Page 76 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

3.2 Eigenfunctions and eigenvalues 67
^
^
^ ÿ1
and B ˆ A . If the operator A possesses a reciprocal, it is non-singular,in
^
^ ÿ1
which case the expression ö ˆ Aø may be solved for ø, giving ø ˆ A ö.If
^
^
A possesses no reciprocal, it is singular and the expression ö ˆ Aø may not be
inverted.
3.2 Eigenfunctions and eigenvalues

Consider a ®nite set of functions f i and the relationship
c 1 f 1 ‡ c 2 f 2 ‡ ‡ c n f n ˆ 0
where c 1 , c 2 , ... are complex constants. If an equation of this form exists, then
the functions are linearly dependent. However, if no such relationship exists,
except for the trivial one with c 1 ˆ c 2 ˆ ˆ c n ˆ 0, then the functions are
linearly independent. This de®nition can be extended to include an in®nite set
of functions.
^
In general, the function ö obtained by the application of the operator A on
an arbitrary function ø, as expressed in equation (3.1), is linearly independent
of ø. However, for some particular function ø 1 , it is possible that
^
Aø 1 ˆ á 1 ø 1
where á 1 is a complex number. In such a case ø 1 is said to be an eigenfunction
^
^
of A and á 1 is the corresponding eigenvalue. For a given operator A, many
eigenfunctions may exist, so that
^
Aø i ˆ á i ø i (3:5)
where ø i are the eigenfunctions, which may even be in®nite in number, and á i
^
are the corresponding eigenvalues. Each eigenfunction of A is unique, that is to
say, is linearly independent of the other eigenfunctions.
Sometimes two or more eigenfunctions have the same eigenvalue. In that
situation the eigenvalue is said to be degenerate. When two, three, ... , n
eigenfunctions have the same eigenvalue, the eigenvalue is doubly, triply, ... ,
n-fold degenerate. When an eigenvalue corresponds only to a single eigenfunc-
tion, the eigenvalue is non-degenerate.
^
A simple example of an eigenvalue equation involves the operator D x
^
kx
mentioned in Section 3.1. When D x operates on e , the result is
d
^ kx ˆ e kx ˆ ke kx
D x e
dx
^
Thus, the exponentials e kx are eigenfunctions of D x with corresponding
eigenvalues k. Since both the real part and the imaginary part of k can have
any values from ÿ1 to ‡1, there are an in®nite number of eigenfunctions
and these eigenfunctions form a continuum of functions.