## 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

Source:
Practical Quantum Mechanics
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)
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To derive the last step we have used the identity $cos2θ=1−sin2θ$. Therefore, we find that

(B.2)
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Applying this relationship for $l→l−1$ we find that

(B.3)
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Now, we apply the above equation for $l→l−2$

(B.4)
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and so on until we reach $l→l−(l−1)$

(B.5)
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Now we multiply these l equations together to find (after the cancellation)

(B.6)
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which implies that

(B.7)
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where (p.326)

(B.8)
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Putting the various pieces that we found together, we conclude that the normalization constant C is given by

(B.9)
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where we have added by hand the phase factor $(−1)l$ following the traditional convention.