Файл: Doicu A., Wriedt T., Eremin Y.A. Light scattering by systems of particles (OS 124, Springer, 2006.pdf

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B.3 Rotations

273

Pmn −m (x) = (−1)nPmmn (−x),

P−nmm (x) = (−1)nPmmn (−x),

P−nm−m (x) = Pmmn (x),

Pmmn (x) = Pmn m(x),

where x = cos β. The orthogonality relation for the generalized spherical functions follows from the orthogonality relation for the Wigner d-function:

π	n	n	( 1)m+m
Pmm (cos β) Pmm (cos β) sin βdβ =			−	δnn .
Pmm (cos β) Pmm (cos β) sin βdβ =			2n + 1	δnn .
0			2n + 1

In practice, the generalized spherical functions can be found from the recurrence relation [162]

(n + 1)2

−

(n + 1)2

−

P n+1

(x)

= (2n + 1) [n(n + 1)x

−

mm ] P n

(x)

−(n + 1) n2 − m2

n2 − m 2Pmmn−1 (x)

with the initial values

Pmmn0−1(x) = 0,

(−j)|m−m |

P n0

(x) =

(2n0)!

2n0

(|m − m |)! (|m + m |)!

m−m

m+m

× (1 − x)

(1 + x)

and n0 = max(|m|, |m |). From (B.42) and (B.44) we see that it is su cient to

compute the generalized spherical functions for positive values of the indices m and m .

If the Euler angles (α1, β1, γ1) and (α2, β2, γ2) describe two consecutive rotations of a coordinate system and the Euler angles (α, β, γ) describes the resulting rotation, the addition theorem for the D-functions is [169, 239]

	n
Dmmn (α, β, γ) =	Dmmn (α1, β1, γ1)Dmn m (α2, β2, γ2)	(B.45)
	m =−n

and the unitarity condition read as

Dmmn (α1, β1, γ1)Dmn m (−γ1, −β1, −α1) = Dmmn (0, 0, 0) = δmm .

m =−n

(B.46)

274 B Wave Functions

If in (B.45) we set α1 = α2 = 0 and γ1 = γ2 = 0, then β = β1 + β2, and we obtain the addition theorem for the d-functions

			n
	dmmn (β) =	dmmn (β1)dmn m (β2).
		m =−n
In particular, when β2 = −β1 we derive the unitarity condition
n	(β1)dmn m (−β1) =		n
dmmn	(β1)dmn m (−β1) =		(−1)m +m dmmn (β1)dmn m (β1)
m =−n			m =−n

=dnmm (β1)dnm m (β1) = dnmm (0) = δmm .

m =−n

The product of two d-functions can be expanded in terms of the Clebsch–
Gordan coe cients Cm+m1u
		mn,m1n1
		n+n1		m +m1u
n	n1		m+m1u u	m +m1u	, (B.47)
dmm (β)dm1m1		(β) =	Cmn,m1n1 dm+m1m +m1	(β) Cm n,m1n1	, (B.47)

u=|n−n1|

and note that the Clebsch–Gordan coe cients are nonzero only when |n − n1| ≤ u ≤ n + n1. The following symmetry properties of the Clebsch–Gordan coe cients are used in our analysis [169, 239]:

m+m1u

= (−1)

n+n1+u

−m−m1u

Cmn,m1n1

C−mn,−m1n1 ,

m+m1u

= (−1)

n+n1+u

m+m1u

Cmn,m1n1

Cm1n1,mn,

m+m1u

= (−1)

2u + 1

−m1n1

m+n

Cmn,m1n1

Cmn,−m−m1u,

2n1 + 1

m1+n+u

m+m1u

= (−1)

2u + 1

Cmn,m1n1

Cm+m1u,−m1n1 .

2n + 1

(B.48)

(B.49)

(B.50)

(B.51)

To compute the Clebsch–Gordan coe cients we ﬁrst deﬁne the coe cients Smn,mu 1n1 by the relation [77]

	Cm+m1u		= gu		Su	,
	mn,m1n1			mn,m1n1	mn,m1n1
where

				(u + m + m1)!(u − m − m1)!
gu	= √
		2u + 1
		2u + 1		(n − m)!(n + m)!
mn,m1n1

×(n + n1 − u)! (u + n − n1)! (u − n + n1)! (n1 − m1)!(n1 + m1)! (n + n1 + u + 1)!

