400 Chapter 14Partial differential equations
The radial equation
WithC 1 = 1 n
2, the radial equation (14.37) is
(14.41)
The equation is transformed into the Bessel equation (Section 13.7) by means of the
change of variablex 1 = 1 αr. Then
and (14.41) becomes the Bessel equation (13.47),
When nis a positive integer or zero, the solution of this equation is the Bessel function
J
n(x)given by (13.50), so that the solutions of the radial equation are
R
n(r) 1 = 1 J
n(αr), n 1 = 1 0, 1, 2, 3, = (14.42)
These solutions are subject to the condition that the wave function vanish at the
boundary of the box, whenr 1 = 1 a. Therefore
R
n(a) 1 = 1 J
n(αa) 1 = 10 (14.43)
and the possible values of αare determined by the zeros of the Bessel function,
examples of which are given in (13.53) of Section 13.7. If the zeros ofJ
n(x)are labelled
x
n,1, x
n,2, x
n,3, =, the allowed values of αare
, k 1 = 1 1, 2, 3, = (14.44)
and the solutions of the radial equation that satisfy the boundary condition are
R
n,k(r) 1 = 1 J
n(α
n,kr) (14.45)
By equation (14.33), the energy of the system is given byE 1 = 1 α
2A
222 m, so that the
energy is quantized, with values
, n 1 = 1 0, ±1, ±2, = (14.46)
and the corresponding total wave functions are
ψ
n,k(r, 1 θ) 1 = 1 R
|n|,k(r)Θ
n(θ) (14.47)
E
m
x
ma
nknk nk,,,==
α
22 2222
2α
nknkx
a
,,=
x
dR
dx
x
dR
dx
xnR
22222++− =() 0
dR
dr
dR
dx
dx
dr
dR
dx
dR
dr
dR
dx
==,αα=
22222r
dR
dr
r
dR
dr
rnR
22222 2++ − =()α 0