Clebsch-Gordan Coefficients¶
Clebsch-Gordon Coefficients.
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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 [R411]:
\[C^{j_1,m_1}_{j_2,m_2,j_3,m_3} = \langle j_1,m_1;j_2,m_2 | j_3,m_3\rangle\]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
[R411] (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
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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 [R412].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
[R412] (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
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class
sympy.physics.quantum.cg.
Wigner6j
[source]¶ Class for the Wigner-6j symbols
See also
Wigner3j
- Wigner-3j symbols
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class
sympy.physics.quantum.cg.
Wigner9j
[source]¶ Class for the Wigner-9j symbols
See also
Wigner3j
- Wigner-3j symbols
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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 [R413].
See also
CG
- Clebsh-Gordan coefficients
References
[R413] (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