B.3 Rotations

275

The S-coe cients obey the three-term downward recurrence relation
				Su−1	= puSu	+ quSu+1
				mn,m1n1	mn,m1n1		mn,m1n1
for u = n + n1, n + n1 − 1, . . . , max(\|m + m1\|, \|n − n1\|) with
pu =			(2u + 1) {(m − m1)u(u + 1) − (m + m1) [n (n + 1) − n1 (n1 + 1)]}					,
				(u + 1) (n + n1 − u + 1) (n + n1 + u + 1)
q	u	=	−	u(u + n − n1 + 1) (u − n + n1 + 1)
	u		−	(u + 1) (n + n1 − u + 1)
			×	(u + m + m1 + 1) (u − m − m1 + 1)
			×	n + n1 + u + 1
and the starting values
					Sn+n1+1	= 0,
					mn,m1n1
					Sn+n1	= 1.
					mn,m1n1

The rotation addition theorem for vector spherical wave functions is [213]

	n
M mn1,3 (kr, θ, ϕ) =	Dmmn	(α, β, γ)M m1,3n(kr, θ1, ϕ1),						(B.52)
m =−n
	n
N mn1,3 (kr, θ, ϕ) =	Dmmn	(α, β, γ)N m1,3n(kr, θ1, ϕ1) ,						(B.53)
m =−n
and in matrix form we have
M mn1,3 (kr, θ, ϕ)	= R (α, β, γ)		M m1,3n (kr, θ1, ϕ1)				,
N 1,3 (kr, θ, ϕ)	= R (α, β, γ)		N 1,3	(kr, θ	, ϕ	)	,
mn			m n	1	1

where R is the rotation matrix. The rotation matrix has a block-diagonal structure and is given by

R (α, β, γ) =	Rmn,m n (α, β, γ)	0
R (α, β, γ) =	0	Rmn,m n (α, β, γ)	,
where
Rmn,m n (α, β, γ) = Dmmn		(α, β, γ)δnn .
The unitarity condition for the D-functions yields
R (−γ, −β, −α) = R−1 (α, β, γ)			(B.54)

276 B Wave Functions

and since

Dmn m (−γ, −β, −α) = Dmmn (α, β, γ),

we also have

R (−γ, −β, −α) = R† (α, β, γ) ,

(B.55)

where X† stands for the complex conjugate transpose of the matrix X. Properties of Wigner functions and generalized spherical functions (which

have been introduced in the quantum theory of angular momentum) are also discussed in [27, 160].

B.4 Translations

We consider two coordinate systems Oxyz and Ox1y1z1 having identical spatial orientations but di erent origins (Fig. B.4). The vectors r and r1 are the position vectors of the same ﬁeld point in the coordinate systems Oxyz and Ox1y1z1, respectively, while the vector r0 connects the origins of both coordinate systems and is given by r0 = r − r1.

In general, the addition theorem for spherical wave functions can be written as [27, 70]

∞	n

umn (kr) =	Cmn,m n (kr0) um n (kr1) .

n =0 m =−n

Integral and series representations for the translation coe cients Cmn,m n can be obtained by using the integral representations for the spherical wave

z
z1		M
r	r1
r0	O1	y1	y
O

x x1

Fig. B.4. Coordinate translation

B.4 Translations

277

functions. For regular spherical wave functions, we use the integral representation (B.1), the relation r = r0 + r1, and the spherical wave expansion of the plane wave exp(jk · r1) to obtain

∞

(kr

) u1

(kr

)

(B.56)

(kr) =

mn,m n

m n

n =0 m =−n

with

jn −n 2π π

jk(β,α) r

Cmn,m n (kr0) =

Ymn (β, α) Y−m n (β, α) e

sin β dβ dα .

2π

(B.57) Taking into account the spherical wave expansion of the plane wave exp(jk·r0) and integrating over α, we ﬁnd the series representation

Cmn,m1 n (kr0) = 2jn −n jn a (m, m | n , n, n ) u1m−m n (kr0) (B.58)

with the expansion coe cients

a (m, m | n , n, n ) = π Pn|m| (cos β) Pn|m | (cos β) Pn|m−m | (cos β) sin βdβ.

We note that the expansion coe cients a(·) are deﬁned by the spherical harmonic expansion theorem [70]

Pn|m| (cos β) Pn|m | (cos β) = a (m, −m | n , n, n ) Pn|m+m | (cos β) ,

where the summation over n is ﬁnite covering the range |n − n |, |n − n | + 2, ..., n + n . These coe cients can be identiﬁed with a product of two Wigner 3j symbols, which are associated with the coupling of two angular momentum eigenvectors. The azimuthal integration in (B.57) can be analytically performed by using the identity

k · r0 = kρ0 sin β cos (α − ϕ0) + kz0 cos β

and the standard integral

2π

ejx cos(α−ϕ0)ej(m−m )α dα = 2πjm −mJm −m(x)e−j(m −m)ϕ0 ,