Research proof index

The nth eigenvalues for a Harmonic oscillator is defined as (n + 1/2) * ℏ * ω. -/ noncomputable def eigenValue (n : ℕ) : ℝ := (n + 1/2) * ℏ * Q.ω /-! ## Derivatives of the eigenfunctions -/ lemma deriv_eigenfunction_zero : deriv (Q.eigenfunction 0) = Complex.ofReal (- 1 / Q.ξ ^2) • Complex.ofReal * Q.eigenfunction 0 := by rw [eigenfunction_zero] simp...

The nth eigenvalues for a Harmonic oscillator is defined as (n + 1/2) * ℏ * ω. -/ noncomputable def eigenValue (n : ℕ) : ℝ := (n + 1/2) * ℏ * Q.ω /-! ## Derivatives of the eigenfunctions

The nth eigenfunction satisfies the time-independent Schrodinger equation with respect to the nth eigenvalue. That is to say for Q a harmonic oscillator, Q.schrodingerOperator (Q.eigenfunction n) x = Q.eigenValue n * Q.eigenfunction n x. The proof of this result is done by explicit calculation of derivatives.