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When taking the superposition of wavefunctions with definite values of any observable (I'll just use momentum, but I am assuming it would work for any variable), I have seen the integral be used:

##\psi = \int_{-\infty}^{\infty}\phi(k)e^{ikx}dk##

and the sum be used:

##\psi = \sum_{j=0}^{\infty}c_je^{ik_jx}##

where k is the wavenumber (##\frac{p}{\hbar}##). It seems to me like these two are one-and-the-same no mater if the possible values of momentum are discrete or continuous, though the integral is used more when the allowed momenta are continuous and the sum is used when momenta are discrete. Is this correct? I know that if we were dealing with some other observable, I would have to replace the plane wave with a different function.

##\psi = \int_{-\infty}^{\infty}\phi(k)e^{ikx}dk##

and the sum be used:

##\psi = \sum_{j=0}^{\infty}c_je^{ik_jx}##

where k is the wavenumber (##\frac{p}{\hbar}##). It seems to me like these two are one-and-the-same no mater if the possible values of momentum are discrete or continuous, though the integral is used more when the allowed momenta are continuous and the sum is used when momenta are discrete. Is this correct? I know that if we were dealing with some other observable, I would have to replace the plane wave with a different function.

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