Next:
Numerical Solutions of Schrödinger
Up:
Eigenvalue Problems
Previous:
Eigenvalue Problems
Contents
Standing Waves on a String
The wave equation and boundary conditions in a string of length
with both fixed ends are as follows:
Firstly, transform this quadratic equation into a system of linear (first degree) equations:
with these values (
), the system of equations to be solved: (
Now, we have a set of equations.
)
and the boundary conditions are:
0
0
Here, the trial-and-error approach differs from the previously discussed boundary value problem.
Different estimates for
values do not make it zero at the other boundary.
Instead, an
estimated value for the
eigenvalue
is taken and a solution search is initiated.
Searching continues by increasing the value of
until the boundary condition (here
) at the other end is satisfied.
For example, the solution at the other boundary is
for a given value of
.
Accordingly, the next step is find the root of the following equation:
When we encountered an eigenvalue
, then
will change sign as indicating the root.
(
Example py-file:
The program to find the 5 smallest of the
eigenvalues in a string:
standingwaves.py
)
The program can find the eigenvalues
on a string of length L=1 m.
However, the error margin is to be increased by increasing eigenvalues (see
values).
Figure 5.13:
Solution for the Eigenvalue Problem for the ODE:
.
Next:
Numerical Solutions of Schrödinger
Up:
Eigenvalue Problems
Previous:
Eigenvalue Problems
Contents