Problem Set 4 Solution

2 2 next the definition of the harmonic frequency can

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Unformatted text preview: 0 ( x ) is an eigenfunction of the harmonic oscillator H amiltonian o perator a nd d etermine t he e nergy e igenvalue. In order to show that the function is an eigenfunction, we start by operating the Hamiltonian operator on it, € Ⱥ 2 d 2 Ⱥ ȹ α ȹ1 / 4 2 1 ˆ H ψ 0 ( x ) = Ⱥ − + k x 2 Ⱥ ȹ ȹ e −α x / 2 2 2 Ⱥ 2µ dx Ⱥ ȹ π Ⱥ 1/ 4 2 ȹ α ȹ1 / 4 2 d 2 ȹ −α x 2 / 2 ȹ k ȹ α ȹ ȹ e ȹ + ȹ ȹ x 2 e −α x / 2 . = − ȹ ȹ 2 ȹ Ⱥ ȹ π Ⱥ 2µ dx 2 ȹ π Ⱥ Evaluating the first derivative, € 2 d ȹ −α x 2 / 2 ȹ ȹ e ȹ = − α x e −α x / 2 . Ⱥ dx ȹ Evaluating the second derivative, €2 2 d ȹ −α x 2 / 2 ȹ d ȹ e ȹ = −α x e −α x / 2 2 ȹ Ⱥ dx dx { = − α e −α x = € ( −α + 2 /2 } − α x (−α x ) e −α x ) α 2 x 2 e −α x 2 /2 . 2 /2 8 5 . continued Substituting into the Hamiltonian equation, 1/ 4 2 2 ȹ α ȹ1 / 4 2 k ȹ α ȹ ˆ H ψ 0 ( x ) = − ȹ ȹ − α + α 2 x 2 e −α x / 2 + ȹ ȹ x 2 e −α x / 2...
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This document was uploaded on 12/05/2013.

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