### 2.5 Eigenvalue Problems

To analyze quantum mechanical systems, it is normally necessary to find so-called eigenvalues and eigenvectors or eigenfunctions. This section defines what they are.

A nonzero vector is called an eigenvector of a matrix if is a multiple of the same vector:

 (2.13)

The multiple is called the eigenvalue. It is just a number.

A nonzero function is called an eigenfunction of an operator if is a multiple of the same function:

 (2.14)

For example, is an eigenfunction of the operator with eigenvalue 1, since 1 . Another simple example is illustrated in figure 2.8; the function is not an eigenfunction of the first derivative operator . However it is an eigenfunction of the second derivative operator , and with eigenvalue 4.

Eigenfunctions like are not very common in quantum mechanics since they become very large at large , and that typically does not describe physical situations. The eigenfunctions of the first derivative operator that do appear a lot are of the form , where and is an arbitrary real number. The eigenvalue is :

Function does not blow up at large ; in particular, the Euler formula (2.5) says:

The constant is called the “wave number.”

Key Points
If a matrix turns a nonzero vector into a multiple of that vector, then that vector is an eigenvector of the matrix, and the multiple is the eigenvalue.

If an operator turns a nonzero function into a multiple of that function, then that function is an eigenfunction of the operator, and the multiple is the eigenvalue.

2.5 Review Questions
1.

Show that , above, is also an eigenfunction of , but with eigenvalue . In fact, it is easy to see that the square of any operator has the same eigenfunctions, but with the square eigenvalues.

2.

Show that any function of the form and any function of the form , where is a constant called the wave number, is an eigenfunction of the operator , though they are not eigenfunctions of .

3.

Show that and , with a constant, are eigenfunctions of the inversion operator , which turns any function into , and find the eigenvalues.