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rrrHrHrNrmzJ2Op._eval_commutator_JxOpcCrrXrrrHrHrNrprzJ2Op._eval_commutator_JyOpcCrrXrrrHrHrNrsrzJ2Op._eval_commutator_JzOpcCrrXrrrHrHrNrvrzJ2Op._eval_commutator_JplusOpcCrrXrrrHrHrNryrzJ2Op._eval_commutator_JminusOpcK|j}td||d|SNrFrGrMr&rerrzrMrHrHrNr|zJ2Op._apply_operator_JxKetcKrrrrrHrHrNrrz!J2Op._apply_operator_JxKetCoupledcKrrrrrHrHrNrrzJ2Op._apply_operator_JyKetcKrrrrrHrHrNrrz!J2Op._apply_operator_JyKetCoupledcKrrrrrHrHrNrrzJ2Op._apply_operator_JzKetcKrrrrrHrHrNrrz!J2Op._apply_operator_JzKetCoupledcCs4td||d}|t||9}|t||9}|SrrrrHrHrNrxszJ2Op.matrix_elementcKrrrrrHrHrNrrnzJ2Op._represent_default_basiscKrrXrrrHrHrNrrzJ2Op._represent_JzOpcGstt|j}td}||S)Nr)rrorfrqrHrHrNrtszJ2Op._print_contents_prettycGsdt|jS)Nz%s^2)rorfrkrHrHrNrurzJ2Op._print_contents_latexcOs0t|ddt|ddt|ddSr)rrr<rrHrHrNrs0zJ2Op._eval_rewrite_as_xyzcOs:|d}t|dtjt|t|t|t|Sr)r<r rwrr)rercrrrrHrHrNrs $zJ2Op._eval_rewrite_as_plusminusN)rrrrrjrrrrrrrrrrrrxrrrtrurrrHrHrHrNr=hs* r=c@seZdZdZeddZeddZeddZedd Z ed d Z d d Z ddZ ddZ ddZeddZeddZddZddZddZddZd d!d"d#Zd$d%Zd&d'Zd(d)Zd d!d*d+Zd,d-Zd.d/Zd0d1Zd2S)3r4amWigner D operator in terms of Euler angles. Defines the rotation operator in terms of the Euler angles defined by the z-y-z convention for a passive transformation. That is the coordinate axes are rotated first about the z-axis, giving the new x'-y'-z' axes. Then this new coordinate system is rotated about the new y'-axis, giving new x''-y''-z'' axes. Then this new coordinate system is rotated about the z''-axis. Conventions follow those laid out in [1]_. Parameters ========== alpha : Number, Symbol First Euler Angle beta : Number, Symbol Second Euler angle gamma : Number, Symbol Third Euler angle Examples ======== A simple example rotation operator: >>> from sympy import pi >>> from sympy.physics.quantum.spin import Rotation >>> Rotation(pi, 0, pi/2) R(pi,0,pi/2) With symbolic Euler angles and calculating the inverse rotation operator: >>> from sympy import symbols >>> a, b, c = symbols('a b c') >>> Rotation(a, b, c) R(a,b,c) >>> Rotation(a, b, c).inverse() R(-c,-b,-a) See Also ======== WignerD: Symbolic Wigner-D function D: Wigner-D function d: Wigner small-d function References ========== .. [1] Varshalovich, D A, Quantum Theory of Angular Momentum. 1988. cCs&t|}t|dkrtd||S)Nz 3 Euler angles required, got: %r)r _eval_argsrrR)r\rcrHrHrNrs   zRotation._eval_argscCrWrXrYr[rHrHrNr^r_zRotation._eval_hilbert_spacecCr`rar]rdrHrHrNalphargzRotation.alphacCr`rrrdrHrHrNbetargz Rotation.betacCr`rrrdrHrHrNgammargzRotation.gammacGdS)NRrHrkrHrHrN_print_operator_namezRotation._print_operator_namecGs|jrtdStdS)Nuℛ zR ) _use_unicoderrkrHrHrN_print_operator_name_prettysz$Rotation._print_operator_name_prettycGr)Nz \mathcal{R}rHrkrHrHrN_print_operator_name_latexrz#Rotation._print_operator_name_latexcCst|j |j |j SrX)r4rrrrdrHrHrN _eval_inverse zRotation._eval_inversecCst||||||S)agWigner D-function. Returns