗ[ℂ] schwartzSubmodule d :=
schwartzEquiv.toLinearMap ∘ₗ 𝐩[i].toLinearMap ∘ₗ schwartzEquiv.symm.toLinearMap

@[sorryful]
lemma momentumOperatorSchwartz_isSymmetric : (momentumOperatorSchwartz i).IsSymmetric := by
lemma momentumOperatorSchwartz_isSymmetric :
(momentumOperatorSchwartz i).IsSymmetric := by
intro ψ ψ'
obtain ⟨f, rfl⟩ := schwartzEquiv.surjective ψ
obtain ⟨f', rfl⟩ := schwartzEquiv.surjective ψ'
unfold momentumOperatorSchwartz
simp only [LinearMap.coe_comp, LinearEquiv.coe_coe, ContinuousLinearMap.coe_coe,
Function.comp_apply, LinearEquiv.symm_apply_apply, schwartzEquiv_inner, momentumOperator_apply,
neg_mul, map_neg, map_mul, Complex.conj_I, Complex.conj_ofReal, neg_neg, mul_neg]
-- Need integration by parts and `starRingEnd ∂[i] = ∂[i] starRingEnd`:
-- ⊢ ∫ (x : Space d), Complex.I * ↑↑ℏ * (starRingEnd ℂ) (Space.deriv i (⇑f) x) * f' x =
-- ∫ (x : Space d), -((starRingEnd ℂ) (f x) * (Complex.I * ↑↑ℏ * Space.deriv i (⇑f') x))
sorry

/-- The symmetric momentum unbounded operators with domain the Schwartz submodule
of the Hilbert space. -/
@[sorryful]
def momentumUnboundedOperator : UnboundedOperator (SpaceDHilbertSpace d) (SpaceDHilbertSpace d) :=
UnboundedOperator.ofSymmetric (schwartzSubmodule_dense d) (momentumOperatorSchwartz_isSymmetric i)
simp only [momentumOperatorSchwartz, LinearMap.coe_comp, LinearEquiv.coe_coe,
ContinuousLinearMap.coe_coe, Function.comp_apply, LinearEquiv.symm_apply_apply,
schwartzEquiv_inner, momentumOperator_apply, neg_mul, map_neg, map_mul, Complex.conj_I,
Complex.conj_ofReal, neg_neg, mul_neg, integral_neg]
simp_rw [show ∀ x : Space d, Complex.I * ↑↑ℏ *
(starRingEnd ℂ) (Space.deriv i (⇑f) x) * f' x = (Complex.I * ↑↑ℏ) *
((starRingEnd ℂ) (Space.deriv i (⇑f) x) * f' x) from fun x => by ring,
show ∀ x : Space d, (starRingEnd ℂ) (f x) *
(Complex.I * ↑↑ℏ * Space.deriv i (⇑f') x) = (Complex.I * ↑↑ℏ) *
((starRingEnd ℂ) (f x) * Space.deriv i (⇑f') x) from fun x => by ring]
rw [integral_const_mul, integral_const_mul, neg_mul_eq_mul_neg]; congr 1
let dS (g : 𝓢(Space d, ℂ)) :=
(SchwartzMap.evalCLM ℂ _ ℂ (basis i)) ((SchwartzMap.fderivCLM ℂ _ ℂ) g)
have hsd : ∀ x, (starRingEnd ℂ) (Space.deriv i (⇑f) x) * f' x =
fderiv ℝ (fun y => star (f y)) x (basis i) * f' x := fun x => by
congr 1; change star (fderiv ℝ (⇑f) x (basis i)) = _
rw [fderiv_star (𝕜 := ℝ)]; simp
simp_rw [hsd]
change ∫ x, fderiv ℝ (fun y => star (f y)) x (basis i) * f' x =
-(∫ x, star (f x) * fderiv ℝ (⇑f') x (basis i))
have ibp := integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable
(f := fun x => star (f x)) (g := ⇑f') (v := basis i) (μ := volume)
(hf'g := ?_) (hfg' := ?_) (hfg := ?_)
(.star (SchwartzMap.differentiable f)) (SchwartzMap.differentiable f')
· rw [ibp, neg_neg]
· have h : ∀ x, (fderiv ℝ (fun x => star (f x)) x) (basis i) =
star ((dS f) x) := fun x => by
rw [fderiv_star (𝕜 := ℝ)]; simp [dS]
simp_rw [h]; exact .mul_of_top_left
(((starL' ℝ : ℂ ≃L[ℝ] ℂ).integrable_comp_iff).mpr (SchwartzMap.integrable _))
(SchwartzMap.memLp_top f' volume)
· have h : ∀ x, (fderiv ℝ (⇑f') x) (basis i) = (dS f') x := fun _ => rfl
simp_rw [h]; exact .mul_of_top_left
(((starL' ℝ : ℂ ≃L[ℝ] ℂ).integrable_comp_iff).mpr (SchwartzMap.integrable f))
(SchwartzMap.memLp_top _ volume)
· exact .mul_of_top_left
(((starL' ℝ : ℂ ≃L[ℝ] ℂ).integrable_comp_iff).mpr (SchwartzMap.integrable f))
(SchwartzMap.memLp_top f' volume)

/-- The symmetric momentum unbounded operators with domain the Schwartz
submodule of the Hilbert space. -/
def momentumUnboundedOperator :
UnboundedOperator (SpaceDHilbertSpace d) (SpaceDHilbertSpace d) :=
UnboundedOperator.ofSymmetric (schwartzSubmodule_dense d)
(momentumOperatorSchwartz_isSymmetric i)

end
end QuantumMechanics
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