`BookProof.ChapterObservableOperator.observable_expectation_born` (b : OrthonormalBasis (Fin m) ℂ (EuclideanSpace ℂ (Fin n))) (v : Fin m → ℝ) (q : EuclideanSpace ℂ (Fin n)) (hq : ‖

Proved

BookProof.ChapterObservableOperator.observable_expectation_born

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterObservableOperator.

BookProof.ChapterObservableOperator.observable_expectation_born (b : OrthonormalBasis (Fin m) ℂ (EuclideanSpace ℂ (Fin n))) (v : Fin m → ℝ) (q : EuclideanSpace ℂ (Fin n)) (hq : ‖q‖ = 1) : (∀ j, 0 ≤ bornProb (fun j => b j) q j) ∧ (∑ j, bornProb (fun j => b j) q j = 1) ∧ expectation (observableOp (fun j => b j) v) q = ((∑ j, bornProb (fun j => b j) q j * v j : ℝ) : ℂ)

Formalization note: Lean 4 identifier BookProof.ChapterObservableOperator.observable_expectation_born.

Preamble

-- Generated from ChapterObservableOperator.lean — theorem BookProof.ChapterObservableOperator.observable_expectation_born
import Mathlib
import Definitions.Def_ChapterObservableOperator
import Definitions.Def_ChapterDoubleSlit
import Definitions.Def_ChapterObservableExpectation
open BookProof.ChapterDoubleSlit
open BookProof.ChapterObservableExpectation
open BookProof.ChapterObservableOperator


open scoped BigOperators

noncomputable section


variable {n m : ℕ}

Formal statement

theorem BookProof.ChapterObservableOperator.observable_expectation_born (b : OrthonormalBasis (Fin m) ℂ (EuclideanSpace ℂ (Fin n)))
    (v : Fin m → ℝ) (q : EuclideanSpace ℂ (Fin n)) (hq : ‖q‖ = 1) :
    (∀ j, 0 ≤ bornProb (fun j => b j) q j) ∧
      (∑ j, bornProb (fun j => b j) q j = 1) ∧
      expectation (observableOp (fun j => b j) v) q
        = ((∑ j, bornProb (fun j => b j) q j * v j : ℝ) : ℂ) := by sorry

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