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Multipole Theory in ElectromagnetismClassical, quantum, and symmetry aspects, with applications$

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

(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)
ΔLijk=L¯ijkLijk=2ħ1ΣsωsnZsnℛe{<q¯ij>ns<m¯k>sn<qij>ns<mk>sn}=2ħ1ΣsωsnZsnℛe{<qij+Δqij>ns<mk+Δmk>sn<qij>ns<mk>sn}=2ħ1ΣsωsnZsnℛe{<qij>ns<Δmk>sn+<Δqij>ns<mk>sn+<Δqij>ns<Δmksn}=2ħ1ΣsωsnZsn{<qij>ns[i2εklmdlωsn<pk>sn]+[di<pj>nsdj<pj>ns]<mk>ns+[di<pj>ns]×[i2εklmdlωsn<pm>sn]}
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)
ΔLijk=1ħεklmdtΣs<1+ω2Zsn>ℐm{<qij><pk>sn}2ħΣsZsnℛe{di<pj>ns<mk>sn+dj<pi>ns<mk>sn}1ħεklmdlΣs<1+ω2Zsn>ℐm{[di<pj>ns+dj<pi>ns]<pm>sn}.
From the closure relatio
(F.3)
Σs<qij>ns<pk>sn=<qijpk>nn
(F.4)
Σs<pi>ns<pj>sn=<pipk>nn.

(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)
ΔLijk=12ωεklmdlakijdiGjkdjGik+12ωεklmdl(diαim)=diGjkdjGik+12ωεklmdl(ajm+diαjm+djαim).

This is (3.76).