Jump to ContentJump to Main Navigation
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

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2017. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in OSO for personal use (for details see http://www.oxfordscholarship.com/page/privacy-policy). Subscriber: null; date: 24 February 2017

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

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

Practical Quantum Mechanics

Efstratios Manousakis

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:

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


Applying this relationship for ll1 we find that


Now, we apply the above equation for ll2


and so on until we reach ll(l1)


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


which implies that


where (p.326)


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


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