The even-parity bound states of a finite square well of strength satisfy z = z_0^2…

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 even-parity bound states of a finite square well of strength

z_0=5

satisfy

\tan z = \sqrt{\dfrac{z_0^2}{z^2}-1},\qquad 0<z<z_0,

with one root, if it exists, in each bracket

[k\pi,\,k\pi+\pi/2)

for

k=0,1,2,\dots

, since

\tan z

runs from

0

+\infty

on each such bracket while the right-hand side is finite and decreasing. ((a)) State how many such brackets lie inside

0<z<5

, and hence how many even-parity roots to expect. ((b)) Using Brent's method on each bracket, find every root to four decimal places.