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##\psi (x, t)## is the representation of the state vector in the position eigenbasis; ##\psi (p, t)## in the momentum eigenbasis et cetera. That is, ##\psi (x, t) = \langle x|S\rangle##; ##\psi (p, t) = \langle p|S\rangle##.

Now, suppose i expand ##\psi(x,t)## as ##\psi(x,t) = \sum_{n=0}^\infty a_n \cos(\frac{n\pi x} {L})##.

Similarly for ##\psi (p, t)##.

[Assuming Dirichlet conditions hold, ##a\leq x\leq b## etc]

I can now interpret this as: ##\psi(x, t)## is an element of a Hilbert space, with the ##\cos(\frac{n\pi x} {L})##'s as basis. [The ##\cos(\frac{n\pi p} {L'})##'s as basis for ##\psi (p, t)##].

How do i reconcile these two interpretations? ##\psi (x, t)## as the coefficient in the expansion of ##| S\rangle## in the ##|x\rangle## eigenbasis as opposed to an element of a Hilbert space expanded in the ##\cos(\frac{n\pi x} {L})## basis with coefficients ##a_n##.

Should i look upon it as an expansion of an expansion (whatever that means)? Please help!