Mathematical Methods for Physicists: A concise introduction - Site Map

LINEAR VECTOR SPACES
Multiplying Eq. (5.33) from the left by B
~
, we obtain
B
~
…A
~
jvi† ˆ …B
~
jvi†;
which using the fact A
~
B
~
ˆ B
~
A
~
, can be rewritten as
A
~
…B
~
jvi† ˆ …B
~
jvi†:
Thus, B jvi is an eigenvector of A belonging to eigenvalue . If is non-degenerate,
then B jvi should be linearly dependent on jvi, so that
~ ~
~
It follows that
a…B
~
jvi† ‡ bjvi ˆ0; with a 6ˆ 0 and b 6ˆ 0:
B
~
jvi ˆ…b=a†jvi ˆjvi:
If A is degenerate, then the matter becomes a little complicated. We now state
the results without proof. There are three possibilities:
(1) The degenerate eigenvectors (that is, the linearly independent eigenvectors
belonging to a degenerate eigenvalue) of A are degenerate eigenvectors of B
~ ~
also.
(2) The degenerate eigenvectors of A belong to di€erent eigenvalues of B . In this
~ ~
case, we say that the degeneracy is removed by the Hermitian operator B .
~
(3) Every degenerate eigenvector of A is not an eigenvector of B . But there are
~ ~
linear combinations of the degenerate eigenvectors, as many in number as
the degrees of degeneracy, which are degenerate eigenvectors of A but
~
are non-degenerate eigenvectors of B . Of course, the degeneracy is removed
~
by B .
~
Function spaces
We have seen that functions can be elements of a vector space. We now return to
this theme for a more detailed analysis. Consider the set of all functions that are
continuous on some interval. Two such functions can be added together to construct
a third function h…x†:
h…x† ˆf …x†‡g…x†; a x b;
where the plus symbol has the usual operational meaning of `add the value of f at
the point x to the value of g at the same point.'
A function f …x† can also be multiplied by a number k to give the function p…x†:
p…x† ˆk f …x†; a x b:
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