public void ClebschGordonOrthonormalityMM()
{
int t00 = 0;
int t01 = 0;
int t1 = 1;
foreach (Spin j1 in GenerateRandomSpins(0.0, 50.0, 4))
{
foreach (Spin j2 in GenerateRandomSpins(0.0, 50.0, 4))
{
foreach (SpinState s1a in GenerateRandomSpinStates(j1, 4))
{
foreach (SpinState s1b in GenerateRandomSpinStates(j1, 4))
{
foreach (SpinState s2a in GenerateRandomSpinStates(j2, 4))
{
foreach (SpinState s2b in GenerateRandomSpinStates(j2, 4))
{
// if the sum of M's are equal, all terms will be trivially zero
if ((s1a.M + s2a.M) != (s1b.M + s2b.M))
{
continue;
}
// sum over j and m
double s = 0.0;
bool nonZero = false;
double j_min = Math.Abs(j1.J - j2.J);
double j_max = j1.J + j2.J;
for (double j = j_min; j <= j_max; j = j + 1.0)
{
Spin j3 = new Spin(j);
foreach (SpinState s3 in j3.States())
{
double ds = SpinMath.ClebschGodron(s1a, s2a, s3) * SpinMath.ClebschGodron(s1b, s2b, s3);
if (ds != 0.0)
{
nonZero = true;
}
s += ds;
}
}
if ((s1a.M == s1b.M) && (s2a.M == s2b.M))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(s, 1.0));
t1++;
}
else
{
Assert.IsTrue(Math.Abs(s) < TestUtilities.TargetPrecision);
if (nonZero)
{
t01++;
}
else
{
t00++;
}
}
}
}
}
}
}
}
Console.WriteLine("Trivial zeros: {0}", t00);
Console.WriteLine("Non-trivial zerios: {0}", t01);
Console.WriteLine("Ones: {0}", t1);
}
public void ClebschGordonSepcialCase()
{
// 0 x 0 => 0
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(0, 0), new SpinState(0, 0), new SpinState(0, 0)), 1.0));
// 1/2 x 1/2 => 1
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(0.5, 0.5), new SpinState(0.5, 0.5), new SpinState(1.0, 1.0)), 1.0));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(0.5, 0.5), new SpinState(0.5, -0.5), new SpinState(1.0, 0.0)), Math.Sqrt(1.0 / 2.0)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(0.5, -0.5), new SpinState(0.5, 0.5), new SpinState(1.0, 0.0)), Math.Sqrt(1.0 / 2.0)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(0.5, -0.5), new SpinState(0.5, -0.5), new SpinState(1.0, -1.0)), 1.0));
// 1/2 x 1/2 => 0
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(0.5, 0.5), new SpinState(0.5, -0.5), new SpinState(0.0, 0.0)), Math.Sqrt(1.0 / 2.0)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(0.5, -0.5), new SpinState(0.5, 0.5), new SpinState(0.0, 0.0)), -Math.Sqrt(1.0 / 2.0)));
// 1 x 1/2 => 3/2
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, 1.0), new SpinState(0.5, 0.5), new SpinState(1.5, 1.5)), 1.0));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, 1.0), new SpinState(0.5, -0.5), new SpinState(1.5, 0.5)), Math.Sqrt(1.0 / 3.0)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, 0.0), new SpinState(0.5, 0.5), new SpinState(1.5, 0.5)), Math.Sqrt(2.0 / 3.0)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, 0.0), new SpinState(0.5, -0.5), new SpinState(1.5, -0.5)), Math.Sqrt(2.0 / 3.0)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, -1.0), new SpinState(0.5, 0.5), new SpinState(1.5, -0.5)), Math.Sqrt(1.0 / 3.0)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, -1.0), new SpinState(0.5, -0.5), new SpinState(1.5, -1.5)), 1.0));
// 1 x 1/2 => 1/2
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, 1.0), new SpinState(0.5, -0.5), new SpinState(0.5, 0.5)), Math.Sqrt(2.0 / 3.0)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, 0.0), new SpinState(0.5, 0.5), new SpinState(0.5, 0.5)), -Math.Sqrt(1.0 / 3.0)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, 0.0), new SpinState(0.5, -0.5), new SpinState(0.5, -0.5)), Math.Sqrt(1.0 / 3.0)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, -1.0), new SpinState(0.5, 0.5), new SpinState(0.5, -0.5)), -Math.Sqrt(2.0 / 3.0)));
// 1 x 1 => 2
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, 1.0), new SpinState(1.0, 1.0), new SpinState(2.0, 2.0)), 1.0));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, 1.0), new SpinState(1.0, 0.0), new SpinState(2.0, 1.0)), Math.Sqrt(1.0 / 2.0)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, 0.0), new SpinState(1.0, 1.0), new SpinState(2.0, 1.0)), Math.Sqrt(1.0 / 2.0)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, 1.0), new SpinState(1.0, -1.0), new SpinState(2.0, 0.0)), Math.Sqrt(1.0 / 6.0)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, 0.0), new SpinState(1.0, 0.0), new SpinState(2.0, 0.0)), Math.Sqrt(2.0 / 3.0)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, -1.0), new SpinState(1.0, 1.0), new SpinState(2.0, 0.0)), Math.Sqrt(1.0 / 6.0)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, 0.0), new SpinState(1.0, -1.0), new SpinState(2.0, -1.0)), Math.Sqrt(1.0 / 2.0)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, -1.0), new SpinState(1.0, 0.0), new SpinState(2.0, -1.0)), Math.Sqrt(1.0 / 2.0)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, -1.0), new SpinState(1.0, -1.0), new SpinState(2.0, -2.0)), 1.0));
