Let's say I'm looking at the infinite square well. Typically, given some arbitrary initial (normalized) wavefunction, we can decompose it into a linear combination of components of the complete set (on the interval [-a,a] or whatever) of sin's and cos's. Then, if you measure something like the energy, you get one of the eigenstates (one of the sin's or cos's), and you measure the energy associated with that eigenstate. But there are many complete sets, sin and cos are just one of them. So, let's say we chose some other one. Obviously, because it's complete, you could decompose the initial wavefunction into a linear combo of this set with the same average energy. But this set might have a different spectrum of energy eigenvalues. But this seems like a contradiction, because nature doesn't care what math you're using. Could this happen? If not with the infinite square well, with an unbound particle?