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Question: a write down the timeindependent schrdinger equation for psix b...

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6. Harmonic oscillator. 35 points. A particle in the harmonic-oscillator is in the super- position state where to(r) and a(z)

a) Write down the time-independent Schrödinger equation for psi(x).

b) Write down the Hamiltonian operator H in terms of the ladder operators a+ and a_. (You do not need to derive it)

c) Use your result from (b) to calculate a+a_psi_n(x). Give your answer in terms of n, psi_n(x), psi_n+1(x), and psi_n-1(x).

d) Normalize psi(x).

e) If you were to measure the energy of the particle, what values could you obtain, and with what probability? (You should be able to write down the answer without any further calculations).

f) Calculate the expectation value of the Hamiltonian, <H>. Is your result consistent with part (e)?

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