It is often convenient to solve **eigenvalue problems** like
using matrices.
Many problems in Quantum Mechanics are solved by limiting the calculation to a finite, manageable, number
of states, then finding the linear combinations which are the energy eigenstates. The calculation
is simple in principle but large dimension matrices are difficult to work with by hand. Standard
computer utilities are readily available to help solve this problem.

Subtracting the right hand side of the equation, we have

For the product to be zero, the

* Example:
Eigenvectors of .*

The eigenvectors computed in the above example show that the x axis is not really any different
than the z axis.
The **eigenvalues** are
,
, and
, the same as for z.
The normalized **eigenvectors** of
are

These vectors, and any vectors, can be written in terms of the eigenvectors of .

We can check whether the **eigenvectors are orthogonal,** as they must be.

The others will also prove orthogonal.

Should
and
be orthogonal?

NO. They are eigenvectors of *different* hermitian operators.

The eigenvectors may be used to compute the probability or amplitude of a particular measurement.
For example, if a particle is in a angular momentum state
and the angular momentum in the
x direction is measured, the probability to measure
is

Jim Branson 2013-04-22