Sum of two planewaves using momentum operator

[DevoBoy]

Hi,

I'm baffled by a problem in a quantum physics course I'm taking.

Problem: "Use the momentum operator to find an expression for the sum of two planewaves moving in opposite directions. Both planewaves have the same kinetic energy."

It's in one dimension only.

I know the momentum operator is (h_bar/i)(d/dx), and one planewave is exp(kx-wt). The other is exp(kx+wt) ??

My main problem is that I don't know quite where to start. How should I use the momentum operator? The answer should be on the same form as a solution to the Scrodinger equation..

 

[country boy]

Try writing the expression expi(kx-wt) + exp-i(kx+wt). The idea is that the wavenumber changes sign, not the frequency. This simplifies to 2coskx exp-iwt. Now apply the momentum operator twice. You'll see that p^2=(hk)^2. This means that p=+/-hk.

This is a standing wave, which describes a particle moving in either +x or -x. Supposedly, a measurement of the momentum will produce either +hk or -hk.