[voidSetup]

I'm not sure if I'm getting this stuff right.  The general equation for simple harmonic motion is y = Asin(squrt(k/m)t + b) right?  It looks like in my notes the proposed wave equation for the quantum harmonic oscillator is Aexp(-(C^2)(x^2)/2, which is plugged into the Hamiltonian to get the energy.  Is this necessary in order to normalize the wave equation?  I tried doing the normalization with the sine equation and it's divergent.

[Juan R.]

I'm not sure if I'm getting this stuff right.  The general equation for simple harmonic motion is y = Asin(squrt(k/m)t + b) right?

That is for classical (Newtonian) harmonic motion.

It looks like in my notes the proposed wave equation for the quantum harmonic oscillator is Aexp(-(C^2)(x^2)/2, which is plugged into the Hamiltonian to get the energy.

The stationary wavefunction for the fundamental state (with energy E_0) has the form that you write but not the rest of wavefunctions for the harmonic oscillator do not. For example for the level E_1 the stationary wavefunction is Psi_1 = B x exp(-(C^2)(x^2)/2)

Is this necessary in order to normalize the wave equation?

All the stationary wavefunctions for the quantum harmonic oscillator are normalized. E.g. the constant A is

A=(alpha / pi)^1/4

with alpha = 2 pi w m / h.

I tried doing the normalization with the sine equation and it's divergent.

Why do you want to use the equation that gives the position of a Newtonian particle for a given time t to normalize a quantum stationary wavefunction?

[voidSetup]

I think I got it.  My teacher asked about it but the wave equation is not shown in the textbook, they only show the derivation for the permitted energy levels by plugging the Hamiltonian into the Schroedinger eqn. I still don't really get why that eqn is given as the wave equation though.  Conceptually what is different about the quantum oscillator versus the simple harmonic oscillator that makes one a trig function and the other exponential.

[Juan R.]

I think I got it.  My teacher asked about it but the wave equation is not shown in the textbook, they only show the derivation for the permitted energy levels by plugging the Hamiltonian into the Schroedinger eqn. I still don't really get why that eqn is given as the wave equation though.

Solving the time-independent Schrödinger equation (H Psi = E Psi) for the harmonic oscillator Hamiltonian gives both the energy levels and the stationary wavefunctions. Check quantum chemistry (e.g. Levine) or spectroscopy (e.g. Hollas) textbooks for the harmonic oscillator stationary wavefunctions. They are usually given in terms of the Hermite polynomials.