o à8Va–zã@sÐdZddlmZmZmZmZmZmZmZm Z m Z m Z m Z m Z mZmZmZmZmZmZdgZdd„Zdd„Zdd „Zdd d „Zdd d„Zddd„Zddd„Zddd„ZGdd„de ƒZdd„Zdd„Zdd„Z d S)a  Wigner, Clebsch-Gordan, Racah, and Gaunt coefficients Collection of functions for calculating Wigner 3j, 6j, 9j, Clebsch-Gordan, Racah as well as Gaunt coefficients exactly, all evaluating to a rational number times the square root of a rational number [Rasch03]_. Please see the description of the individual functions for further details and examples. References ========== .. [Regge58] 'Symmetry Properties of Clebsch-Gordan Coefficients', T. Regge, Nuovo Cimento, Volume 10, pp. 544 (1958) .. [Regge59] 'Symmetry Properties of Racah Coefficients', T. Regge, Nuovo Cimento, Volume 11, pp. 116 (1959) .. [Edmonds74] A. R. Edmonds. Angular momentum in quantum mechanics. Investigations in physics, 4.; Investigations in physics, no. 4. Princeton, N.J., Princeton University Press, 1957. .. [Rasch03] J. Rasch and A. C. H. Yu, 'Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients', SIAM J. Sci. Comput. Volume 25, Issue 4, pp. 1416-1428 (2003) .. [Liberatodebrito82] 'FORTRAN program for the integral of three spherical harmonics', A. Liberato de Brito, Comput. Phys. Commun., Volume 25, pp. 81-85 (1982) Credits and Copyright ===================== This code was taken from Sage with the permission of all authors: https://groups.google.com/forum/#!topic/sage-devel/M4NZdu-7O38 Authors ======= - Jens Rasch (2009-03-24): initial version for Sage - Jens Rasch (2009-05-31): updated to sage-4.0 - Oscar Gerardo Lazo Arjona (2017-06-18): added Wigner D matrices Copyright (C) 2008 Jens Rasch é)ÚIntegerÚpiÚsqrtÚsympifyÚDummyÚSÚSumÚYnmÚzerosÚFunctionÚsinÚcosÚexpÚIÚ factorialÚbinomialÚAddÚImmutableMatrixécCsR|ttƒkrtttƒt|dƒƒD] }t t|d|¡qtdt|ƒd…S)a1 Function calculates a list of precomputed factorials in order to massively accelerate future calculations of the various coefficients. Parameters ========== nn : integer Highest factorial to be computed. Returns ======= list of integers : The list of precomputed factorials. Examples ======== Calculate list of factorials:: sage: from sage.functions.wigner import _calc_factlist sage: _calc_factlist(10) [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800] rN)ÚlenÚ _FactlistÚrangeÚintÚappend)ÚnnÚii©rú6/usr/lib/python3/dist-packages/sympy/physics/wigner.pyÚ_calc_factlist:s rcCsBt|dƒ|dkst|dƒ|dkst|dƒ|dkr"tdƒ‚t|dƒ|dks@t|dƒ|dks@t|dƒ|dkrDtdƒ‚|||dkrNdStdt|||ƒƒ}| }|||}|dkridS|||}|dkrudS| ||} | dkr‚dSt|ƒ|ks”t|ƒ|ks”t|ƒ|kr–dSt|||d|t|ƒ|t|ƒ|t|ƒƒ} tt| ƒƒttt|||ƒtt|||ƒtt| ||ƒtt||ƒtt||ƒtt||ƒtt||ƒtt||ƒtt||ƒƒtt|||dƒ} t| ƒ} | js| j