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Numerical Analysis Problem

The numerical analysis problem we must solve is to find the eigenvalues of an infinite banded matrix given by

\begin{displaymath}
h^{\rm anh}_{ij} = (i+1/2)\delta_{ij} + \epsilon g_{ij}
\end{displaymath} (3)

for $i,j = 0,1,\ldots{}$, where
\begin{displaymath}
\delta_{ij} = \left\{\begin{array}{ll}
1 & {\rm for }\ \ \ i=j \\
0 & {\rm otherwise}
\end{array}\right.
\end{displaymath} (4)

and
\begin{displaymath}
g_{ij} = \left\{\begin{array}{ll}
\frac{3}{2}(m^2 + m + \f...
...or }   i=j\pm 4\\
0 & {\rm otherwise}
\end{array}\right.
\end{displaymath} (5)

where $m = \min(i,j)$.

The first five rows and columns of the matrix $h^{\rm anh}_{ij}$ are

\begin{displaymath}
\left(
\begin{array}{ccccc}
\frac{1}{2}+ \frac{3\epsilon}{4}...
...& 0 & \frac{9}{2} + \frac{123 \epsilon}{4}
\end{array}\right)
\end{displaymath}

Obviously we do not wish to solve this problem for an infinite matrix. Fortunately, for small enough $\epsilon$, the expansion in the harmonic oscillator basis is convergent, so we may consider a truncated problem, stopping at a finite number of rows and columns. We may get a numeric measure of the effect of truncation by increasing the size of the truncated matrix and observing the change in the eigenvalues.

For a discussion of this problem and a solution using the Lanczos method of tridiagonalization, see S.S.M. Wong, Computational Methods in Physics and Engineering (Prentice Hall, Englewood Cliffs, New Jersey, 1992), pp 293ff.


next up previous
Next: About this document ... Up: anharm_osc Previous: Quantum Mechanics Background
Carleton DeTar 2002-10-18