Clebsch-Gordan Coefficients

Clebsch-Gordon Coefficients.

class sympy.physics.quantum.cg.CG(j1, m1, j2, m2, j3, m3)[source]

Class for Clebsch-Gordan coefficient

Clebsch-Gordan coefficients describe the angular momentum coupling between two systems. The coefficients give the expansion of a coupled total angular momentum state and an uncoupled tensor product state. The Clebsch-Gordan coefficients are defined as [R428]:

\[C^{j_1,m_1}_{j_2,m_2,j_3,m_3} = \langle j_1,m_1;j_2,m_2 | j_3,m_3\rangle\]
Parameters

j1, m1, j2, m2, j3, m3 : Number, Symbol

Terms determining the angular momentum of coupled angular momentum systems.

See also

Wigner3j

Wigner-3j symbols

References

R428(1,2)

Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.

Examples

Define a Clebsch-Gordan coefficient and evaluate its value

>>> from sympy.physics.quantum.cg import CG
>>> from sympy import S
>>> cg = CG(S(3)/2, S(3)/2, S(1)/2, -S(1)/2, 1, 1)
>>> cg
CG(3/2, 3/2, 1/2, -1/2, 1, 1)
>>> cg.doit()
sqrt(3)/2
doit(**hints)[source]

Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via ‘hints’ or unless the ‘deep’ hint was set to ‘False’.

>>> from sympy import Integral
>>> from sympy.abc import x

>>> 2*Integral(x, x)
2*Integral(x, x)

>>> (2*Integral(x, x)).doit()
x**2

>>> (2*Integral(x, x)).doit(deep=False)
2*Integral(x, x)
class sympy.physics.quantum.cg.Wigner3j(j1, m1, j2, m2, j3, m3)[source]

Class for the Wigner-3j symbols

Wigner 3j-symbols are coefficients determined by the coupling of two angular momenta. When created, they are expressed as symbolic quantities that, for numerical parameters, can be evaluated using the .doit() method [R429].

Parameters

j1, m1, j2, m2, j3, m3 : Number, Symbol

Terms determining the angular momentum of coupled angular momentum systems.

See also

CG

Clebsch-Gordan coefficients

References

R429(1,2)

Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.

Examples

Declare a Wigner-3j coefficient and calcualte its value

>>> from sympy.physics.quantum.cg import Wigner3j
>>> w3j = Wigner3j(6,0,4,0,2,0)
>>> w3j
Wigner3j(6, 0, 4, 0, 2, 0)
>>> w3j.doit()
sqrt(715)/143
doit(**hints)[source]

Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via ‘hints’ or unless the ‘deep’ hint was set to ‘False’.

>>> from sympy import Integral
>>> from sympy.abc import x

>>> 2*Integral(x, x)
2*Integral(x, x)

>>> (2*Integral(x, x)).doit()
x**2

>>> (2*Integral(x, x)).doit(deep=False)
2*Integral(x, x)
class sympy.physics.quantum.cg.Wigner6j(j1, j2, j12, j3, j, j23)[source]

Class for the Wigner-6j symbols

See also

Wigner3j

Wigner-3j symbols

doit(**hints)[source]

Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via ‘hints’ or unless the ‘deep’ hint was set to ‘False’.

>>> from sympy import Integral
>>> from sympy.abc import x

>>> 2*Integral(x, x)
2*Integral(x, x)

>>> (2*Integral(x, x)).doit()
x**2

>>> (2*Integral(x, x)).doit(deep=False)
2*Integral(x, x)
class sympy.physics.quantum.cg.Wigner9j(j1, j2, j12, j3, j4, j34, j13, j24, j)[source]

Class for the Wigner-9j symbols

See also

Wigner3j

Wigner-3j symbols

doit(**hints)[source]

Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via ‘hints’ or unless the ‘deep’ hint was set to ‘False’.

>>> from sympy import Integral
>>> from sympy.abc import x

>>> 2*Integral(x, x)
2*Integral(x, x)

>>> (2*Integral(x, x)).doit()
x**2

>>> (2*Integral(x, x)).doit(deep=False)
2*Integral(x, x)
sympy.physics.quantum.cg.cg_simp(e)[source]

Simplify and combine CG coefficients

This function uses various symmetry and properties of sums and products of Clebsch-Gordan coefficients to simplify statements involving these terms [R430].

See also

CG

Clebsh-Gordan coefficients

References

R430(1,2)

Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.

Examples

Simplify the sum over CG(a,alpha,0,0,a,alpha) for all alpha to 2*a+1

>>> from sympy.physics.quantum.cg import CG, cg_simp
>>> a = CG(1,1,0,0,1,1)
>>> b = CG(1,0,0,0,1,0)
>>> c = CG(1,-1,0,0,1,-1)
>>> cg_simp(a+b+c)
3