Chapter 7. The Eigenvalue Problem

7. The Eigenvalue Problem, December 17, 2009 2
aspects of the eigenvalue problem for matrices and do not discuss numerical
methods for solving it. When we need a numerical solution, we use
Mathematica. For demanding calculations, we must use Fortran, C, orC++,
since Mathematica is less efficient. These computer languages have libraries
of programs designed to give you the eigenvalues and the eigenvectors when
you give them the matrix.
§ 2 Some general considerations. It turns out that the eigenvalue problem
A|ψ = λ|ψ has many solutions and not all of them are physically meaningful.
We are interested in this eigenvalue problem because, if it arises
from an observable, the eigenvalues are the spectrum of the observable and
some of the eigenfunctions are the pure states of the observable. In addition,
the operator representing the observable is Hermitian, because its spectrum,
which consists of all the values the observable can take, just contain only real
numbers.
Before we try to solve the eigenvalue problem, we need to decide how
we recognize which eigenstates correspond to pure states: these are the only
eigenstates of interest to us.
In Chapter 2, we examined the definition of the probability and concluded
that kets |a i and |a j representing pure states must satisfy the orthonormalization
condition
a i | a j = δ ij (4)
§ 3 Normalization. Let us assume that we are interested in solving the
eigenvalue equation
Â|ψ = λ|ψ (5)
It is easy to see that if α is a complex number and |ψ satisfies Eq. 5 then
|η = α|ψ satisfies the same equation. For every given eigenvalue we have
as many eigenstates as complex numbers. The one that is a pure state must
be normalized. We can therefore determine the value of α by requiring that
|η satisfies
η | η =1 (6)
Using Eq. 5 in Eq. 6 (and the properties of the scalar product) leads to
α ∗ αψ | ψ =1 (7)
or
α ∗ α = 1
ψ | ψ
(8)