y(x,t)=Asin(ωt+kx) is the equation of motion for a simple harmonic oscillator. You get this by solving Newton's force law.. [itex] F=ma=-kx \\ ma+kx=0 \\ a+\frac{k}{m}x=0 [/itex] Or you can write [itex] \ddot{x}+\frac{k}{m}x=0 [/itex] This is a differential equation, solved by Asin(ωt+kx), where [itex] \omega = \sqrt{\frac{k}{m}} [/itex]. I'm not sure if this answers your question?
He was asking for the wave function. You need to solve it with the Schrodinger equation, not Newtons laws.
This is posted in classical physics, however.. and in any case if it is undergoing simple harmonic motion then it isn't a quantum harmonic oscillator, so I don't see any reason to be messing around with Schrodingers.