r$|   ¡d} t| ||| ||dƒ} t |||||||ƒ}d}t t| ƒt|ƒdƒD]I}t|tt||||ƒtt|||ƒtt|||ƒtt||||ƒtt||||ƒ}|td|ƒ|}qO| ||}|S)a– Calculate the Wigner 3j symbol `\operatorname{Wigner3j}(j_1,j_2,j_3,m_1,m_2,m_3)`. Parameters ========== j_1, j_2, j_3, m_1, m_2, m_3 : Integer or half integer. Returns ======= Rational number times the square root of a rational number. Examples ======== >>> from sympy.physics.wigner import wigner_3j >>> wigner_3j(2, 6, 4, 0, 0, 0) sqrt(715)/143 >>> wigner_3j(2, 6, 4, 0, 0, 1) 0 It is an error to have arguments that are not integer or half integer values:: sage: wigner_3j(2.1, 6, 4, 0, 0, 0) Traceback (most recent call last): ... ValueError: j values must be integer or half integer sage: wigner_3j(2, 6, 4, 1, 0, -1.1) Traceback (most recent call last): ... ValueError: m values must be integer or half integer Notes ===== The Wigner 3j symbol obeys the following symmetry rules: - invariant under any permutation of the columns (with the exception of a sign change where `J:=j_1+j_2+j_3`): .. math:: \begin{aligned} \operatorname{Wigner3j}(j_1,j_2,j_3,m_1,m_2,m_3) &=\operatorname{Wigner3j}(j_3,j_1,j_2,m_3,m_1,m_2) \\ &=\operatorname{Wigner3j}(j_2,j_3,j_1,m_2,m_3,m_1) \\ &=(-1)^J \operatorname{Wigner3j}(j_3,j_2,j_1,m_3,m_2,m_1) \\ &=(-1)^J \operatorname{Wigner3j}(j_1,j_3,j_2,m_1,m_3,m_2) \\ &=(-1)^J \operatorname{Wigner3j}(j_2,j_1,j_3,m_2,m_1,m_3) \end{aligned} - invariant under space inflection, i.e. .. math:: \operatorname{Wigner3j}(j_1,j_2,j_3,m_1,m_2,m_3) =(-1)^J \operatorname{Wigner3j}(j_1,j_2,j_3,-m_1,-m_2,-m_3) - symmetric with respect to the 72 additional symmetries based on the work by [Regge58]_ - zero for `j_1`, `j_2`, `j_3` not fulfilling triangle relation - zero for `m_1 + m_2 + m_3 \neq 0` - zero for violating any one of the conditions `j_1 \ge |m_1|`, `j_2 \ge |m_2|`, `j_3 \ge |m_3|` Algorithm ========= This function uses the algorithm of [Edmonds74]_ to calculate the value of the 3j symbol exactly. Note that the formula contains alternating sums over large factorials and is therefore unsuitable for finite precision arithmetic and only useful for a computer algebra system [Rasch03]_. Authors ======= - Jens Rasch (2009-03-24): initial version éz(j values must be integer or half integerz(m values must be integer or half integerréÿÿÿÿr) rÚ ValueErrorrÚabsÚmaxrrrZ is_complexZ is_infiniteÚ as_real_imagÚminr)Új_1Új_2Új_3Úm_1Úm_2Úm_3ZprefidÚa1Úa2Úa3ÚmaxfactÚargsqrtÚressqrtÚiminÚimaxÚsumresrÚdenÚresrrrÚ wigner_3j[s†(Vÿ(ÿ  $$ ÿ ÿþýüûúùø ÷   ÿþýüû r7cCs<dt|||ƒtd|dƒt|||||| ƒ}|S)a± Calculates the Clebsch-Gordan