In recent posts I have been doing some numerical work based on a series of papers on the ‘Exact Computation and Asymptotic Approximations of 6j Symbols’.
and a review of their use in spin networks
In this posts I’ll be looking at a paper which looks and their semiclassical properties.
In this paper the authors describe a direct method for the exact computation of 3nj symbols from the defining series. The properties and asymptotic formulas of the 6j symbols or Racah coefficients are discussed. Relationships with families of hypergeometric orthogonal polynomials are presented and the asymptotic behaviour is studied both from the viewpoints of the basic geometrical significance and as a source of accurate approximation formulas.
Numerical aspects are specifically investigated in detail, regarding the relationship between functions of discrete and of continuous variables, exhibiting the transition in the limit of large angular momenta toward both Wigner’s reduced rotation matrices or Jacobi
polynomials and harmonic oscillators or Hermite polynomials.
This paper contains a presentation of properties, useful formulas, and illustrations for a basic mathematical tool, the 6j-symbol, also known in quantum mechanical angular momentum theory as Racah coefficient and the building block of 3nj-symbols and spin networks.
There are basic connections among the 6j symbols of angular momentum theory with both the theory of superposition coefficients of hyperspherical harmonics and the theory of discrete orthogonal polynomials. There is a connection between the Askey
scheme of orthogonal polynomials and the tools of angular momentum theory such as 6j, 3j, rotation d-matrices. Going down the Askey scheme corresponds in quantum mechanics to the semiclassical limit, while going up provides discretization algorithms for quantum mechanical calculations for example the hyperquantization algorithm.
Explicit expressions for the 6j coefficients can be written according to the series expressions of the Racah type, or as generalized hypergeometric series, or in connection with the so-called Racah polynomials. Orthogonal polynomials of a discrete variable are important tools of numerical analysis for the representation of functions on grids.
Computation of Mathematical Functions and Angular Momentum
We can calculate the 3nj-symbols and Wigner d functions
by directly summing the defining series using multiprecision arithmetic. The multiple precision arithmetic allows convenient calculation of hypergeometric functions, pFq, of small
and large argument by their series definition.
d functions with 2F1:
Clebsch–Gordan coefficients and 3j-symbols with 3F2
6j-symbols with 4F3
Semiclassical Limits and Schulten–Gordon Approach
In the Askey scheme for orthogonal polynomials
of hyperspherical family and its counterpart for the
tools of angular momentum theory shown arrows pointing out downwards are asymptotic connections.
A basic role is played by the relationship which relates
three 6j symbols with an argument differing by one
In this formula, one can introduce a quantity R such
that either and m1=j23−j, m2=j3−j23, m3 = j −j3.
So that when R goes to infinity, we obtain a three terms
recurrence relationship for the 3j symbols as a function of j1.
when R goes to infinity, we obtain another three term recurrence relationship for the 3j symbols as function of m2.
Wigner Reduced Rotation Matrix Elements as Limits of 6j Symbols
The post Numerical work with sagemath 23: Wigner Reduced Rotation Matrix Elements as Limits of 6j Symbols deals with this section.
has an interesting geometrical interpretation, based on the vector model visualization of quantum angular momentum coupling by
the triangle of vectors that we would draw in classical mechanics.
In view of this when we consider 6j properties as correlated to those of the tetrahedron,
we use the substitution
which greatly improves all asymptotic formulas down to surprisingly low values of the entries.
The square of the area of each triangular face is given by the formula:
where a, b, c are the sides of the face. Similarly, the square of the volume of an irregular tetrahedron,can be written as the Cayley-Menger determinant:
When the values of J1, J2, J12, J3, and J are fixed, the maximum value for the volume as a function of J23 is given at:
The corresponding volume is
Therefore, the two values of J23 for which the volume is zero are:
They mark the boundaries between classical and nonclassical regions.
