Problem Set 4 Solution

Problem Set 4 Solution

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Unformatted text preview: ȹ π Ⱥ 2µ 2 ȹ π Ⱥ 1/ 4 Ⱥȹ 2α ȹ Ⱥ Ⱥ 2 k 2 Ⱥ ȹ α ȹ = Ⱥȹ x Ⱥ ȹ ȹ e −α x / 2 ȹ 1 − α x 2 + Ⱥ ȹ 2µ Ⱥ 2 Ⱥ ȹ π Ⱥ Ⱥ Ⱥ ( ) ( ) Ⱥȹ 2α ȹ Ⱥ = Ⱥȹ ȹ 1 − α x 2 Ⱥ ȹ 2µ Ⱥ Ⱥ ( ) Ⱥ k 2 Ⱥ x Ⱥ ψ 0 ( x ) . 2 Ⱥ Ⱥ + Using the definition of α, € Ⱥȹ 2α ȹ Ⱥ ˆ H ψ 0 ( x ) = Ⱥȹ ȹ 1 − α x 2 Ⱥ ȹ 2µ Ⱥ Ⱥ Ⱥ ȹ 2 ȹ ȹ µ k ȹ1 / 2 Ⱥ = Ⱥ ȹ ȹ ȹ 2 ȹ − Ⱥ ȹ 2µ Ⱥ ȹ Ⱥ Ⱥ 1/ 2 Ⱥ Ⱥ 1 ȹ k ȹ k = Ⱥ ȹ ȹ − 2 ȹ µ Ⱥ 2 Ⱥ Ⱥ ( = ) + Ⱥ k 2 Ⱥ x Ⱥ ψ 0 ( x ) 2 Ⱥ Ⱥ ȹ 2 ȹ ȹ µ k ȹ Ⱥ k 2 Ⱥ x Ⱥ ψ 0 ( x ) ȹ ȹ ȹ 2 ȹ x 2 + 2 Ⱥ ȹ 2µ Ⱥ ȹ Ⱥ Ⱥ Ⱥ k 2 Ⱥ x2 + x Ⱥ ψ 0 ( x ) 2 Ⱥ Ⱥ 1/ 2 1 ȹ k ȹ ȹ ȹ ψ 0 ( x ) 2 ȹ µ Ⱥ 1/ 2 1 h ȹ k ȹ ˆ H ψ 0 ( x) = ȹ ȹ ψ 0 ( x ) . 2 2 π ȹ µ Ⱥ Next, the definition of the harmonic frequency can be used, € ν0 = 1/ 2 1 ȹ k ȹ ȹ ȹ . 2π ȹ µ Ⱥ Substituting, € 1/ 2 1 h ȹ k ȹ ˆ H ψ 0 ( x) = ȹ ȹ ψ 0 ( x ) , 2 2 π ȹ µ Ⱥ 1 ˆ or H ψ 0 ( x ) = hν 0 ψ 0 ( x ) . 2 Thus, we see that the function ψ 0 ( x ) is an eigenfunction of the Schrödinger equation for the harmonic oscillator. ˆ Since H ψ 0 ( x ) = E0...
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This document was uploaded on 12/05/2013.

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