On the phase space representation for spin systems

PHYSICS LETTERS

Volume 4 1A, number 5

ON THE PHASE SPACE REPRESENTATION

23 October 1972

FOR SPIN SYSTEMS

J. KUTZNER Fachbereich

Physik, Freie Universitat Berlin, Germany

Received 28 August 1972 Some results of a phase space representation for spin systems are presented, starting from the definition of two mapping operators. One of them is the projection operator onto coherent spin states.

During the last few years systems [l-4]. In this letter coherent spin states [S] . To we use a similar notation. Coherent spin states [5]

lz> =

there has been much interest in deriving phase space methods to describe physical we present some results of a phase space representation for spin systems based on facilitate comparison of our results to the mapping theorems for Bose-operators [l] are defined by

(l+;,2)s$o Qzyznin).

where z is a c-number, S(S+ 1) is the eigenvalue of the operator S2 and In> is the eigenstate of S, such that Szln) = (S- n)ln>, 0 < n < 2S. We now introduce two mapping operators A$)( z, z*) where t = + 1. They are defined by the formulae ( 1)(z, z*) = ]z)(zl , AsAg”(z,

z*)

=

2sc-l) (2s+

5

InHkl BEZEL-”

l)! n,k=,,

X

m=O r=O

Using these operators one can derive the following theorem: Theorem I. Every operator G in spin space can be expanded in the following manner: G=2S+l d2z 71 s (1ttz12)2

z*) A$)(z I z*) 9

Fg)(z

F$z,

z*) = Tr [GA’,-“(z 9z*)] 9

’

where the integration extends over the whole complex z-plane. Theorem 1 may be regarded as defining a correspondence G * F(z, z*) between the operators and c-number functions of the class F(z, z*) =

l

2s

c

cnkznz*k .

(1 tlz12)2s n,k=O It can be shown that this correspondence is one-to-one. Using theorem 1 it is easy to verify the following theorem: Theorem 2. The trace of the product of two operators G, and G2 in spin space is given by

475

Volume

41A, number

Tr [G, Gzl =2s

PHYSICS

5

1972

d2z2 2 F~)(z, z*)$-')(z. z*),

s 77

23 October

LETTERS

(1+lzl )

where $)(z, z*) and F$-‘)(z, z*) are the c-number equivalents of the operators G, and G,. If G2 is the density operator p of the spin system, then theorem 2 expresses the expectation value of G, , in the state represented by p as an average over phase space. To transcribe the equations of motion for operators into equivalent equations for the c-number functions. we need the following theorem: Theorem 3. The c-number equivalent F{$l’(z, z*) of the p roduct of two operators G, and G2 is given by @(z,

z*) = Dt2$+1’(z

1, q> 4+1’(z2,

qz,

=z* =z >

where $+‘)(z, z*) and $+‘)( z, z*) are the c-number operator defined by 1 D12=(,t,zl,2)2S

D;:‘D\‘:

..

. D;;n(l

a$+‘)

is the differential

a2 1

2

Using this theorem for the case of S =3, the normalized ing Fokker-Planck equation

(1 t ]z]2)2+@(+1)

of the two operators and D,,

+ Iz,I~)~~(~ + Iz~I~)~‘,

t+l,zl,;)azaz” k2

equivalents

distribution

function

a(+‘)(~, z*) must obey the follow-

1+C.C.,

where F$‘)(z, z*) is the c-number equivalent of the Hamiltonian. A more detailed discussion of the phase space representation described here will be published elsewhere References [l] G.S. Agarwal and E. Wolf, Phys. Lett. 268 (1968) 485; Phys. Rev. Lett. 21 (1968) [2] [3] [4] (S]

476

(1970) 2187. K.E. Cahilland R.J. Glauber, Phys. Rev. 177 (1969) 1857, H. Haken, Z.Phys. 219(1969)411. D.M. Kaplan, Transp. Theory and Stat. Phys. 1 (1971) 81. J.M. Radcliffe, J. Phys. A4 (1971) 313.

177 (1969)

1882.

180; Phys. Rev. D2 (1970)

2161,

D2