14.8 Exercises 415
(i)Show that the equation is separable in spherical polar coordinates, with the same
angular wave functions, the spherical harmonics (14.68), as for the hydrogen atom.
(ii)Show that the radial equation reduces to the Bessel equation (13.60) for spherical
Bessel functionsj
l(x)where, as in Section 14.5 for the particle in a circular box,
.(iii)Use the boundary condition to find an expression for the quantized
energy in terms of the zeros of the Bessel functions. (iv)Find the wave function and
energy of the ground state.
Section 14.7
16.Find the solution of the wave equation for the vibrating string that satisfies the initial
conditions
y(x, 1 0) 1 = 131 sin 1 πx 2 l,(∂y 2 ∂t)
t= 01 = 1 0.
17.A homogeneous thin bar of length land constant cross-section is perfectly insulated
along its length with the ends kept at constant temperatureT 1 = 10 (on some temperature
scale). The temperature profile of the bar is a functionT(x, t)of position x( 01 ≤ 1 x 1 ≤ 1 l) and
of time t, and satisfies the heat-conduction (diffusion) equation where Dis
the thermal diffusivity of the material. The boundary conditions areT(0, 1 t) 1 = 1 T(l, 1 t) 1 = 10.
Find the solution of the equation for initial temperature profileT(x, 1 0) 1 = 131 sin 1 πx 2 l.
- (i) Find the general solution of the Laplace equation in the rectangle
01 ≤ 1 x 1 ≤ 1 a, 01 ≤ 1 y 1 ≤ 1 bsubject to the boundary conditions
u(0,y) 1 = 1 0, u(a, y) 1 = 10
u(x, 0) 1 = 1 0, u(x, b) 1 = 1 f(x)
wheref(x)is an arbitrary function of x. (ii)Find the particular solution for
.
fxx
a
() sin=
3 π
∂
∂
∂
∂
=
2
2
2
2
0
u
x
u
y
∂
∂
=
∂
∂
T
t
D
T
x
2
2
xmEr= 2