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I'm currently watching MIT 8.04 (Quantum Physics I) on MIT open courseware. I have just finished lecture 5. In the past 2 lectures, they introduced opperators, specifically momentum, energy and position. To prove/derive (I'm not sure what the correct term is) the momentum and energy opperators, they operated on ##e^{ikx}## and got back to the De Broglie equations. So, my question is, do wavefunctions have to be in the form ##Ae^{ikx}##? I would think no, but if I am correct, would the momentum and energy opperators work for other types of wavefunctions? Why is it enough to prove it just with ##e^{ikx}##? This especially confuses me because ##e^{ikx}## isn't even normalizeable, and thus can't even be a wavefunction.

Thanks!