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Sum of two planewaves using momentum operator

  1. Mar 6, 2007 #1
    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. :rofl: How should I use the momentum operator? The answer should be on the same form as a solution to the Scrodinger equation.. :uhh:
     
  2. jcsd
  3. Mar 6, 2007 #2
    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.
     
    Last edited: Mar 6, 2007
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