coefficient. `\left\langle j_1 m_1 \; j_2 m_2 | j_3 m_3 \right\rangle`. The reference for this function is [Edmonds74]_. Parameters ========== j_1, j_2, j_3, m_1, m_2, m_3 : Integer or half integer. Returns ======= Rational number times the square root of a rational number. Examples ======== >>> from sympy import S >>> from sympy.physics.wigner import clebsch_gordan >>> clebsch_gordan(S(3)/2, S(1)/2, 2, S(3)/2, S(1)/2, 2) 1 >>> clebsch_gordan(S(3)/2, S(1)/2, 1, S(3)/2, -S(1)/2, 1) sqrt(3)/2 >>> clebsch_gordan(S(3)/2, S(1)/2, 1, -S(1)/2, S(1)/2, 0) -sqrt(2)/2 Notes ===== The Clebsch-Gordan coefficient will be evaluated via its relation to Wigner 3j symbols: .. math:: \left\langle j_1 m_1 \; j_2 m_2 | j_3 m_3 \right\rangle =(-1)^{j_1-j_2+m_3} \sqrt{2j_3+1} \operatorname{Wigner3j}(j_1,j_2,j_3,m_1,m_2,-m_3) See also the documentation on Wigner 3j symbols which exhibit much higher symmetry relations than the Clebsch-Gordan coefficient. Authors ======= - Jens Rasch (2009-03-24): initial version r rr)rrr7)r&r'r(r)r*r+r6rrrÚclebsch_gordanês"2ÿr8NcCs`t|||ƒ|||krtdƒ‚t|||ƒ|||kr$tdƒ‚t|||ƒ|||kr6tdƒ‚|||dkr@dS|||dkrJdS|||dkrTdSt||||||||||||dƒ}t|ƒttt|||ƒtt|||ƒtt|||ƒƒttt|||dƒƒ}t|ƒ}|r®| |¡ ¡d}|S)a˜ Calculates the Delta coefficient of the 3 angular momenta for Racah symbols. Also checks that the differences are of integer value. Parameters ========== aa : First angular momentum, integer or half integer. bb : Second angular momentum, integer or half integer. cc : Third angular momentum, integer or half integer. prec : Precision of the ``sqrt()`` calculation. Returns ======= double : Value of the Delta coefficient. Examples ======== sage: from sage.functions.wigner import _big_delta_coeff sage: _big_delta_coeff(1,1,1) 1/2*sqrt(1/6) zJj values must be integer or half integer and fulfill the triangle relationrr) rr!r#rrrrZevalfr$)ÚaaÚbbÚccÚprecr/r0r1rrrÚ_big_delta_coeff!s22ÿþýr=cCsøt||||ƒt||||ƒt||||ƒt||||ƒ}|dkr"dSt||||||||||||ƒ}t||||||||||||ƒ} t| d||||||||||||ƒ} t| ƒd} tt|ƒt| ƒdƒD]l} tt| |||ƒtt| |||ƒtt| |||ƒtt| |||ƒtt||||| ƒtt||||| ƒtt||||| ƒ} | td| t| dƒ| } q}|| dt||||ƒ}|S)aÓ Calculate the Racah symbol `W(a,b,c,d;e,f)`. Parameters ========== a, ..., f : Integer or half integer. prec : Precision, default: ``None``. Providing a precision can drastically speed up the calculation. Returns ======= Rational number times the square root of a rational number (if ``prec=None``), or real number if a precision is given. Examples ======== >>> from sympy.physics.wigner import racah >>> racah(3,3,3,3,3,3) -1/14 Notes ===== The Racah symbol is related to the Wigner 6j