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# Clebsch-Gordan Coefficients¶

Clebsch-Gordon Coefficients.

class sympy.physics.quantum.cg.CG[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 [R67]:

$C^{j_1,m_1}_{j_2,m_2,j_3,m_3} = \langle j_1,m_1;j_2,m_2 | j_3,m_3\rangle$
Parameters : j1, m1, j2, m2, j3, m3 : Number, Symbol Terms determining the angular momentum of coupled angular momentum systems.

Wigner3j
Wigner-3j symbols

References

 [R67] (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

class sympy.physics.quantum.cg.Wigner3j[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 [R68].

Parameters : j1, m1, j2, m2, j3, m3 : Number, Symbol Terms determining the angular momentum of coupled angular momentum systems.

CG
Clebsch-Gordan coefficients

References

 [R68] (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

class sympy.physics.quantum.cg.Wigner6j[source]

Class for the Wigner-6j symbols

Wigner3j
Wigner-3j symbols
class sympy.physics.quantum.cg.Wigner9j[source]

Class for the Wigner-9j symbols

Wigner3j
Wigner-3j symbols
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 [R69].

CG
Clebsh-Gordan coefficients

References

 [R69] (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


Anticommutator

Commutator