This demonstration program pre-calculates binomial coefficients and Stirling numbers of the first kind and then invokes two routines for testing the correctness of the the reduction formula for six-term products of angular momentum ladder operators to the normal form:

J~6 = J(z)^a J(+)^b J(-)^c J(z)^d J(+)^e J(-)^f = SUM[J=1..N] A(j) J(z)^k(j) J(+)^l(j) J(-)^m(j); (1)

and the reduction formula for the normal-ordered product of rotational ladder operators and the Wigner function D1(0,eps), eps = -1,0,+1 to the normal form:

J(z)^a J(+)^b J(-)^c D1(0,eps) = SUM[i=1..N] SUM[j=-1,0,+1] C(abcd,ij) D1(0,j) J(z)^r(ij) J(+)^s(ij) J(-)^t(ij). (2)

Both tests are based on comparing the results of action of operators before and after normal ordering on rigid rotor rotational wave functions |J,K>, J=0..J(max), K=-J..+J. Routines ROTPOKET and WIGOPKET can be individually and trivially used for evaluation of matrix elements of types <J'K'| JzJ+J- |JK> or <J'K'| DJzJ+J- |JK> wherev J-operators can have any power.

During these tests very small relative errors can occur due to the limited accuracy of floating point numbers (currently Real(16)). An individual test is marked as `dubious' if the difference between the supposedly exact results is not zero, but a small number.

The user can specify the max powers a,b,c,d,e,f (MAXPOW) and J(max) (JMAX). As the derived coefficients can become very large, the recommended values for case (1) are 6/9 and for case (2) are 8/14.

LIBRARY ROUTINES:

BINOM_TABLE -- Pre-calculate binomial coefficients.

STIRL_TABLE -- Pre-calculate Stirling numbers of the first kind.

JREDUCE -- Reduction of six-term products of angular momentum ladder operators to the normal form;

ROTOPKET -- Evaluate the result of action of a rotational normal-ordered operator product on rigid rotor rotational wave functions |J,K>;

DREDUCE -- Reduction of the normal-ordered product of rotational ladder operators and the Wigner function D1(0,eps), eps = -1,0,+1 to the normal form:

WIGOPKET -- Evaluate the result of action of a composite Wigner D-function and the rotational normal-ordered operator product on wave functions |J,K>.

EXTERNAL ROUTINE: Wigner3j -- For comparison/verification of the correctness of Wigner 3-j symbols.

COMPILATION:

Intel Fortran (IFORT) with an essential Visual Studio option: Project Properties (Alt-Enter) > Configuration Properties > Fortran > Data > Default Real KIND := 16.

Alternatively, use Gfortran:

gfortran -c -O3 -fdefault-real-16 *.f90

gfortran *.o -o JDNORD