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Practical Quantum MechanicsModern Tools and Applications$

Efstratios Manousakis

Print publication date: 2015

Print ISBN-13: 9780198749349

Published to Oxford Scholarship Online: December 2015

DOI: 10.1093/acprof:oso/9780198749349.001.0001

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(p.325) Appendix B Normalization integral of spherical harmonics

(p.325) Appendix B Normalization integral of spherical harmonics

Source:
Practical Quantum Mechanics
Author(s):

Efstratios Manousakis

Publisher:
Oxford University Press

In this section of the appendix we calculate the normalization integral I2l+1 for spherical harmonics defined by Eq. 15.61.

First, by integration by parts we find a recursion relation between such integrals:

(B.1)
I2l+1=0πdθ(sin θ)2l+1=0πdθd cosθdθ(sin θ)2l =2l 0πdθ(cos θ)2(sin θ)2l1=2lI2l12lI2l+1.

To derive the last step we have used the identity cos2θ=1sin2θ. Therefore, we find that

(B.2)
(2l+1)I2l+1=2lI2l1.

Applying this relationship for ll1 we find that

(B.3)
(2l1)I2l1=2(l1)I2l3.

Now, we apply the above equation for ll2

(B.4)
(2l3)I2l3=2(l2)I2l5.

and so on until we reach ll(l1)

(B.5)
3I3=2I1.

Now we multiply these l equations together to find (after the cancellation)

(B.6)
(2l+1)(2l1)(2l3).3I2l+1=[2l][2(l1)]2I1,

which implies that

(B.7)
I2l+1=[2ll!]2I1(2l+1)!,

where (p.326)

(B.8)
I1=0πdθsinθ=2.

Putting the various pieces that we found together, we conclude that the normalization constant C is given by

(B.9)
C=(1)l2ll!(2l+1)!4π,

where we have added by hand the phase factor (1)l following the traditional convention.