Where does the sinusoidal form of Schrödinger equation solutions come from?

[pzona]

I'm learning how to solve the Schrödinger wave equation and I know that the solutions are of the general form:

$$\Psi$$(x) = A sin kx + B cos kx

This was given to us, but where did it come from? Is it the fact that the wave could be a sine or cosine wave, and the other constant multiplier (A or B) can then be set to zero using boundary conditions? Or is there something more complicated going on here?

 

[sweet springs]

Hi.
ψ (x) = A sin kx + B cos kx
is derived from
ψ" (x) = -k^2 ψ (x)
which is derived from
(-i hbar ∂/∂x)^2/2m ψ (x) = hbar^2 k^2/2m ψ (x)
which is stationary SWE in the system of no potential energy with energy eigenvalue hbar^2 k^2/2m.
A and B are chosen to meet the boundary conditions.
Regards.

 

[pzona]

Ooooookay, that makes sense. I wasn't sure where (or even if) -k^2 came into play here, but now that you showed me, I should have seen it. I didn't even think of reducing h, but that makes a lot more sense. Thanks a lot for the response.