Fourier Integral zero of mem orthogonal
QuantumMechanics.OneDimension.HarmonicOscillator.fourierIntegral_zero_of_mem_orthogonal
Plain-language statement
If f is a function ℝ → ℂ satisfying MemHS f such that it is orthogonal to all eigenfunction n then the fourier transform of f (x) * e ^ (- x ^ 2 / (2 ξ^2)) is zero. The proof of this result relies on orthogonal_exp_of_mem_orthogonal.
Exact Lean statement
lemma fourierIntegral_zero_of_mem_orthogonal (f : ℝ → ℂ) (hf : MemHS f)
(hOrth : ∀ n : ℕ, ⟪HilbertSpace.mk (Q.eigenfunction_memHS n), HilbertSpace.mk hf⟫_ℂ = 0) :
𝓕 (fun x => f x * Real.exp (- x^2 / (2 * Q.ξ^2))) = 0Formal artifact
Lean source
lemma fourierIntegral_zero_of_mem_orthogonal (f : ℝ → ℂ) (hf : MemHS f) (hOrth : ∀ n : ℕ, ⟪HilbertSpace.mk (Q.eigenfunction_memHS n), HilbertSpace.mk hf⟫_ℂ = 0) : 𝓕 (fun x => f x * Real.exp (- x^2 / (2 * Q.ξ^2))) = 0 := by funext c rw [Real.fourier_eq] simp only [RCLike.inner_apply, conj_trivial, ofReal_exp, ofReal_div, ofReal_neg, ofReal_mul, ofReal_pow, ofReal_ofNat, Pi.zero_apply] rw [← Q.orthogonal_exp_of_mem_orthogonal f hf hOrth (- 2 * Real.pi * c)] congr funext x simp only [fourierChar, Circle.exp, ContinuousMap.coe_mk, ofReal_mul, ofReal_ofNat, AddChar.coe_mk, ofReal_neg, mul_neg, neg_mul, ofReal_exp, ofReal_div, ofReal_pow] change cexp (-(2 * ↑π * (↑c * ↑x) * I)) * (f x * Complex.exp (- x ^ 2 / (2 * Q.ξ^2))) = _ congr 2 ring- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QuantumMechanics/HarmonicOscillator/OneDimension/Completeness.lean:370-385
Reuse this declaration
Bring the exact result into your workflow
The import identifies the source module. Your project still needs the pinned package dependency shown on this page.
What this badge means
This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.
Continue in this project
Related declarations
Adiabatic relation log
adiabatic_relation_log
Plain-language statement
Adiabatic relation in logarithmic form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then c * log (Ua/Ub) + log (Va/Vb) = 0.
Source project: Physlib
Person-level attribution pending.
Adiabatic relation Ua Ub Va Vb
adiabatic_relation_UaUbVaVb
Plain-language statement
Adiabatic relation in product form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then (Ua/Ub)^c * (Va/Vb) = 1.
Source project: Physlib
Person-level attribution pending.
Deriv Within mean Energy Beta eq neg variance
CanonicalEnsemble.derivWithin_meanEnergy_Beta_eq_neg_variance
Plain-language statement
(∂U/∂β) = -Var(E) for finite systems.
Source project: Physlib
Person-level attribution pending.