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 [R621]:
\[C^{j_3,m_3}_{j_1,m_1,j_2,m_2} = \left\langle j_1,m_1;j_2,m_2 | j_3,m_3\right\rangle\]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 [R622].
See also
Wigner3j
Wigner-3j symbols
References
- R621(1,2)
Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.
- R622(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 [R623].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
- 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 [R624].
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