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Eigenvalues and eigenvectors
In general, the ket is not a constant multiple of .
However, there are some special kets
known as the eigenkets of operator . These are denoted

(42) 
and have the property

(43) 
where , , are numbers called
eigenvalues. Clearly, applying to one of its
eigenkets yields the same eigenket multiplied by the associated eigenvalue.
Consider the eigenkets and eigenvalues of a Hermitian operator . These are
denoted

(44) 
where is the eigenket associated with the eigenvalue .
Three important results are readily deduced:
(i) The eigenvalues are all real numbers, and the eigenkets corresponding
to different eigenvalues are orthogonal.
Since is Hermitian, the dual equation to Eq. (44) (for the eigenvalue
) reads

(45) 
If we leftmultiply Eq. (44) by
, rightmultiply the above
equation by , and take the difference, we obtain

(46) 
Suppose that the eigenvalues and are the same. It follows from the
above that

(47) 
where we have used the fact that is not the null ket. This proves
that the eigenvalues are real numbers. Suppose that the eigenvalues
and are different. It follows that

(48) 
which demonstrates that eigenkets corresponding to different eigenvalues are
orthogonal.
(ii) The eigenvalues associated with eigenkets are the same as the eigenvalues
associated with eigenbras. An eigenbra of corresponding to an eigenvalue
is defined

(49) 
(iii) The dual of any eigenket is an eigenbra belonging to the same eigenvalue,
and conversely.
Next: Observables
Up: Fundamental concepts
Previous: The outer product
Richard Fitzpatrick
20060216