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Eigenfunctions and Eigenvalues
An eigenfunction of an operator
is a function f such
that the application of
on f gives f again, times a constant.

(49) 
where k is a constant called the eigenvalue. It is easy to show
that if
is a linear operator with an eigenfunction g, then
any multiple of g is also an eigenfunction of .
When a system is in an eigenstate of observable A (i.e., when
the wavefunction is an eigenfunction of the operator )
then
the expectation value of A is the eigenvalue of the wavefunction.
Thus if

(50) 
then
assuming that the wavefunction is normalized to 1, as is generally
the case. In the event
that
is not or cannot be normalized (free particle, etc.)
then we may use the formula

(52) 
What if the wavefunction is a combination of eigenstates? Let us
assume that we have a wavefunction which is a linear combination of
two eigenstates of
with eigenvalues a and b.

(53) 
where
and
.
Then what is the
expectation value of A?
<A> 
= 

(54) 

= 



= 



= 



= 
a c_{a}^{2} + b c_{b}^{2} 

assuming that
and
are orthonormal (shortly we will
show that eigenvectors of Hermitian operators are orthogonal). Thus
the average value of A is a weighted average of eigenvalues, with the
weights being the squares of the coefficients of the eigenvectors in
the overall wavefunction.
Next: Hermitian Operators
Up: Operators
Previous: Linear Operators