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HarmOsc

HarmOsc - Harmonic Oscillator-Ground State 2 d2 h V(x 2m...

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Harmonic Oscillator-Ground State h 2 2m d 2 dx 2 Ψ + V(x) Ψ=ΕΨ V(x) = 1 / 2 kx 2 Guess: Ψ (x) = A e 1 / 2 α 2 x 2 ν o = 1 2 π f8e5 k m ϖ o =2 πν o ϖ o = f8e5 k m V(x) x E= h 1 2 ν o d Ψ dx = A (– 1 / 2 α 2 )(2x) e 1 / 2 α 2 x 2 = –A α 2 x e 1 / 2 α 2 x 2 d 2 Ψ dx 2 = –A α 2 ( ) x(– 1 / 2 α 2 )(2x) e 1 / 2 α 2 x 2 + e 1 / 2 α 2 x 2 = A α 4 x 2 e 1 / 2 α 2 x 2 A α 2 e 1 / 2 α 2 x 2 = α 4 x 2 Ψ (x) – α 2 Ψ (x) h 2 2m ( ) α 4 x 2 Ψ (x)– α 2 Ψ (x) + 1 / 2 kx 2 Ψ=ΕΨ h 2 2m α 4 x 2 + h 2 2m α 2 + 1 / 2 kx 2 E = h 2 α 2 2m or α = f8e5f8e5f8e5f8e5 2mE h h 2 2m α 4 x 2 + 1 / 2 kx 2 = 0 or h 2 m α 4 + k = 0 h
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