Mathematical Methods for Physics and Engineering : A Comprehensive Guide

CALCULUS OF VARIATIONS

22.23 For the boundary conditions given below, obtain a functional Λ(y) whose sta-
tionary values give the eigenvalues of the equation

(1 +x)

d^2 y

dx^2

+(2+x)

dy

dx

+λy=0,y(0) = 0,y′(2) = 0.

Derive an approximation to the lowest eigenvalueλ 0 using the trial function

y(x)=xe−x/^2. For what value(s) ofγwould

y(x)=xe−x/^2 +βsinγx

be a suitable trial function for attempting to obtain an improved estimate ofλ 0?
22.24 This is an alternative approach to the example in section 22.8. Using the notation
of that section, the expectation value of the energy of the state∫ ψis given by
ψ∗Hψ dv. Denote the eigenfunctions ofHbyψi,sothatHψi=Eiψi, and, since
His self-adjoint (Hermitian),

∫

ψ∗jψidv=δij.

(a) By writing any functionψas

∑

cjψjand following an argument similar to

that in section 22.7, show that

E=

∫

∫ψ∗Hψ dv

ψ∗ψdv

≥E 0 ,

the energy of the lowest state. This is the Rayleigh–Ritz principle.

(b) Using the same trial function as in section 22.8,ψ=exp(−αx^2 ), show that

the same result is obtained.

22.25 This is an extension to section 22.8 and the previous question. With the ground-
state (i.e. the lowest-energy) wavefunction as exp(−αx^2 ), take as a trial function
the orthogonal wave functionx^2 n+1exp(−αx^2 ), using the integernas a variable
parameter. Use either Sturm–Liouville theory or the Rayleigh–Ritz principle to
show that the energy of the second lowest state of a quantum harmonic oscillator
is≤ 3
ω/2.
22.26 The HamiltonianHfor the hydrogen atom is

−

^2

2 m

∇^2 −

q^2

4 π 0 r

.

For a spherically symmetric state, as may be assumed for the ground state, the

only relevant part of∇^2 is that involving differentiation with respect tor.

(a) Define the integralsJnby

Jn=

∫∞

0

rne−^2 βrdr

and show that, for a trial wavefunction of the form exp(∫ −βr)withβ>0,

ψ∗Hψ dvand

∫

ψ∗ψdv(see exercise 22.24(a)) can be expressed asaJ 1 −bJ 2

andcJ 2 respectively, wherea,bandcare factors which you should determine.

(b) Show that the estimate ofEis minimised whenβ=mq^2 /(4π 0 
^2 ).

(c) Hence find an upper limit for the ground-state energy of the hydrogen atom.

In fact, exp(−βr) is the correct form for the wavefunction and the limit gives

the actual value.

22.27 The upper and lower surfaces of a film of liquid, which has surface energy per
unit area (surface tension)γand densityρ, have equationsz=p(x)andz=q(x),
respectively. The film has a given volumeV(per unit depth in they-direction)
and lies in the region−L<x<L,withp(0) =q(0) =p(L)=q(L)=0.The