May 4, 1976 - View the table of contents for this issue, or go to the journal ... where Kk is the square of a Clebsch-Gordan coefficient (Gilmore et a1 1975) and.
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Q and P representatives for spherical tensors
This article has been downloaded from IOPscience. Please scroll down to see the full text article. 1976 J. Phys. A: Math. Gen. 9 L65 (http://iopscience.iop.org/0305-4470/9/7/001) View the table of contents for this issue, or go to the journal homepage for more
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J . Phys. A: Math. Gen., Vol. 9 , No. 7. 1Y7h. Printed in Great Britain. @ 1976
LETl’ER TO THE EDITOR
Q and P representatives for spherical tensors R Gilmore? Institut de Physique Thtorique, UniversitC Catholique de Louvain, B-1348 Louvain-laNeuve, Belgique Received 4 May 1976 Abstract. The Q and Prepresentatives for the irreducible spherical tensor operators SkJ, are proportional to the corresponding spherical harmonics Y36,&). In the coherent state basis associated with the ( 2 j + 1)-dimensional unitary irreducible representation of SU(2), theproportionalityfactorsare (2j)!/(2j-L)!2L and ( 2 j + 1 +L)!/(2j+ 1)!2L,respectively.
Let 8be a symmetrized polynomial function of the angular momentum operators J for in the a single spin. Then 8 has a unique 0 representative (Glauber 1963) 8%f((n) (2j 1)-dimensional space which carries the irreducible representation D’[SU(2)]:
+
f(W=(film.
(IQ)
Here R = (O,c$) represents a point on the surface of the Bloch sphere, with 8 measured from the north pole, and /a)is an atomic coherent state (Arecchi et a1 1972):
The operator
8 also possesses
a non-unique (Arecchi et a1 1972) P representative
$5 F(R) determined by
The Q and P representatives of 2j+1
f ( W =- 4T
8 are related by the convolution
F(Rf)((R’lR)I2dp(RR.
(3)
This convolution annihilates the non-unique part of the P representative, and provides a one-to-one correspondence between f(R) and the unique part of F(R). The relation betweenf(R) and F(R) is made explicit by expanding these functions in terms of spherical harmonics as follows:
t Permanent address: Physics Department, University of South Florida, Tampa, Florida 33620, USA.
L65
L66
Letter to the Editor
If the Fourier coefficients of F(R) and the kernel
2j+ 1 K(R) = -p 4T
= 0,4 = oln)1*
are defined similarly, then (Gilmore 1972) f h=F h K k
(5)
where Kk is the square of a Clebsch-Gordan coefficient (Gilmore et a1 1975) and
(2j + 1)!(2j)! KLO-(2j+l+L)!(2j-L)!' The Q and P representatives of
3 are proportional if either is homogeneous of degree
L.
The generating function (Arecchi er a1 1972) foi moments of angular momentum operators can be used to compute the 0 representative for (J+)":
The operator (J+)" is proportional to the spherical tensor 9i(J)and (ei'sin 8)" is proportional to the spherical harmonic Y;(8,4)with the same proportionality factor. and Y a 8 , 4) transform identically under SU(2), we have a very simple Since expression for the Q representative of 9 a J ) :
9aJ)
The unique part of the P representative follows directly from (5) and (6)
For example, for L = 1 the Q representatives of J*,J, are j e*'" sin 8, j cos 8, while their P representatives are ( j + 1) e*i' sin 8, ( j + 1)cos 8. For L = 2, the Q and Prepresenta) as previously tives of 35:-J' are j ( j - $ ) (3cos'O-1) and ( j + l ) ( j + $(3cos28-l), given by Lieb (1973).
References Arecchi F T, Courtens E, Gilmore R and Thomas H 1972 Phys. Rev. A 6 221 1-39 Gilmore R 1972 Ann. Phys. NY 74 391-463 Gilmore R, Bowden C M and Narducci L M 1975 Phys. Rev. A 12 1019-31 Glauber R J 1963 Phys. Rev. 131 2766-88 Lieb E H 1973 Commun. Math. Phys. 31 327-40