an instance of the WignerD class corresponding to the Wigner-D function specified by the parameters. Parameters =========== j : Number Total angular momentum m : Number Eigenvalue of angular momentum along axis after rotation mp : Number Eigenvalue of angular momentum along rotated axis alpha : Number, Symbol First Euler angle of rotation beta : Number, Symbol Second Euler angle of rotation gamma : Number, Symbol Third Euler angle of rotation Examples ======== Return the Wigner-D matrix element for a defined rotation, both numerical and symbolic: >>> from sympy.physics.quantum.spin import Rotation >>> from sympy import pi, symbols >>> alpha, beta, gamma = symbols('alpha beta gamma') >>> Rotation.D(1, 1, 0,pi, pi/2,-pi) WignerD(1, 1, 0, pi, pi/2, -pi) See Also ======== WignerD: Symbolic Wigner-D function r5)r\rMrrrrrrHrHrND s)z Rotation.DcCst|||d|dS)aWigner small-d function. Returns an instance of the WignerD class corresponding to the Wigner-D function specified by the parameters with the alpha and gamma angles given as 0. Parameters =========== j : Number Total angular momentum m : Number Eigenvalue of angular momentum along axis after rotation mp : Number Eigenvalue of angular momentum along rotated axis beta : Number, Symbol Second Euler angle of rotation Examples ======== Return the Wigner-D matrix element for a defined rotation, both numerical and symbolic: >>> from sympy.physics.quantum.spin import Rotation >>> from sympy import pi, symbols >>> beta = symbols('beta') >>> Rotation.d(1, 1, 0, pi/2) WignerD(1, 1, 0, 0, pi/2, 0) See Also ======== WignerD: Symbolic Wigner-D function rr)r\rMrrrrHrHrNd7s&z Rotation.dcCs.|j||||j|j|j}|t||9}|SrX)rrrrrr%rrHrHrNrx_s zRotation.matrix_elementc Kst|dtj}t|d}t|\}}t||}t|D]&}t|D]} ||||||| } |r>| ||| f<q%| ||| f<q%q|S)NrMdoit) rrvr rwr-rrSrxr ) reryrzrMevaluaterUr{r|r}r~rrHrHrNrfs    zRotation._represent_basecKrrrrrHrHrNrurnz!Rotation._represent_default_basiscKrrXrrrHrHrNrxrzRotation._represent_JzOpT)dummyc Ks|j}|j}|j}|j}|j} |jr=g} t|} | d} | D]} t|| | |||}| }| |||| qt | S|rDt d} nt d} tt|| | |||||| | | |fSNrGr)rrrrMr is_numberr-r4rr rrrrr)rerrr rzrrrsgrMrrrUszrrrrHrHrN_apply_operator_uncoupled{s$ ,z"Rotation._apply_operator_uncoupledcK|jt|fi|SrX)rr6rrHrHrNrrzRotation._apply_operator_JxKetcKrrX)rr8rrHrHrNrrzRotation._apply_operator_JyKetcKrrX)rr:rrHrHrNrrzRotation._apply_operator_JzKetc Ks|j}|j}|j}|j}|j} |j} |j} |jrEg} t|} | d}|D]}t || ||||}| }| ||||| | q$t | S|rLtd}ntd}tt || |||||||| | || |fSr )rrrrMrrrr r-r4rr rrrrr)rerrr rzrrrsrrMrrrrrUrrrrrHrHrN_apply_operator_coupleds0  z Rotation._apply_operator_coupledcKrrX)rr>rrHrHrNrrz%Rotation._apply_operator_JxKetCoupledcKrrX)rr@rrHrHrNrrz%Rotation._apply_operator_JyKetCoupledcKrrX)rrBrrHrHrNrrz%Rotation._apply_operator_JzKetCoupledN)rrrrrrr^rrrrrrrrrrrxrrrrrrrrrrrrHrHrHrNr4s@3      * ' r4c@seZdZdZdZddZeddZeddZed d Z ed d Z ed dZ eddZ ddZ ddZddZddZdS)r5aWigner-D function The Wigner D-function gives the