## Roger E. Raab and Owen L. de Lange

Print publication date: 2004

Print ISBN-13: 9780198567271

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198567271.001.0001

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# (p.218) APPENDIX F ORIGIN DEPENDENCE OF A POLARIZABILITY TENSOR

Source:
Multipole Theory in Electromagnetism
Publisher:
Oxford University Press

We derive here the origin dependence of the polarizability Lijk as an example of how to obtain those in (3.64)–(3.79). Using the notation in (3.55) for a quantity relative to origin Ō, and from the expression for Lijk in (2.127), we have

(F.1)
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in which we used (3.57), (3.59), the canonical commutation relation, and <q (α)>ns=q (α)<n|s> = 0 for orthogonal kets. Then from (2.75) and (2.131), (F.1) becomes
(F.2)
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From the closure relatio
(F.3)
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(F.4)
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(p.219) The product operators in the expectation values in (F.3) and (F.4) are Hermitian because their factors are commuting Hermitian operators. Thus (F.3) and (F.4) are real. Using this, and (2.116), (2.118), and (2.119), we can express (F.2) as

(F.5)
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This is (3.76).