symbol: .. math:: \operatorname{Wigner6j}(j_1,j_2,j_3,j_4,j_5,j_6) =(-1)^{j_1+j_2+j_4+j_5} W(j_1,j_2,j_5,j_4,j_3,j_6) Please see the 6j symbol for its much richer symmetries and for additional properties. Algorithm ========= This function uses the algorithm of [Edmonds74]_ to calculate the value of the 6j symbol exactly. Note that the formula contains alternating sums over large factorials and is therefore unsuitable for finite precision arithmetic and only useful for a computer algebra system [Rasch03]_. Authors ======= - Jens Rasch (2009-03-24): initial version rrr )r=r#r%rrrrr)r9r:r;ÚddÚeeZffr<Úprefacr2r3r/r4Úkkr5r6rrrÚracah[sB 5 ÿ þ ý.0$ÿÿþýüûú" rBc Cs0dt||||ƒt|||||||ƒ}|S)a+ Calculate the Wigner 6j symbol `\operatorname{Wigner6j}(j_1,j_2,j_3,j_4,j_5,j_6)`. Parameters ========== j_1, ..., j_6 : Integer or half integer. prec : Precision, default: ``None``. Providing a precision can drastically speed up the calculation. Returns ======= Rational number times the square root of a rational number (if ``prec=None``), or real number if a precision is given. Examples ======== >>> from sympy.physics.wigner import wigner_6j >>> wigner_6j(3,3,3,3,3,3) -1/14 >>> wigner_6j(5,5,5,5,5,5) 1/52 It is an error to have arguments that are not integer or half integer values or do not fulfill the triangle relation:: sage: wigner_6j(2.5,2.5,2.5,2.5,2.5,2.5) Traceback (most recent call last): ... ValueError: j values must be integer or half integer and fulfill the triangle relation sage: wigner_6j(0.5,0.5,1.1,0.5,0.5,1.1) Traceback (most recent call last): ... ValueError: j values must be integer or half integer and fulfill the triangle relation Notes ===== The Wigner 6j symbol is related to the Racah symbol but exhibits more symmetries as detailed below. .. math:: \operatorname{Wigner6j}(j_1,j_2,j_3,j_4,j_5,j_6) =(-1)^{j_1+j_2+j_4+j_5} W(j_1,j_2,j_5,j_4,j_3,j_6) The Wigner 6j symbol obeys the following symmetry rules: - Wigner 6j symbols are left invariant under any permutation of the columns: .. math:: \begin{aligned} \operatorname{Wigner6j}(j_1,j_2,j_3,j_4,j_5,j_6) &=\operatorname{Wigner6j}(j_3,j_1,j_2,j_6,j_4,j_5) \\ &=\operatorname{Wigner6j}(j_2,j_3,j_1,j_5,j_6,j_4) \\ &=\operatorname{Wigner6j}(j_3,j_2,j_1,j_6,j_5,j_4) \\ &=\operatorname{Wigner6j}(j_1,j_3,j_2,j_4,j_6,j_5) \\ &=\operatorname{Wigner6j}(j_2,j_1,j_3,j_5,j_4,j_6) \end{aligned} - They are invariant under the exchange of the upper and lower arguments in each of any two columns, i.e. .. math:: \operatorname{Wigner6j}(j_1,j_2,j_3,j_4,j_5,j_6) =\operatorname{Wigner6j}(j_1,j_5,j_6,j_4,j_2,j_3) =\operatorname{Wigner6j}(j_4,j_2,j_6,j_1,j_5,j_3) =\operatorname{Wigner6j}(j_4,j_5,j_3,j_1,j_2,j_6) - additional 