Question

Simple Harmonic Oscillator Schrodinger Equation h2 d For a simple harmonic oscillator with wavefunction ψη(x) mo v21-Hn(f)e-f/2 V2nn! Hermite Polynomials H.(E) Ho-I H.-1654-484° + 12 H,-3255-1604° + 120E Beginning with the Time dependent Schrodinger equation ih ot 2m dx2 and Ψ(x, t) ψ(x)φ(t) Explicitly show how the following two equations are obtained h2 d 2m dx2

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Answer 1

Time dependent Schrodinger equation I h partial differential psi/partial differential t (x, t) = -h^2/2m partial differential^2 psi/partial differential x^2 + v_(x) psi: psi_(x, t) = psi (x) phi (t) I h psi (x) partial differential phi (t)/partial differential t = -h^2/2m phi (t) partial differential^2 psi/partial differential x^2 (x) + v(x) psi (x) phi (t) = E psi (x) p(t) Divide both sides by psi (x) psi(t) I h/phi(t) partial differential phi (t)/partial differential t = -h^2/2m psi(x) partial differential^2 psi (x)/ partial differential x^2 + v(x) = E therefore ih/phi (t) partial differential phi(t)/partial differential phi (t)/partial differential t = E implies d phi (t)/dt = E/I h phi (t) \r\n d phi (t)/dt = -i^2 E phi (t)/I h = -I E phi (t)/h d phi/dt = -i/h E phi now - h^2/2m d^2 psi (x)/dx^2 + v(x) psi(x) = E psi(x)
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