If we have an initial state given by $ \Psi(x,0) $ and we want to find $ \Psi(x,t) $, we would expand the function in the basis of eigenstates of the Hamiltonian, $\{\psi_n\}$:
$ \Psi(x,t)=\sum _nC_n \psi _n(x)e^{-i E_nt/\hbar}$, with $C_n=(\psi_n(x), \Psi(x,0))$.
However, in the case of a free particle, the eigenstates of the Hamiltonian are
$\psi _k=Ae^{ikx}+Be^{-ikx}$
So, now, the basis is not discrete. Then, how could we find out the time-dependent state, $ \Psi=(x,t) $? How could we expand the wavefunction in the basis of eigenfunctions of the free particle?
Best Answer
With a continuous spectrum, we can take the integral, i.e. $$\Psi(x, t) = \int_{-\infty}^{\infty} \phi(k) e^{ikx - i\frac{\hbar k^2}{2m} t} dk,$$ where $\phi(k)$ are analogous to the coefficients $C_n$. This is also the well-known Fourier transform.
Actually, this is pretty hand-wavey, because the eigenfunctions of the free particle aren't normalizable, so can't be the "basis" of our Hilbert space.
See, for example, Rigged Hilbert space and QM