
Numerical Techniques:
Differential Equations -
Eigenvalue Problems
Dr. Cem Özdo
˘
gan
LOGIK
Differential Equations -
Eigenvalue Problems
Eigenvalue Problems
Standing Waves on a
String
Numerical Solutions of
Schrödinger Equation
Hydrogen Atom
8.9
Standing Waves on a String II
•
Instead, an estimated value for the k eigenvalue is
taken and a solution search is initiated.
•
Searching continues by increasing the value of k until the
boundary condition (here y(1) = 0) at the other end is
satisfied.
•
For example, the solution at the other boundary is y
1k
(1)
for a given value of k.
•
Accordingly, the next step is find the root of the following
equation:
F (k) = y
1k
(1) − y(1) = 0
When we encountered an eigenvalue k, then F (k) will
change sign as indicating the root.
•
(Example py-file: The program to find the 5 smallest of
the k eigenvalues in a string:
standingwawes.py)
•
The program can find the eigenvalues
k = nπ/L = π, 2π, 3π, . . . on a str ing of length L=1 m.
•
However, the error margin is to be increased by increasing
eigenvalues (see k
n
/π values).