Schrodinger Operator eigenfunction
QuantumMechanics.OneDimension.HarmonicOscillator.schrodingerOperator_eigenfunction
Plain-language statement
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.
Exact Lean statement
lemma schrodingerOperator_eigenfunction (n : ℕ) (x : ℝ) :
Q.schrodingerOperator (Q.eigenfunction n) x = Q.eigenValue n * Q.eigenfunction n xFormal artifact
Lean source
lemma schrodingerOperator_eigenfunction (n : ℕ) (x : ℝ) : Q.schrodingerOperator (Q.eigenfunction n) x = Q.eigenValue n * Q.eigenfunction n x := by simp only [schrodingerOperator_eq_ξ, one_div] rw [Q.deriv_deriv_eigenfunction] have hm' := Complex.ofReal_ne_zero.mpr (Ne.symm (_root_.ne_of_lt Q.hm)) have hℏ' := Complex.ofReal_ne_zero.mpr ℏ_ne_zero rw [eigenValue] simp only [← Complex.ofReal_pow, ξ_sq] simp only [Complex.ofReal_pow, Complex.ofReal_div, Complex.ofReal_mul, inv_div, one_div, Complex.ofReal_add, Complex.ofReal_natCast, Complex.ofReal_inv, Complex.ofReal_ofNat] field_simp ring- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QuantumMechanics/HarmonicOscillator/OneDimension/TISE.lean:187-198
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