Introduction to Quantum Mechanics · Problem 4.10

Quantum Mechanics in Three Dimensions — David J. Griffiths

Physics — Quantum Mechanics · infinite_spherical_well

Problem

(a) Check that

Ar j_1(kr)

satisfies the radial equation with

V(r) = 0

and

\ell = 1

. (b) Determine graphically the allowed energies for the infinite spherical well, when

\ell = 1

. Show that for large

N

E_{N1} \approx \left(\hbar^2\pi^2/2ma^2\right)(N + 1/2)^2

. Hint: First show that

j_1(x) = 0 \Rightarrow x = \tan x

. Plot

x

and

\tan x

on the same graph, and locate the points of intersection.