Alternate form of wave equation

[Daniel Sellers]

Apologies if this question is better posed in the mathematics section, it is for a quantum mechanics class so I decided to post it here:

We are asked to verify that the following equation is a solution to the Schrödinger wave equation for a free particle:

Psi(x,t) = Ae^i(kx-wt) - Ae^-i(kx+wt)

I did that part just fine, but then we are asked to show that this wave equation can be rewritten as

Psi(x,t) = 2AiSinkxe^(-iwt)

and answer the question: What kind of wave of this?

So far the only thing I can come up with is

Psi(x,t) = 2AiSin(kx)cost(wt)

Which doesn't seem compatible at all.

Can anyone show me how this is the same equation? and if you know 'what kind of wave' this is that would be very helpful too.

 

[PeterDonis]

Can anyone show me how this is the same equation?

Try rewriting $e^{i(kx - wt)}$ and $e^{-i(kx + wt)}$ as products of exponentials. Pay careful attention to the signs of the $wt$ terms in the exponentials.

 

[Daniel Sellers]

Try rewriting $e^{i(kx - wt)}$ and $e^{-i(kx + wt)}$ as products of exponentials. Pay careful attention to the signs of the $wt$ terms in the exponentials.

In fact, that works! Thanks very much!

Now I just need to figure out what the problem means when it asks what type of wave this is and I'll be set.