6 symmetries [Regge59]_ giving rise to 144 symmetries in total - only non-zero if any triple of `j`'s fulfill a triangle relation Algorithm ========= This function uses the algorithm of [Edmonds74]_ to calculate the value of the 6j symbol exactly. Note that the formula contains alternating sums over large factorials and is therefore unsuitable for finite precision arithmetic and only useful for a computer algebra system [Rasch03]_. r )rrB)r&r'r(Új_4Új_5Új_6r<r6rrrÚ wigner_6j¬s\ÿrFc  Csžtt||||||ƒdƒ} | d} d} t| t| ƒddƒD],} | | dt|||||| d| ƒt|||||| d| ƒt|||||| d| ƒ} q | S)a? Calculate the Wigner 9j symbol `\operatorname{Wigner9j}(j_1,j_2,j_3,j_4,j_5,j_6,j_7,j_8,j_9)`. Parameters ========== j_1, ..., j_9 : Integer or half integer. prec : precision, default ``None``. Providing a precision can drastically speed up the calculation. Returns ======= Rational number times the square root of a rational number (if ``prec=None``), or real number if a precision is given. Examples ======== >>> from sympy.physics.wigner import wigner_9j >>> wigner_9j(1,1,1, 1,1,1, 1,1,0 ,prec=64) # ==1/18 0.05555555... >>> wigner_9j(1/2,1/2,0, 1/2,3/2,1, 0,1,1 ,prec=64) # ==1/6 0.1666666... It is an error to have arguments that are not integer or half integer values or do not fulfill the triangle relation:: sage: wigner_9j(0.5,0.5,0.5, 0.5,0.5,0.5, 0.5,0.5,0.5,prec=64) Traceback (most recent call last): ... ValueError: j values must be integer or half integer and fulfill the triangle relation sage: wigner_9j(1,1,1, 0.5,1,1.5, 0.5,1,2.5,prec=64) Traceback (most recent call last): ... ValueError: j values must be integer or half integer and fulfill the triangle relation Algorithm ========= This function uses the algorithm of [Edmonds74]_ to calculate the value of the 3j symbol exactly. Note that the formula contains alternating sums over large factorials and is therefore unsuitable for finite precision arithmetic and only useful for a computer algebra system [Rasch03]_. rrr)rr%rrB)r&r'r(rCrDrEZj_7Zj_8Zj_9r<r3r2r4rArrrÚ wigner_9j s 3ÿþýrGcCst|ƒ|kst|ƒ|kst|ƒ|krtdƒ‚t|ƒ|ks(t|ƒ|ks(t|ƒ|kr,tdƒ‚|||}|d}|||} | dkrBdS|||} | dkrNdS| ||} | dkr[dS|dradS|||dkrkdSt|ƒ|ks}t|ƒ|ks}t|ƒ|krdSt| ||| ||dƒ} t|||||||ƒ} t|||d| dƒ}t|ƒd|dd|dd|dt||t||t||t||t||t||dt}t|ƒ}t t|t|||t|||t|||ƒtd|dt||t||t||}d}t t| ƒt| ƒdƒD]?