// 1 x 1=> 1
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, 1.0), new SpinState(1.0, 0.0), new SpinState(1.0, 1.0)), Math.Sqrt(1.0 / 2.0)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, 0.0), new SpinState(1.0, 1.0), new SpinState(1.0, 1.0)), -Math.Sqrt(1.0 / 2.0)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, 1.0), new SpinState(1.0, -1.0), new SpinState(1.0, 0.0)), Math.Sqrt(1.0 / 2.0)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, 0.0), new SpinState(1.0, 0.0), new SpinState(1.0, 0.0)), 0.0));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, -1.0), new SpinState(1.0, 1.0), new SpinState(1.0, 0.0)), -Math.Sqrt(1.0 / 2.0)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, 0.0), new SpinState(1.0, -1.0), new SpinState(1.0, -1.0)), Math.Sqrt(1.0 / 2.0)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, -1.0), new SpinState(1.0, 0.0), new SpinState(1.0, -1.0)), -Math.Sqrt(1.0 / 2.0)));
// 1 x 1=> 0
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, 1.0), new SpinState(1.0, -1.0), new SpinState(0.0, 0.0)), Math.Sqrt(1.0 / 3.0)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, 0.0), new SpinState(1.0, 0.0), new SpinState(0.0, 0.0)), -Math.Sqrt(1.0 / 3.0)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(SpinMath.ClebschGodron(new SpinState(1.0, -1.0), new SpinState(1.0, 1.0), new SpinState(0.0, 0.0)), Math.Sqrt(1.0 / 3.0)));
}
public void ClebschGordonOrthonormalityJM()
{
int t00 = 0;
int t01 = 0;
int t1 = 0;
foreach (Spin j1 in GenerateRandomSpins(0.0, 50.0, 4))
{
foreach (Spin j2 in GenerateRandomSpins(0.0, 50.0, 4))
{
foreach (Spin j3a in GenerateRandomCombinedSpins(j1, j2, 4))
{
foreach (Spin j3b in GenerateRandomCombinedSpins(j1, j2, 4))
{
foreach (SpinState s3a in GenerateRandomSpinStates(j3a, 4))
{
foreach (SpinState s3b in GenerateRandomSpinStates(j3b, 4))
{
// skip the trivial zeros; if the sums of the Ms for each coefficient
// are not the same, no term can be non-zero
if (s3a.M != s3b.M)
{
continue;
}
// sum over m1 and m2
double s = 0.0;
bool nonZero = false;
foreach (SpinState s1 in j1.States())
{
foreach (SpinState s2 in j2.States())
{
double ds = SpinMath.ClebschGodron(s1, s2, s3a) * SpinMath.ClebschGodron(s1, s2, s3b);
if (ds != 0.0)
{
nonZero = true;
}
s += ds;
}
}
// check orthonormality
if ((s3a.J == s3b.J) && (s3a.M == s3b.M))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(s, 1.0));
t1++;
}
else
{
Assert.IsTrue(Math.Abs(s) <= TestUtilities.TargetPrecision);
if (nonZero)
{
t01++;
}
else
{
t00++;
}
}
}
}
}
}
}
}
Console.WriteLine("Trivial zeros: {0}", t00);
Console.WriteLine("Non-trivial zerios: {0}", t01);
Console.WriteLine("Ones: {0}", t1);
}
public void SixJThreeJRelation()
{
SixJSymbol[] symbols = GenerateRandomSixJSymbols(50.0, 5);
foreach (SixJSymbol symbol in symbols)
{
Spin a = symbol.J1;
Spin b = symbol.J2;
Spin c = symbol.J3;
Spin d = symbol.J4;
Spin e = symbol.J5;
Spin f = symbol.J6;
SpinState[] ams = GenerateRandomSpinStates(a, 2);
SpinState[] bms = GenerateRandomSpinStates(b, 2);
foreach (SpinState am in ams)
{
foreach (SpinState bm in bms)
{
if (Math.Abs(am.M + bm.M) > c.J)
{
continue;
}
SpinState cm = new SpinState(c, -(am.M + bm.M));
double g1 = SpinMath.ThreeJ(am, bm, cm);
double g2 = SpinMath.SixJ(a, b, c, d, e, f);
double p = g1 * g2;
double q = 0.0;
List <double> ts = new List <double>();
SpinState[] dms = d.States();
foreach (SpinState dm in dms)
{
if (Math.Abs(dm.M + cm.M) > e.J)
{
continue;
}
SpinState em = new SpinState(e, dm.M + cm.M);
SpinState mem = new SpinState(e, -em.M);
double f1 = SpinMath.ThreeJ(dm, mem, cm);
if (Math.Abs(em.M + am.M) > f.J)
{
continue;
}
SpinState fm = new SpinState(f, em.M + am.M);
SpinState mfm = new SpinState(f, -fm.M);
double f2 = SpinMath.ThreeJ(em, mfm, am);
SpinState mdm = new SpinState(d, -dm.M);
double f3 = SpinMath.ThreeJ(fm, mdm, bm);
double t = f1 * f2 * f3;
int s = (int)Math.Round(dm.J + dm.M + em.J + em.M + fm.J + fm.M);
if (s % 2 != 0)
{
t = -t;
}
q += t;
ts.Add(t);
}
Console.WriteLine("{0} v. {1}", p, q);
//Assert.IsTrue(TestUtilities.IsNearlyEqual(p, q));
Assert.IsTrue(TestUtilities.IsSumNearlyEqual(ts, p));
}
}
}
}