matrix elements of the rotation operator in the jm-representation. For the Euler angles `\alpha`, `\beta`, `\gamma`, the D-function is defined such that: .. math :: = \delta_{jj'} D(j, m, m', \alpha, \beta, \gamma) Where the rotation operator is as defined by the Rotation class [1]_. The Wigner D-function defined in this way gives: .. math :: D(j, m, m', \alpha, \beta, \gamma) = e^{-i m \alpha} d(j, m, m', \beta) e^{-i m' \gamma} Where d is the Wigner small-d function, which is given by Rotation.d. The Wigner small-d function gives the component of the Wigner D-function that is determined by the second Euler angle. That is the Wigner D-function is: .. math :: D(j, m, m', \alpha, \beta, \gamma) = e^{-i m \alpha} d(j, m, m', \beta) e^{-i m' \gamma} Where d is the small-d function. The Wigner D-function is given by Rotation.D. Note that to evaluate the D-function, the j, m and mp parameters must be integer or half integer numbers. Parameters ========== j : Number Total angular momentum m : Number Eigenvalue of angular momentum along axis after rotation mp : Number Eigenvalue of angular momentum along rotated axis alpha : Number, Symbol First Euler angle of rotation beta : Number, Symbol Second Euler angle of rotation gamma : Number, Symbol Third Euler angle of rotation Examples ======== Evaluate the Wigner-D matrix elements of a simple rotation: >>> from sympy.physics.quantum.spin import Rotation >>> from sympy import pi >>> rot = Rotation.D(1, 1, 0, pi, pi/2, 0) >>> rot WignerD(1, 1, 0, pi, pi/2, 0) >>> rot.doit() sqrt(2)/2 Evaluate the Wigner-d matrix elements of a simple rotation >>> rot = Rotation.d(1, 1, 0, pi/2) >>> rot WignerD(1, 1, 0, 0, pi/2, 0) >>> rot.doit() -sqrt(2)/2 See Also ======== Rotation: Rotation operator References ========== .. [1] Varshalovich, D A, Quantum Theory of Angular Momentum. 1988. TcOsXt|dks td|t|}|dd}|r#tj|g|RStj|g|RS)Nz6 parameters expected, got %sr F)rrRrrvr__new__ _eval_wignerd)r\rchintsr rHrHrNrs   zWignerD.__new__cCr`rarbrdrHrHrNrMrgz WignerD.jcCr`rrbrdrHrHrNrrgz WignerD.mcCr`rrbrdrHrHrNrrgz WignerD.mpcCr`NrrbrdrHrHrNr#rgz WignerD.alphacCr`)NrbrdrHrHrNr'rgz WignerD.betacCr`)NrbrdrHrHrNr+rgz WignerD.gammac Gs|jdkr"|jdkr"d||j||j||j||jfSd||j||j||j||j||j||jfS)Nrzd^{%s}_{%s,%s}\left(%s\right)z#D^{%s}_{%s,%s}\left(%s,%s,%s\right))rr_printrMrrrrkrHrHrN_latex/s"  zWignerD._latexcGs||j}||j}t|d}t|||j}t||}t|d}t|d}||krHt|d||}||kr[t|d||}|j dkrt|j dkrt||j }t dd|}n2||j }t|d}t|||j }t|d}t|||j }t dd|}t| }t||}t||}t||}|S)N, rrr)rrMrrrightrmaxwidthleftrrrrparensabovebelow)rerlrctopbotpadrrHrHrN_pretty<s2       zWignerD._prettycKsd|d<t|ji|S)NTr )r5rc)rerrHrHrNr \sz WignerD.doitc Cs|j}|j}|j}|j}|j}|j}|dkr#|dkr#|dkr#t||S|js,td|d}|t dkrt d|dD].