Introduce a parameter λ indicating the growth of the angular momentum. Consider the Schulten-Gordon relationships:
For λ =1
where F(a, b, c) is area of abc triangle from
The coefficients in
are connected to the geometry of the tetrahedron:
In terms of the finite difference operator:
From these formulas, and from that of the volume,
we have that
- V=0 implies cosθ1 =±1 and establishes the classical domain between J1min and J1max
- F(J1, J2, J3)=0 or F(J1, L2, L3)=0 establish the definition limits j1min and j1max
For a Schrodinger type equation
its discrete analog in a grid having one as a step
and so comparing this with
allowing the identification
Plots corresponding to the three cases are given below:
On the closed loop, we can enforce Bohr–Sommerfeld phase space
where the role of q is played by j12. This is an eigenvalue equation for allowed L1. The illustration of these formulas is below:
Illustration of phase space for semiclassical quantization with j1=92, j2 =47, j3 =80, j =121 for j12 =139 (dots), j12 =134 (triangles) ,j12 =129 (plus signs)
Values for the integral for different
number of nodes n.
- j12 = 139 ⇒ n = 0,
- j12 =134 ⇒n =5,
- j12 =129 ⇒n =10.
The values of the integrals are connected by the line while the dots are evaluated with p given by – see table below:
The 6j symbol and the oscillator wavefuncions
The Askey scheme and its counterpart point out at the connection in the angular momentum case between the top, the 6j symbol, and
the bottom, the harmonic oscillator. The geometrical insight of the Ponzano–Regge theory and its implementation in the Schulten–Gordon asymptotic formulas consistently lead to the expected Airy function behaviour astride of the transitions between classical and nonclassical regions of the ranges of elongations of the oscillator.
There is a connection between the harmonic oscillator wavefunctions and 6j coefficients for large angular momentum arguments. Ponzano and Regge have approximated 6j-coefficients with sine and cosine functions as well as Airy functions. The formulas and extensions by Schulten and Gordon are excellent for the uniform semi-classical approximation for 6j-coefficients.
These semi-classical approximations rely on the volume, surface areas, and angles that characterize the tetrahedron that corresponds to each 6j-coefficient that is required. A simple method connects a large set of special 6j-coefficients to harmonic oscillator wavefunctions by using only three parameters that are uniquely given from a simple algebraic analysis of the volumes of some tetrahedra related to the desired set of 6j-coefficients.
Consider the approximation of the 6j-coefficients:
as a function of j12 and j23. These discrete functions are orthonormal with relations:
Compare the 6j-coefficients with one dimensional quantum mechanical harmonic oscillator wavefunctions which belong to
an orthogonal set. Consider weighted 6j-coefficients:
Draw the connection with the harmonic oscillator wave-functions by noting that for given j1, j2, j12, j3, j there will be a value of j23max that will yield the maximum volume for the corresponding tetrahedron. The volume is given by
The maximum V² is obtained by finding the appropriate value of j23
max that gives d (V²)/dj23=0. Consider the particular 6j-symbol:
where j2 = j1 and j= j3. This symbol has n nodes as j23 is varied. Set up the approximation using harmonic oscillator wave functions:
Looking at the 6j-symbol:
which gives the values for j23max and α. The figures
show n = 0, n =2, and n =7.
Representation of the 6j symbols by the harmonic oscillator wavefuntion for the case j1=1000, j2=1000, j12 =200, j3 =100,
Representation of the 6j symbols by the harmonic oscillator wavefunction for the case j1 = 1000, j2 = 1000, j12 =198, j3 = 100, j =100.
Representation of the 6j symbols by the harmonic oscillator wavefunction for the case j1 = 1000, j2 =1000, j12 =193, j3 = 100, j =100.
Representation of the 6j symbols by the harmonic oscillator wavefunction for the case for j1 = 4000, j2 = 4000, j12 = 200, j3 =100, j =100
Representation of the 6j symbols by the harmonic oscillator wavefunction for the case for j1 = 8000, j2 =8000, j12 = 200, j3 =100, j = 100.
The harmonic oscillator parameters obtained from the two parameters: j23max and α provide a representation of the behaviour of specific 6j-symbols by harmonic oscillator wavefunctions. The present state of the theory shows that the agreement should get better with increasing j.