Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Radius Reg Pow tendsto radius Pow

QuantumMechanics.radiusRegPow_tendsto_radiusPow

physics quantum field theory relativity position

Plain-language statement

𝐫[ε,s] ψ converges pointwise to 𝐫[s] ψ as ε → 0 except perhaps at x = 0.

Exact Lean statement

lemma radiusRegPow_tendsto_radiusPow {d : ℕ} (s : ℝ) (ψ : 𝓢(Space d, ℂ)) {x : Space d}
    (hx : x ≠ 0) : Tendsto (fun ε ↦ 𝐫₀ ε s ψ x) nhdsZeroUnits (nhds (𝐫 s ψ x))

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma radiusRegPow_tendsto_radiusPow {d : ℕ} (s : ℝ) (ψ : 𝓢(Space d, ℂ)) {x : Space d}    (hx : x ≠ 0) : Tendsto (fun ε ↦ 𝐫₀ ε s ψ x) nhdsZeroUnits (nhds (𝐫 s ψ x)) := by  have hpow : ‖x‖ ^ s = (‖x‖ ^ 2 + 0 ^ 2) ^ (s / 2) := by    simp [← Real.rpow_natCast_mul, mul_div_cancel₀]  simp only [radiusRegPowCLM_apply, radiusPowLM_apply, Complex.real_smul, hpow]  refine Tendsto.mul_const (ψ x) <| Tendsto.ofReal ?_  refine Tendsto.rpow_const ?_ (Or.inl <| by simp [hx])  exact Tendsto.const_add _ <| Tendsto.pow tendsto_comap 2

Project: Physlib
License: Apache-2.0
Commit: dd43e9e65791
Source: Physlib/QuantumMechanics/Operators/Position.lean:269-276

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

Project-declaredLean 4.32.0

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.

physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

View proof record

Project-declaredLean 4.32.0

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.

physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

View proof record

Project-declaredLean 4.32.0

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.

physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

View proof record