K ir~dZddlmZddlmZddlmZddlmZddl m Z ddl m Z dgZ d Zd ZGd d e Zd Zy)a This module implements Pauli algebra by subclassing Symbol. Only algebraic properties of Pauli matrices are used (we do not use the Matrix class). See the documentation to the class Pauli for examples. References ========== .. [1] https://en.wikipedia.org/wiki/Pauli_matrices )Add)Mul)I)Pow)Symbol) TensorProductevaluate_pauli_productc||k(ryy)z Returns 1 if ``i == j``, else 0. This is used in the multiplication of Pauli matrices. Examples ======== >>> from sympy.physics.paulialgebra import delta >>> delta(1, 1) 1 >>> delta(2, 3) 0 r)ijs `/mnt/ssd/data/python-lab/Trading/venv/lib/python3.12/site-packages/sympy/physics/paulialgebra.pydeltars Avc$|||fdvry|||fdvryy)ak Return 1 if i,j,k is equal to (1,2,3), (2,3,1), or (3,1,2); -1 if ``i``,``j``,``k`` is equal to (1,3,2), (3,2,1), or (2,1,3); else return 0. This is used in the multiplication of Pauli matrices. Examples ======== >>> from sympy.physics.paulialgebra import epsilon >>> epsilon(1, 2, 3) 1 >>> epsilon(1, 3, 2) -1 )r )rrr )rr rr ))r rr)rrr )rr rrr )r rks repsilonr,s+" 1ay55 Q7 7rcDeZdZdZdZddZdZdZfdZfdZ xZ S) Paulia The class representing algebraic properties of Pauli matrices. Explanation =========== The symbol used to display the Pauli matrices can be changed with an optional parameter ``label="sigma"``. Pauli matrices with different ``label`` attributes cannot multiply together. If the left multiplication of symbol or number with Pauli matrix is needed, please use parentheses to separate Pauli and symbolic multiplication (for example: 2*I*(Pauli(3)*Pauli(2))). Another variant is to use evaluate_pauli_product function to evaluate the product of Pauli matrices and other symbols (with commutative multiply rules). See Also ======== evaluate_pauli_product Examples ======== >>> from sympy.physics.paulialgebra import Pauli >>> Pauli(1) sigma1 >>> Pauli(1)*Pauli(2) I*sigma3 >>> Pauli(1)*Pauli(1) 1 >>> Pauli(3)**4 1 >>> Pauli(1)*Pauli(2)*Pauli(3) I >>> from sympy.physics.paulialgebra import Pauli >>> Pauli(1, label="tau") tau1 >>> Pauli(1)*Pauli(2, label="tau") sigma1*tau2 >>> Pauli(1, label="tau")*Pauli(2, label="tau") I*tau3 >>> from sympy import I >>> I*(Pauli(2)*Pauli(3)) -sigma1 >>> from sympy.physics.paulialgebra import evaluate_pauli_product >>> f = I*Pauli(2)*Pauli(3) >>> f I*sigma2*sigma3 >>> evaluate_pauli_product(f) -sigma1 r labelc||dvr tdtj|d||fzdd}||_||_|S)NrzInvalid Pauli indexz%s%dFT) commutative hermitian) IndexErrorr__new__r r)clsr robjs rr!z Pauli.__new__sG I 23 3nnS&5)"3RVW  rc6|j|jfifSNrselfs r__getnewargs_ex__zPauli.__getnewargs_ex__s #R''rc2|j|jfSr%rr&s r_hashable_contentzPauli._hashable_contents ##rct|tr|j}|j}|j}|j}||k(rrt ||t t ||dztd|zzt t ||dztd|zzt t ||dztd|zzSt|!|S)Nr rr) isinstancerr rrrrsuper__mul__)r'otherrrjlabklab __class__s rr.z Pauli.__mul__s eU #AA::D;;Dt|Q{1a((q671a((q671a((q677wu%%rcn|jr(|jrt| t |dzSyy)Nr) is_Integer is_positiver-__pow__int)ber2s r _eval_powerzPauli._eval_powers. <>> from sympy.physics.paulialgebra import Pauli, evaluate_pauli_product >>> from sympy import I >>> evaluate_pauli_product(I*Pauli(1)*Pauli(2)) -sigma3 >>> from sympy.abc import x >>> evaluate_pauli_product(x**2*Pauli(2)*Pauli(1)) -I*x**2*sigma3 rr ) r,rargsris_oddrr rr as_coeff_mulis_commutative) argstartendparttmp sigma_product com_productkeeperels rr r s* E C#s 388A; > 88A;  88A; #schhGd+D1GHH#}%Q5d;QRR S ! sleslscz  "  a& "B"e$# &&b#&:bggaj%+Hwwqz((%3 M2#M1%GIwwOt4T:OF%&M#M1"4F$%Mr! # "$!fVmM)+5 #:u J9sleslscz8 JIHR.Ps>H 7H3H N)r?sympy.core.addrsympy.core.mulrsympy.core.numbersrsympy.core.powerrsympy.core.symbolrsympy.physics.quantumr__all__rrrr r rrrWsE  $/ # $*2\/F\/~Cr