Clebsch-Gordan Coefficients¶

Clebsch-Gordon Coefficients.

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

Class for Clebsch-Gordan coefficient.

Parameters

j1, m1, j2, m2 : Number, Symbol

Angular momenta of states 1 and 2.

j3, m3: Number, Symbol

Total angular momentum of the coupled system.

Explanation

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 [R665]:

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

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
>>> CG(j1=S(1)/2, m1=-S(1)/2, j2=S(1)/2, m2=+S(1)/2, j3=1, m3=0).doit()
sqrt(2)/2

Compare [R666].

See also

Wigner3j: Wigner-3j symbols

References

R665(1,2): Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.
R666(1,2): Clebsch-Gordan Coefficients, Spherical Harmonics, and d Functions in P.A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020).

class sympy.physics.quantum.cg.Wigner3j(j1, m1, j2, m2, j3, m3)[source]¶

Class for the Wigner-3j symbols.

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

Terms determining the angular momentum of coupled angular momentum systems.

Explanation

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 [R667].

Examples

Declare a Wigner-3j coefficient and calculate 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

See also

CG: Clebsch-Gordan coefficients

References

R667(1,2): Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.

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

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

sympy.physics.quantum.cg.cg_simp(e)[source]¶

Simplify and combine CG coefficients.

Explanation

This function uses various symmetry and properties of sums and products of Clebsch-Gordan coefficients to simplify statements involving these terms [R668].

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

See also

CG: Clebsh-Gordan coefficients

References

R668(1,2): Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.

Anticommutator

Commutator