Deriv deriv eigenfunction zero
QuantumMechanics.OneDimension.HarmonicOscillator.deriv_deriv_eigenfunction_zero
Project documentation
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...
Exact Lean statement
lemma deriv_deriv_eigenfunction_zero (x : ℝ) : deriv (deriv (Q.eigenfunction 0)) x =
(- 1 / Q.ξ^2) * (1 + ((- 1/ Q.ξ^2) * x ^ 2)) * Q.eigenfunction 0 xFormal artifact
Lean source
lemma deriv_deriv_eigenfunction_zero (x : ℝ) : deriv (deriv (Q.eigenfunction 0)) x = (- 1 / Q.ξ^2) * (1 + ((- 1/ Q.ξ^2) * x ^ 2)) * Q.eigenfunction 0 x := by simp only [deriv_eigenfunction_zero, Complex.ofReal_div, Complex.ofReal_neg, Algebra.smul_mul_assoc] trans deriv (fun x => (- (1/Q.ξ^2)) • (Complex.ofReal x * Q.eigenfunction 0 x)) x · congr funext x simp only [Complex.ofReal_one, Complex.ofReal_pow, Pi.smul_apply, Pi.mul_apply, smul_eq_mul, one_div, neg_smul, Complex.real_smul, Complex.ofReal_inv] ring simp only [Complex.real_smul, Complex.ofReal_neg, Complex.ofReal_div, deriv_const_mul_field'] rw [deriv_fun_mul (by fun_prop) (by fun_prop)] simp only [Complex.deriv_ofReal] rw [deriv_eigenfunction_zero] simp only [Complex.ofReal_div, Complex.ofReal_neg, Pi.mul_apply, Pi.smul_apply, smul_eq_mul, neg_mul] push_cast ring- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QuantumMechanics/HarmonicOscillator/OneDimension/TISE.lean:100-117
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