The antisymmetric (odd-parity) bound states of a finite square well of strength…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Wave mechanics in one dimension

Scientific Computing for Physicists, Ch The numerical toolkit closed: backward error and the audit of a library routine

Problem

The antisymmetric (odd-parity) bound states of a finite square well of strength parameter

z_0 = 8

satisfy

-\cot z = \sqrt{\frac{z_0^2}{z^2} - 1}, \qquad 0 < z < z_0.

The right-hand side is real and positive only for

0 < z < z_0

, and

\cot z

has poles at

z = 0, \pi, 2\pi, \ldots

and passes through zero at

z = \pi/2, 3\pi/2, 5\pi/2, \ldots

. ((a)) Give the three brackets of the form

(k\pi + \pi/2,\ (k+1)\pi)

, restricted to

z < 8

, inside which the left-hand side rises from

0

+\infty

and so must cross the right-hand side exactly once (the right-hand side is finite and continuous there, falling from a finite value to

0

z \to 8^-

in the last bracket). ((b)) Using a safeguarded root finder (Brent's method) in each bracket, find the three roots to three decimal places.