}t|t||||t|||t|||t||||t||||}|t d|ƒ|}q1|||t d||||ƒ}|dur|  |¡}|S) a* Calculate the Gaunt coefficient. Explanation =========== The Gaunt coefficient is defined as the integral over three spherical harmonics: .. math:: \begin{aligned} \operatorname{Gaunt}(l_1,l_2,l_3,m_1,m_2,m_3) &=\int Y_{l_1,m_1}(\Omega) Y_{l_2,m_2}(\Omega) Y_{l_3,m_3}(\Omega) \,d\Omega \\ &=\sqrt{\frac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi}} \operatorname{Wigner3j}(l_1,l_2,l_3,0,0,0) \operatorname{Wigner3j}(l_1,l_2,l_3,m_1,m_2,m_3) \end{aligned} Parameters ========== l_1, l_2, l_3, m_1, m_2, m_3 : Integer. prec - precision, default: ``None``. Providing a precision can drastically speed up the calculation. Returns ======= Rational number times the square root of a rational number (if ``prec=None``), or real number if a precision is given. Examples ======== >>> from sympy.physics.wigner import gaunt >>> gaunt(1,0,1,1,0,-1) -1/(2*sqrt(pi)) >>> gaunt(1000,1000,1200,9,3,-12).n(64) 0.00689500421922113448... It is an error to use non-integer values for `l` and `m`:: sage: gaunt(1.2,0,1.2,0,0,0) Traceback (most recent call last): ... ValueError: l values must be integer sage: gaunt(1,0,1,1.1,0,-1.1) Traceback (most recent call last): ... ValueError: m values must be integer Notes ===== The Gaunt coefficient obeys the following symmetry rules: - invariant under any permutation of the columns .. math:: \begin{aligned} Y(l_1,l_2,l_3,m_1,m_2,m_3) &=Y(l_3,l_1,l_2,m_3,m_1,m_2) \\ &=Y(l_2,l_3,l_1,m_2,m_3,m_1) \\ &=Y(l_3,l_2,l_1,m_3,m_2,m_1) \\ &=Y(l_1,l_3,l_2,m_1,m_3,m_2) \\ &=Y(l_2,l_1,l_3,m_2,m_1,m_3) \end{aligned} - invariant under space inflection, i.e. .. math:: Y(l_1,l_2,l_3,m_1,m_2,m_3) =Y(l_1,l_2,l_3,-m_1,-m_2,-m_3) - symmetric with respect to the 72 Regge symmetries as inherited for the `3j` symbols [Regge58]_ - zero for `l_1`, `l_2`, `l_3` not fulfilling triangle relation - zero for violating any one of the conditions: `l_1 \ge |m_1|`, `l_2 \ge |m_2|`, `l_3 \ge |m_3|` - non-zero only for an even sum of the `l_i`, i.e. `L = l_1 + l_2 + l_3 = 2n` for `n` in `\mathbb{N}` Algorithms ========== This function uses the algorithm of [Liberatodebrito82]_ to calculate the value of the Gaunt coefficient exactly. Note that the formula contains alternating sums over large factorials and is therefore unsuitable for finite precision arithmetic and only useful for a computer algebra system [Rasch03]_. Authors ======= Jens Rasch (2009-03-24): initial version for Sage. zl values must be integerzm values must be integerrrrér N) rr!r"r#r%rrrrrrÚn)Zl_1Zl_2Zl_3r)r*r+r<ZsumLZbigLr,r-r.r2r3r/r0r1r@r4rr5r6rrrÚgauntKsŒ$h$   $ " ÿ ÿ ÿ þ þ þýÿÿþ  ÿ ÿýÿÿþþ$  rJc@seZdZdd„ZdS)ÚWigner3jcKs"tdd„|jDƒƒrt|jŽS|S)Ncss|]}|jVqdS©N)Z is_number)Ú.0ÚobjrrrÚ ís€z Wigner3j.doit..)