Chapter 7. The Eigenvalue Problem

7. The Eigenvalue Problem, December 17, 2009 4
When λ i = λ j ,wemusthave
η i | η j =0,
which is what we wanted to prove.
So, when you solve an eigenvalue equation for an observable, the eigenstates
corresponding to different eigenvalues must be orthogonal. If they are
not, there is something wrong with the program calculating the eigenvectors.
What happens if some eigenvalues are equal to each other Physicists
call such eigenvalues ‘degenerate’, mathematicians call them ‘multiple’. If |η i
and |η j belong to the same eigenvalue (i.e. if Â|η i = λ|η i and Â|η j = λ|η j )
then |η i and |η j are under no obligation to be orthogonal to each other (only
to, according to the fact in the previous §, all eigenvectors belonging to eigenvalues
that differ from λ). Most numerical procedures for solving eigenvalue
problems will provide non-orthogonal degenerate eigenvectors. These can be
orthogonalized with the Gram-Schmidt procedure and then normalized. The
resulting eigenvectors are now pure states. You will see how this is done in
an example given later in this chapter.
It is important to keep in mind this distinction between eigenstates and
pure states. All pure states are eigenstates but not all eigenstates are pure
states.
§ 5 The eigenvalue problem in matrix representation: a review. Let |a 1 ,
|a 2 , ..., |a n , ...be a complete orthonormal basis set. This set need not be
a set of pure states of an observable; we can use any complete basis set that
a mathematician can construct. Orthonormality means that
Because the set is complete, we have
a i | a j = δ ij for all i, j (16)
Î
N
|a n a n | (17)
n=1
Eq. 17 ignores the continuous spectrum and truncates the infinite sum in the
completeness relation.
We can act with each a m |, m =1, 2,...,N, on the eigenvalue equation
to turn it into the following N equations:
a m | Â | ψ = λa m | ψ, m =1, 2,...,N (18)