Mathematical Methods for Physics and Engineering : A Comprehensive Guide

18.13 EXERCISES

Y 00 =

√

1

4 π,Y

0

1 =

√

3

4 πcosθ,

Y 1 ±^1 =∓

√

3

8 πsinθexp(±iφ),Y

0

2 =

√

5

16 π(3 cos

(^2) θ−1),

Y 2 ±^1 =∓

√

15

8 πsinθcosθexp(±iφ),Y

± 2

2 =

√

15

32 πsin

(^2) θexp(± 2 iφ),
to verify for=0, 1 ,2that
∑
m=−
|Ym(θ, φ)|^2 =

2 +1

4 π

,

and so is independent of the values ofθandφ.Thisistrueforany, but
a general proof is more involved. This result helps to reconcile intuition with
the apparently arbitrary choice of polar axis in a general quantum mechanical
system.
18.2 Express the function

f(θ, φ)=sinθ[sin^2 (θ/2) cosφ+icos^2 (θ/2) sinφ]+sin^2 (θ/2)

as a sum of spherical harmonics.
18.3 Use the generating function for the Legendre polynomialsPn(x) to show that
∫ 1

0

P 2 n+1(x)dx=(−1)n

(2n)!

22 n+1n!(n+1)!

and that, except for the casen=0,

∫ 1

0

P 2 n(x)dx=0.

18.4 Carry through the following procedure as a proof of the result

In=

∫ 1

− 1

Pn(z)Pn(z)dz=

2

2 n+1

.

(a) Square both sides of the generating-function definition of the Legendre

polynomials,

(1− 2 zh+h^2 )−^1 /^2 =

∑∞

n=0

Pn(z)hn.

(b) Express the RHS as a sum of powers ofh, obtaining expressions for the

coefficients.

(c) Integrate the RHS from−1 to 1 and use the orthogonality property of the

Legendre polynomials.

(d) Similarly integrate the LHS and expand the result in powers ofh.

(e) Compare coefficients.

18.5 The Hermite polynomialsHn(x) may be defined by

Φ(x, h)=exp(2xh−h^2 )=

∑∞

n=0

1

n!

Hn(x)hn.

Show that

∂^2 Φ

∂x^2

− 2 x

∂Φ

∂x

+2h

∂Φ

∂h

=0,