The algebra of tensor operators 3 6 1 Introduction

As already discussed, the rotational properties of an operator are determined by its commutator with the total angular momentum operator J_. For the case of a scalar operator A, which is by definition invariant under all rotations, the commutator is given by CJ_,A] • 0. A vector operator V can be expressed in terms of its cartesian components V., Vv and V,. However, since most systems of interest possess spherical symmetry, it is more convenient to work with the spherical components defined by

V, = - (V. + iVy)<2)-‘/a

V-i » <V„ - iVy)(2)-‘/z

Vo * V. 3.29

The commutation relations which define the rotational properties of a vector operator are given by

CJt,V,l » (2 - q(q ± 1))*'2 VQ± ,

« qV„ q » 1, 0, -1

3.30

A vector operator transforms under rotations according to the irreducible representation D'*’ of SO(3>. This follows from equation 2.2S where the spherical components of the vector V correspond to the

1 • 1 (m ■ -1, 0, 1) components of the spherical harmonics. When the extension is made to the rotation-inversion group 0(3) it becomes necessary to distinguish between vector operators which transform differently under the inversion operation. A polar vector operator transforms according to D “ ’"’, changing sign under inversion, while an axial vector operator transforms according to D'1'**. The electric dipole moment operator is an example of a polar vector operator and the orbital and spin angular momenta operators are examples of axial vector operators.

The importance of choosing the components of an operator to simplify its representations in the scheme of use becomes more important when considering tensor operators in general. The more usual cartesian tensor operators generally appear in a reducible fora when dealing with

rotationally invariant systems. For example, consider two vector operators U and V transforming separately under rotations according to equation 2.17 with 1 » 1. The second rank tensor operator UV. is the direct product of U and V, a general component of which transforms under rotations according to

R<#,8,T>U,Vj = 2 ^„V., Di.'’, Dl'j <s,6,y) 3.31

The tensor UV consists of nine coeponents end these generate the direct product representation D '‘’ ® D 11’ of S0(3). According to equation 2.30 this is reducible into the following irreducible representations

D 11’ ® D'1’ « D ,0> ® D *‘ * ® D <2> 3.32

1

Therefore it is possible to decompose UV into three irreducible tensor operators T <0> , T “ * and T ,2> which transform as spherical harmonics of degree zero, one and two respectively.

In general an irreducible tensor operator (or spherical tensor operator) L <H> of degree k, where k « 0, 1/2, 1, 3/2, ...., is defined as an operator having 2k+l components T^k>, with q ■ -k, -k + 1, .... . k, called the spherical or standard components which transform under rotations according to

k

R<«,8,Y>T,<k’ * 2 Tik’ DA,"’ (0,6,1) 3.33

m • — k

Only integral values of k occur in physical problems and therefore the components of an Irreducible tensor operator have the same commutation relations with the components of the total angular momentum operator J as the spherical harmonics

tJ±,T;k>] « (k (k + 1) - q(q±l))»'a T^,k* CJ. ,T^*k* 1 ■ q TJk>

3.34

These relations provided the starting point for the derivation of irreducible tensor operator algebra by Racah 121.

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3.6.2 The Wigner-Eckart theorem

The importance of irreducible tenaor operator* becomes apparent when considering the matrix elements of the components of such operators. The dependence of the matrix element of T^k> on m in the lj*> scheme is given by the Wigner-Eckart theorem [161

where <jll T <k> II j1 > is known as the reduced matrix element. The dependence on m is contained totally within the 3-j symbol and phase factor which are both easily calculable. Consideration of the 3-j symbol alone indicates that the matrix element is zero unless a * m'+ q.

3.6.3 Matrix elements of mixed tensor operators

The scalar product of two tensor operators T.<k> and U ,k> is defined by

The symbol a stands for any additional quantum numbers necessary to define the state uniquely.

Consider a system containing two angular momenta ji and j*. If T ,k> and U <k> are of such a form that they act on different parts of the system with basis functions ljiai> and Ij2aa > respectively, then the matrix elements between the basis functions ljtj>JM> of the combined system are

u

T<k *,U<k> . i (-l)i T4k>U4k> _3.36

H

with its matrix element given by C161

<«jml <T,k,.U<k >) I a 'j 'm '>

- I (-11 II T <k> II*" J* II U ,k* ll«'J->

«'• y

x 6<j,j -16<m,m')/[j] 3.37

3.38

<«j>jpJHI (T<k >.y«k > >

')6<M,M')

X Z < « j i l | T ,k> •><«" j2 'II U 1 k ’ II « ' j 3 ' >

Two special cases arise 'from a consideration of equation 3.38:

(i) For an operator T <k> ’ acting on part 1 only and which commutes with J2 it can be shown that

(ii) For an operator U <k» ’ acting on part 2 only and which commutes with ji it can be shown that

< *Jij 2J I U'ki ’ I « ' ji ' j* ' J ' >

The problem of calculating matrix elements of tensor operators ultimately becomes one of calculating the reduced matrix elements. The methods that are employed to obtain these reduced matrix elements will be discussed later in relation to some specific examples.

3.7 The application of tensor operator techniques in atomic spectroscopy