Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: SEPARATION OF VARIABLES AND OTHER METHODS

21.2 A cube, made of material whose conductivity isk, has as its six faces the planes
x=±a,y=±aandz=±a, and contains no internal heat sources. Verify that
the temperature distribution

u(x, y, z, t)=Acos

πx

a

sin

πz

a

exp

(

−

2 κπ^2 t

a^2

)

obeys the appropriate diffusion equation. Across which faces is there heat flow?
What is the direction and rate of heat flow at the point (3a/ 4 ,a/ 4 ,a)attime
t=a^2 /(κπ^2 )?
21.3 The wave equation describing the transverse vibrations of a stretched membrane
under tensionTand having a uniform surface densityρis

T

(

∂^2 u

∂x^2

+

∂^2 u

∂y^2

)

=ρ

∂^2 u

∂t^2

.

Find a separable solution appropriate to a membrane stretched on a frame of

lengthaand widthb, showing that the natural angular frequencies of such a

membrane are given by

ω^2 =

π^2 T

ρ

(

n^2

a^2

+

m^2

b^2

)

,

wherenandmare any positive integers.
21.4 Schrodinger’s equation for a non-relativistic particle in a constant potential region ̈
can be taken as

−

^2

2 m

(

∂^2 u

∂x^2

+

∂^2 u

∂y^2

+

∂^2 u

∂z^2

)

=i

∂u

∂t

.

(a) Find a solution, separable in the four independent variables, that can be

written in the form of a plane wave,

ψ(x, y, z, t)=Aexp[i(k·r−ωt)].

Using the relationships associated with de Broglie (p=
k) and Einstein

(E=
ω), show that the separation constants must be such that

p^2 x+p^2 y+p^2 z=2mE.

(b) Obtain a different separable solution describing a particle confined to a box

of sidea(ψmust vanish at the walls of the box). Show that the energy of

the particle can only take the quantised values

E=

^2 π^2

2 ma^2

(n^2 x+n^2 y+n^2 z),

wherenx,nyandnzare integers.

21.5 Denoting the three terms of∇^2 in spherical polars by∇^2 r,∇^2 θ,∇^2 φin an obvious
way, evaluate∇^2 ru, etc. for the two functions given below and verify that, in each
case, although the individual terms are not necessarily zero their sum∇^2 uis zero.
Identify the corresponding values ofandm.

(a) u(r, θ, φ)=

(

Ar^2 +

B

r^3

)

3cos^2 θ− 1

2

.

(b)u(r, θ, φ)=

(

Ar+

B

r^2

)

sinθexpiφ.

21.6 Prove that the expression given in equation (21.47) for the associated Legendre
functionPm(μ) satisfies the appropriate equation, (21.45), as follows.