ÚallÚargsr7)ÚselfZhintsrrrÚdoitìs z Wigner3j.doitN)Ú__name__Ú __module__Ú __qualname__rSrrrrrKês rKc Cs°t|ƒ}t|ƒ}t|ƒ}t|ƒ}t|ƒ}t|ƒ}tdƒ}dd„}tj||tt|||||ƒ||||||ƒd|d|d|d||||t||ƒ||fƒS)a) Returns dot product of rotational gradients of spherical harmonics. Explanation =========== This function returns the right hand side of the following expression: .. math :: \vec{R}Y{_j^{p}} \cdot \vec{R}Y{_l^{m}} = (-1)^{m+p} \sum\limits_{k=|l-j|}^{l+j}Y{_k^{m+p}} * \alpha_{l,m,j,p,k} * \frac{1}{2} (k^2-j^2-l^2+k-j-l) Arguments ========= j, p, l, m .... indices in spherical harmonics (expressions or integers) theta, phi .... angle arguments in spherical harmonics Example ======= >>> from sympy import symbols >>> from sympy.physics.wigner import dot_rot_grad_Ynm >>> theta, phi = symbols("theta phi") >>> dot_rot_grad_Ynm(3, 2, 2, 0, theta, phi).doit() 3*sqrt(55)*Ynm(5, 2, theta, phi)/(11*sqrt(pi)) Úkc Ss`td|dd|dd|ddtƒt|||tjtjtjƒt|||||| |ƒS)NrrrH)rrrKrZZero)ÚlÚmÚjÚprWrrrÚalphas .ÿþzdot_rot_grad_Ynm..alphar)rrrZ NegativeOnerr r")rZr[rXrYZthetaZphirWr\rrrÚdot_rot_grad_Ynmòs2"ÿÿr]c sê‡fdd„tdˆdƒDƒ}tdˆdƒ}t|ƒD]U\}‰t|ƒD]L\}‰tˆ ˆˆˆgƒ}tdˆˆgƒ}ttˆˆƒtˆˆƒtˆˆƒtˆˆƒƒ}‡‡‡‡fdd„t||dƒDƒ} |t| Ž|||f<q#qt|ƒS)u¥Return the small Wigner d matrix for angular momentum J. Explanation =========== J : An integer, half-integer, or sympy symbol for the total angular momentum of the angular momentum space being rotated. beta : A real number representing the Euler angle of rotation about the so-called line of nodes. See [Edmonds74]_. Returns ======= A matrix representing the corresponding Euler angle rotation( in the basis of eigenvectors of `J_z`). .. math :: \mathcal{d}_{\beta} = \exp\big( \frac{i\beta}{\hbar} J_y\big) The components are calculated using the general form [Edmonds74]_, equation 4.1.15. Examples ======== >>> from sympy import Integer, symbols, pi, pprint >>> from sympy.physics.wigner import wigner_d_small >>> half = 1/Integer(2) >>> beta = symbols("beta", real=True) >>> pprint(wigner_d_small(half, beta), use_unicode=True) ⎡ ⎛β⎞ ⎛β⎞⎤ ⎢cos⎜─⎟ sin⎜─⎟⎥ ⎢ âŽ2⎠ âŽ2⎠⎥ ⎢ ⎥ ⎢ ⎛β⎞ ⎛β⎞⎥ ⎢-sin⎜─⎟ cos⎜─⎟⎥ ⎣ âŽ2⎠ âŽ2⎠⎦ >>> pprint(wigner_d_small(2*half, beta), use_unicode=True) ⎡ 2⎛β⎞ ⎛β⎞ ⎛β⎞ 2⎛β⎞ ⎤ ⎢ cos ⎜─⎟ √2â‹…sin⎜─⎟⋅cos⎜─⎟ sin ⎜─⎟ ⎥ ⎢ âŽ2⎠ âŽ2⎠ âŽ2⎠ âŽ2⎠ ⎥ ⎢ ⎥ ⎢ ⎛β⎞ ⎛β⎞ 2⎛β⎞ 2⎛β⎞ ⎛β⎞ ⎛β⎞⎥ ⎢-√2â‹…sin⎜─⎟⋅cos⎜─⎟ - sin ⎜─⎟ + cos ⎜─⎟ √2â‹…sin⎜─⎟⋅cos⎜─⎟⎥ ⎢ âŽ2⎠ âŽ2⎠ âŽ2⎠ âŽ2⎠ âŽ2⎠ âŽ2⎠⎥ ⎢ ⎥ ⎢ 2⎛β⎞ ⎛β⎞ ⎛β⎞ 2⎛β⎞ ⎥ ⎢ sin ⎜─⎟ -√2â‹…sin⎜─⎟⋅cos⎜─⎟ cos ⎜─⎟ ⎥ ⎣ âŽ2⎠ âŽ2⎠ âŽ2⎠ âŽ2⎠ ⎦ From table 4 in [Edmonds74]_ >>> pprint(wigner_d_small(half, beta).subs({beta:pi/2}), use_unicode=True) ⎡ √2 √2⎤ ⎢ ── ──⎥ ⎢ 2 2 ⎥ ⎢ ⎥ ⎢-√2 √2⎥ ⎢──── ──⎥ ⎣ 2 2 ⎦ >>> pprint(wigner_d_small(2*half, beta).subs({beta:pi/2}), ... use_unicode=True) ⎡ √2 ⎤ ⎢1/2 ── 1/2⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢-√2 √2 ⎥ ⎢──── 0 ── ⎥ ⎢ 2 2 ⎥ ⎢ ⎥ ⎢ -√2 ⎥ ⎢1/2 ──── 1/2⎥ ⎣ 2 ⎦ >>> pprint(wigner_d_small(3*half, beta).subs({beta:pi/2}), ... use_unicode=True) ⎡ √2 √6 √6 √2⎤ ⎢ ── ── ── ──⎥ ⎢ 4 4 4 4 ⎥ ⎢ ⎥ ⎢-√6 -√2 √2 √6⎥ ⎢──── ──── ── ──⎥ ⎢ 4 4 4 4 ⎥ ⎢ ⎥ ⎢ √6 -√2 -√2 √6⎥ ⎢ ── ──── ──── ──⎥ ⎢ 4 4 4 4 ⎥ ⎢ ⎥ ⎢-√2 √6 -√6 √2⎥ ⎢──── ── ──── ──⎥ ⎣ 4 4 4 4 ⎦ >>> pprint(wigner_d_small(4*half, beta).subs({beta:pi/2}), ... use_unicode=True) ⎡ √6 ⎤ ⎢1/4 1/2 ── 1/2 1/4⎥ ⎢ 4 ⎥ ⎢ ⎥ ⎢-1/2 -1/2 0 1/2 1/2⎥ ⎢ ⎥ ⎢ √6 √6 ⎥ ⎢ ── 0 -1/2 0 ── ⎥ ⎢ 4 4 ⎥ ⎢ ⎥ ⎢-1/2 1/2 0 -1/2 1/2⎥ ⎢ ⎥ ⎢ √6 ⎥ ⎢1/4 -1/2 ── -1/2 1/4⎥ ⎣ 4 ⎦ cóg|]}ˆ|‘qSrr©rMÚi©ÚJrrÚ ”óz"wigner_d_small..rrrcs€g|]<}dˆˆ|tˆˆˆˆ|ƒtˆˆ|ƒtˆdƒd|ˆˆtˆdƒdˆd|ˆˆ‘qS)r r)rr r )rMÚs©rbÚMiÚMjÚbetarrrcŸsûÿ þý"ü) rr Ú enumerater#r%rrrr) rbriÚMÚdr`rZZsigmamaxZsigmaminZdijZtermsrrfrÚwigner_d_small"s"r ÿ ÿ ûñrmcsLtˆ|ƒ‰‡fdd„tdˆdƒDƒ‰‡‡‡‡fdd„tˆƒDƒ}t|ƒS)u˜Return the Wigner D matrix for angular momentum J. Explanation =========== J : An integer, half-integer, or sympy symbol for the total angular momentum of the angular momentum space being rotated. alpha, beta, gamma - Real numbers representing the Euler. Angles of rotation about the so-called vertical, line of nodes, and figure axes. See [Edmonds74]_. Returns ======= A matrix representing the corresponding Euler angle rotation( in the basis of eigenvectors of `J_z`). .. math :: \mathcal{D}_{\alpha \beta \gamma} = \exp\big( \frac{i\alpha}{\hbar} J_z\big) \exp\big( \frac{i\beta}{\hbar} J_y\big) \exp\big( \frac{i\gamma}{\hbar} J_z\big) The components are calculated using the general form [Edmonds74]_, equation 4.1.12. Examples ======== The simplest possible example: >>> from sympy.physics.wigner import wigner_d >>> from sympy import Integer, symbols, pprint >>> half = 1/Integer(2) >>> alpha, beta, gamma = symbols("alpha, beta, gamma", real=True) >>> pprint(wigner_d(half, alpha, beta, gamma), use_unicode=True) ⎡ ⅈ⋅α ⅈ⋅γ ⅈ⋅α -ⅈ⋅γ ⎤ ⎢ ─── ─── ─── ───── ⎥ ⎢ 2 2 ⎛β⎞ 2 2 ⎛β⎞ ⎥ ⎢ ℯ ⋅ℯ â‹…cos⎜─⎟ ℯ ⋅ℯ â‹…sin⎜─⎟ ⎥ ⎢ âŽ2⎠ âŽ2⎠ ⎥ ⎢ ⎥ ⎢ -ⅈ⋅α ⅈ⋅γ -ⅈ⋅α -ⅈ⋅γ ⎥ ⎢ ───── ─── ───── ───── ⎥ ⎢ 2 2 ⎛β⎞ 2 2 ⎛β⎞⎥ ⎢-ℯ ⋅ℯ â‹…sin⎜─⎟ ℯ ⋅ℯ â‹…cos⎜─⎟⎥ ⎣ âŽ2⎠ âŽ2⎠⎦ cr^rrr_rarrrcßrdzwigner_d..rrcs.g|]\‰‰‡‡‡‡‡fdd„tˆƒDƒ‘qS)cs<g|]\}}ttˆˆƒˆˆ|ftt|ˆƒ‘qSr)rr)rMrZrh)rgr\rlÚgammar`rrrcàs0ÿz'wigner_d...)rj)rM)rkr\rlrn)rgr`rrcàs ÿ ÿ)rmrrjr)rbr\rirnÚDr)rbrkr\rlrnrÚwigner_d«s 3ÿrprL)!Ú__doc__Zsympyrrrrrrrr r r r r rrrrrrrrr7r8r=rBrFrGrJrKr]rmrprrrrÚs$P/! 7 : Q a > 0