##

Eigenvalue Equations

The time independent Schrödinger Equation is an example of an Eigenvalue equation.

The Hamiltonian operates on
**the eigenfunction**,
giving a constant
**the eigenvalue**, times the same function.
(Eigen just means the same in German.)
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.

In fact, all the derivatives of
must go to zero at infinity in order for
the wave function to stay at zero.
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.

(This can be proven for many of the eigenfunctions we will use.)
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 Hamiltonian**using the expansion of
and the orthonormality of the eigenfunctions.

We can see that the
**coefficients of the eigenstates represent probability amplitudes
to be in those states**,
since the absolute squares of the coefficients
obviously give the probability.

Jim Branson
2013-04-22