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Eigenfunctions of Hermitian Operators are Orthogonal

We wish to prove that eigenfunctions of Hermitian operators are orthogonal.
In fact we will first do this **except in the case of equal eigenvalues**.

Assume we have a **Hermitian operator and two of its eigenfunctions** such that

Now we compute
two ways.

Remember the **eigenvalues are real** so there's no conjugation needed.
Now we **subtract the two equations**.
The left hand sides are the same so they give zero.

If
then

The **eigenfunctions are orthogonal**.
What if two of the eigenfunctions have the **same eigenvalue**?
Then, our proof doesn't work.
Assume
is real, since we can always adjust a
phase to make it so.
Since any linear combination of
and
has the same eigenvalue,
we can use any linear combination.
Our aim will be to **choose two linear combinations which are orthogonal**.
Lets try

so

This is zero under the assumption that the dot product is real.
We have thus found an
**orthogonal set of eigenfunctions even in the case that some of the
eigenvalues are equal** (degenerate).
From now on we will just assume that we are working with an orthogonal set of eigenfunctions.

Jim Branson
2013-04-22