Solution
= Solution
Expand the normalized initial <wavefunction> in the orthonormal energy basis:
$$
c_n=\int_0^a\chi_n(x)^*f(x),dx.
$$
The <Time-dependent Schrödinger equation> then gives the <unitary time evolution>
$$
\boxed{
\psi(x,t)=\sum_{n=1}^\infty
c_n\chi_n(x)e^{-iE_nt/\hbar}}.
$$