Hi, I'm pretty much an amateur in quantum mechanics. If anyone could clarify the following, that would be greatly appreciated! When you write a wave-function (phi or "amplitude" for example) in terms of basis states (either position or momentum), does it undergo a Fourier decomposition? If so, do you actually perform it with respect to position, time, or both? Does this process have anything to do with how momentum and position wave-functions are Fourier transforms of each other? Does this also have anything to do with the de Broglie relations (which one, frequency-energy or wavelength-momentum, or both as related through the constant c)? Finally, regarding the basis states, are they also wave equations? If so, do their wave-numbers and frequencies have any relation to the wave-function undergoing decomposition, or can they be arbitrarily chosen? In either case, does the speed of a basis state wave-equation have any physical implications?