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:
The multiple is called the eigenvalue. It is just a number.
of the eigenfunction concept. Function is shown in
black. Its first derivative , shown in red, is not
just a multiple of . Therefore is
not an eigenfunction of the first derivative operator.
However, the second derivative of is ,
which is shown in green, and that is indeed a multiple of
. So is an eigenfunction of the second
derivative operator, and with eigenvalue 4.
A nonzero function is called an eigenfunction of an operator if
is a multiple of the same function:
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
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.”
- 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
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.
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 .
Show that and , with a constant, are eigenfunctions of the inversion operator , which turns any function into , and find the eigenvalues.