Hermite

Hermite - Hermite Polynomials: Harmonic Oscillator-Excited...

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Hermite Polynomials: Harmonic Oscillator-Excited States I.) h 2 2m d 2 dx 2 Ψ + 1 / 2 kx 2 Ψ=ΕΨ E = h ν o ( v + 1 / 2 ) Ψ v (x) = H v e 1 / 2 α 2 x 2 Ψ v (y) = H v e –y 2 /2 V(x) x d 2 dx 2 Ψ 2mk 2h 2 x 2 Ψ= 2mE h 2 Ψ multiplying I.) by 2m h 2 E = h 2 α 2 m (v+ 1 / 2 ) 2mE h 2 = α 2 (2v + 1) α 4 = mk h 2 d 2 dx 2 Ψ α 4 x 2 Ψ= α 2 (2v + 1) Ψ change variables: y = α x d dx = d dy dy dx = α d dy d 2 dx 2 = α 2 d 2 dy 2 α 2 d 2 dy 2 Ψ α 2 y 2 Ψ= α 2 (2v + 1) Ψ substituting for d 2 dx 2 and x 2 ΙΙ.29 d 2 dy 2 Ψ y 2 Ψ+ (2v + 1) Ψ=0 dividing both sides by α 2 d dy H v e –y 2 /2 = H v (–y) e –y 2 /2 + e –y 2 /2 dH v dy taking the derivatives d 2
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