It is often convenient to solve eigenvalue problems like
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.
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
, the same as for z.
The normalized eigenvectors of
We can check whether the eigenvectors are orthogonal, as they must be.
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
Jim Branson 2013-04-22