Usually, for bound states, there are many eigenfunction solutions (denoted here by the index ).
For states representing one particle (particularly bound states) we must
require that the solutions be normalizable.
Solutions that are not normalizable must be discarded.
A normalizable wave function must go to zero at infinity.
We will prove later that the eigenfunctions are orthogonal to each other.
We will assume that the
eigenfunctions form a complete set
so that any function can be written as a linear combination of them.
Since the eigenfunctions are orthogonal, we can easily
compute the coefficients
in the expansion of an arbitrary wave function
We will later think of the eigenfunctions as unit
vectors in a vector space.
The arbitrary wave function
is then a vector in that space.
It is instructive to compute the expectation value of the Hamiltonianusing the expansion of and the orthonormality of the eigenfunctions.
Jim Branson 2013-04-22