Mathematical Methods for Physicists: A concise introduction - Site Map

EIGENVALUES AND EIGENVECTORS OF AN OPERATOR
By combining the operations of addition and multiplication, functions of operators
can be formed. We can also de®ne functions of operators by their power
series expansions. For example, e A ~ formally means
e A ~ 1 ‡ A~ ‡ 1 2! A 2 ‡ 1
~ 3! A 3 ‡:
~
A function of a linear operator is a linear operator.
Given an operator A
~
that acts on vector ji, we can de®ne the action of the same
operator on vector hj. We shall use the convention of operating on hj from the
right. Then the action of A
~
on a vector hj is de®ned by requiring that for any jui
and hvj, we have
fhujA gji v hujfA jigˆ v hjA u ji: v
~ ~ ~
We may write jvi ˆjvi and the corresponding bra as hvj. However, it is
important to note that hvj ˆA*hvj.
Eigenvalues and eigenvectors of an operator
The result of operating on a vector with an operator A
~
is, in general, a di€erent
vector. But there may be some vector jvi with the property that operating with A
~
on it yields the same vector jvi multiplied by a scalar, say :
A
~
jvi ˆjvi:
This is called the eigenvalue equation for the operator A
~
, and the vector jvi is
called an eigenvector of A
~
belonging to the eigenvalue . A linear operator has, in
general, several eigenvalues and eigenvectors, which can be distinguished by a
subscript
A
~
jv k iˆ k jv k i:
The set f k g of all the eigenvalues taken together constitutes the spectrum of the
operator. The eigenvalues may be discrete, continuous, or partly discrete and
partly continuous. In general, an eigenvector belongs to only one eigenvalue. If
several linearly independent eigenvectors belong to the same eigenvalue, the eigenvalue
is said to be degenerate, and the degree of degeneracy is given by the number
of linearly independent eigenvectors.
Some special operators
Certain operators with rather special properties play very important roles in
physics. We now consider some of them below.
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