Spherical tensor operators



Spherical tensor operators

see the following reference – an appendix to the book:

Computational Atomic Structure: An MCHF Approach

Charlotte Froese Fischer Vanderbilt University, Tomas Brage and Per Jönsson Lund University



Angular momentum operators can not only be used to classify wavefunctions, but also operators acting on these functions. If T denotes a general operator acting on the space spanned by the eigenfunctions of the angular momentum operator [pic], then a spherical tensor operator of rank k [pic]can be defined as a set of 2k+1 operators, [pic], that fulfills the following commutation relations

 

|[pic] | | |(116) |

where

|[pic] |(117) |

As is easily verified, the renormalized spherical harmonics

|[pic] |(118) |

satisfy

|[pic] | | |(119) |

and can thus be regarded as spherical tensor operators of rank k.

Another important example is given by the vector operator [pic]. According to definition, the Cartesian components of a vector operator transform as the spatial co-ordinates x, y and z during a rotation. From these transformation properties it can be shown that the spherical components

|V(1)1 |= |[pic] |(120) |

|V(1)1 |= |Vz |  |

|V(1)-1 |= |[pic] |  |

define a spherical tensor operator [pic]of rank 1. Specifically, the angular momentum operators themselves may by written as spherical tensor operators [pic]of rank 1.

Scalar operators

According to the definition, an operator T that commutes with all components of the angular momentum operator is a scalar, or rank zero, operator. From the relations eq:tensor_eq it is easy to show that the matrix elements of T are diagonal in j and m

|[pic] |(121) |

In addition the matrix elements are independent of the m quantum number.

Coupling of tensor operators

Two spherical tensor operators [pic]and [pic]acting on the space spanned by the eigenfunctions of [pic]may be coupled to a spherical tensor operator [pic]of rank K using the Clebsch-Gordan expansion

 

|[pic] | | |(124) |

If the two tensor operators [pic]and [pic]have the same rank k1=k2=k they can be combined to form a scalar tensor, i.e. a tensor of rank zero

|[pic] | | |  |

|[pic] | | |(125) |

Traditionally, the scalar product of two tensor operators is defined by

|[pic] |(126) |

and thus

|[pic] |(127) |

When k=1 the tensor scalar product reduces to the ordinary cartesian scalar product

|[pic] | | |  |

|[pic] | | |(128) |

The Wigner-Eckart theorem

From the commutation relations eq:tensor_eq it can be shown that the matrix element [pic]is proportional to a 3j-symbol

| [pic] |(129) |

This relation is known as the Wigner-Eckart theorem. The coefficient [pic]is called the reduced matrix element of [pic]and is independent of the magnetic quantum numbers m, m' and q.

From the properties of the 3j-symbol we obtain the following selection rules for the quantum numbers that have to be fulfilled for the matrix element [pic]to be non-zero.

|m=q+m' |(130) |

and

|[pic] |(131) |

As an example we consider the matrix elements of the renormalized spherical harmonic [pic]From the Wigner-Eckart theorem we obtain

|[pic] |(132) |

Noting that

|[pic] | | |  |

|[pic] | | |(133) |

it is seen that

|[pic] |(134) |

Another important example is the matrix elements of the angular momentum operator [pic]. For the j(1)0 = jz component we have

|[pic] |(135) |

Thus,

|[pic] | | |  |

|[pic] | | |(136) |

and

| [pic] |(137) |

Matrix elements of tensor operators between coupled functions

We will now evaluate matrix elements of the coupled tensor operator [pic]eq:couptens. In most cases the tensor operators [pic]and [pic]act on different spaces, spanned by [pic]and [pic], respectively, and so we need to consider matrix elements between coupled functions

 

|[pic] | | |(138) |

In this case

 

|[pic] |

| |= |[pic] |  |

From this relation several special cases can be obtained. Setting K=0 and k1=k2=k and using the properties of the 3j- and 9j-symbols it can be shown that

 

|[pic] |

|[pic] | | |  |

Other special cases of equation eq:tens.pr are obtained by setting k2=0 and k1=0, respectively and using the properties of the 9-j symbol

 

|[pic] |

| | |[pic] |  |

 

|[pic] |

| | |[pic] |  |

Setting k=0, we obtain the reduced matrix element form of the uncoupling formulas eq:unc1 and eq:unc2 for the scalar operator.

Coupling of two angular momenta

Two angular momentum operators [pic]and [pic]acting on different parts of a system, for example two different electrons or, respectively, the space- and spin-part of the electron wave function, can be coupled to form an angular momentum operator

|[pic] |(103) |

for the total system. The operators [pic]and jz define a complete set of commuting observables. The simultaneous eigenfunctions [pic]corresponding to this set can be obtained as a linear combination

 

|[pic] | | |  |

where the coefficients [pic]are known as vector-coupling, or Clebsch-Gordan, coefficients. With the normal phase conventions the the Clebsch-Gordan coefficients are real and define a unitary matrix. Thus we have the following orthogonality relations

|[pic] |(104) |

and

|[pic] |(105) |

In addition, the Clebsch-Gordan, coefficients have the following important symmetry property

|[pic] |(106) |

Often the Clebsh-Gordan coefficients are given in terms of the more symmetric 3j-symbols

|[pic] | | |(107) |

where we have introduced the notation

|[pic] |(108) |

The general 3j-symbol

|[pic] |  |  |(109) |

has the symmetry properties

|[pic] |(110) |

and is nonzero only if

|[pic] | | |(111) |

Coupling of three- and four angular momenta

Three angular momenta can be coupled together in two different ways that are grahically represented in fig:coupling. Corresponding to these couplings there are two sets of eigenfunctions

[pic]

and

[pic]

where J12 and J23 are intermediate quantum numbers.

  

|Figure: Coupling of three angular momenta |

|[pic] |

Both these sets are complete, and thus there exists a transformation

 

|[pic] | | |  |

|[pic] | | |(112) |

Here the summation need only to be done over J12, since all other quantum numbers are common to the two sets of eigenfunctions. The coefficients are called recoupling coefficients and are often given in terms of 6-j symbols

|[pic] |

| |= |[pic] |(113) |

When dealing with four angular momenta, there is a whole set of different coupling schemes. Two possibilities are defined by the following sets of eigenfunctions

[pic]

and

[pic]

The transformation between these two sets is given by

|[pic] | | |  |

|[pic] | | |  |

|[pic] | | |(114) |

In this case the coefficients are given in terms of 9-j symbols

 

|[pic] |

| |= |[pic] |(115) |

where [pic]is the sum of all nine j quantum numbers. The 3-,6- and 9-j symbols are available from a number of tables, for example Rotenberg et. al. (1959).

Computational Atomic Structure: 

An MCHF Approach

Charlotte Froese Fischer Vanderbilt University

Tomas Brage and Per Jönsson Lund University



angular momentum



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