}|||ksP|||ksP|||krQq<|t j |t |||t |||||7}q<|t j ||d|tt||t||t||t||9}nLt|\} } | D]-} |t||| t dt| |tt| |t|| | t d7}q|td|||dd|}t|}|tt ||tt ||9}|S)Nrz1j parameter must be numerical to evaluate, got %srFrG)rMrrrrrr%r rRr rSr NegativeOnerrrr-r4rr rr rrr) rerMrrrrrrkrUr{mpprHrHrNr`s@  $4"  8$(zWignerD._eval_wignerdN)rrrris_commutativerrrMrrrrrrr)r rrHrHrHrNr5s(O        r5Jc@seZdZdZdZddZeddZeddZe d d Z d d Z d dZ ddZ ddZddZddZddZddZddZdS) SpinStatez'Base class for angular momentum states.rcCst|}t|}|jr%d|td|krtd||dkr%td||jr8d|td|kr8td||jr^|jr^t||krLtd||ft||||kr^td||ft|||S)NrFz*j must be integer or half-integer, got: %srzj must be >= 0, got: %sz*m must be integer or half-integer, got: %sz7Allowed values for m are -j <= m <= j, got j, m: %s, %sz>Both j and m must be integer or half-integer, got j, m: %s, %s)rr rTrRabsr$r)r\rMrrHrHrNrs(   zSpinState.__new__cCr`rarrdrHrHrNrMrgz SpinState.jcCr`rrrdrHrHrNrrgz SpinState.mcCstd|ddS)NrFrrGr'r[rHrHrNr^szSpinState._eval_hilbert_spacec Ks|j}|j}t|dd}t|dd}t|dd}t|\}}t|d} t|D]*\} } |jrGt |j| |j||| | | df<q-t |j| |j|||| | df<q-| S)NrrrrrG) rMrrrvr-r enumerater r4rr ) rerzrMrrrrrUr{r|r}mvalrHrHrNrs$    zSpinState._represent_basecO2t|tr|jttfi|S|jttfi|SrX)rr"_rewrite_basisr0r7r6rercrzrHrHrN_eval_rewrite_as_Jx zSpinState._eval_rewrite_as_JxcOr5rX)rr"r6r1r9r8r7rHrHrN_eval_rewrite_as_Jyr9zSpinState._eval_rewrite_as_JycOr5rX)rr"r6r2r;r:r7rHrHrN_eval_rewrite_as_Jzr9zSpinState._eval_rewrite_as_Jzc  sddlm}|j|jddjret|tr2tkr#dndddddnd||fd|i|tfddt ddD}t|trc| d d urct |S|Sd}t d }| |d|kr~|d7}t d |} t|trd|d f}nd|f}t|tr||j||ddjdd} n||j||ddjdjdd} | dkr|S|gR} tj||jg| R} t| | | fS)Nr) representrFrGrrycs.g|]}||gRqSrHrHrIrcevectrMstartvectrHrNrOs.z,SpinState._rewrite_basis..coupledFmizmi%d)rr)ryrr)rrr)sympy.physics.quantum.representr<rMrcr rCoupledSpinStaterTrrSrvrErsubsr#rr4rrr) reryr>rzr<r|rKrB test_argsanglesrltrHr=rNr6sb    $     zSpinState._rewrite_basiscK^t|j|j}||jur||d|j|j9}|S|t|j|jt|j|j9}|SrX)r%rM dual_classr_represent_JxOprrebrarr|rHrHrN_eval_innerproduct_JxBra  z"SpinState._eval_innerproduct_JxBracKrIrX)r%rMrJr_represent_JyOprrLrHrHrN_eval_innerproduct_JyBrarOz"SpinState._eval_innerproduct_JyBracKrIrX)r%rMrJrrrrLrHrHrN_eval_innerproduct_JzBrarOz"SpinState._eval_innerproduct_JzBracKs ||SrX)r )rerMrrHrHrNr&s zSpinState._eval_traceN)rrrr_label_separatorrrrMrrr^rr8r:r;r6rNrQrRrrHrHrHrNr0s&   ,  r0c@HeZdZdZeddZeddZddZdd Zd d Z d d Z dS)r6zEigenket of Jx. See JzKet for the usage of spin eigenstates. See Also ======== JzKet: Usage of spin states cCtSrXr7rdrHrHrNrJBzJxKet.dual_classcCrUrXr>rdrHrHrN coupled_classFrWzJxKet.coupled_classcKrr)rKrrHrHrNrJrnzJxKet._represent_default_basiscK|jdi|SNrHrrrHrHrNrKMrnzJxKet._represent_JxOpcK|jddttddi|SNrrrFrHrr r rrHrHrNrPPzJxKet._represent_JyOpcK|jddtdi|SNrrFrHrr rrHrHrNrSrzJxKet._represent_JzOpN rrrrrrJrYrrKrPrrHrHrHrNr66   r6c@(eZdZdZeddZeddZdS)r7zEigenbra of Jx. See JzKet for the usage of spin eigenstates. See Also ======== JzKet: Usage of spin states cCrUrXr6rdrHrHrNrJcrWzJxBra.dual_classcCrUrXr?rdrHrHrNrYgrWzJxBra.coupled_classNrrrrrrJrYrHrHrHrNr7W  r7c@rT)r8zEigenket of Jy. See JzKet for the usage of spin eigenstates. See Also ======== JzKet: Usage of spin states cCrUrXr9rdrHrHrNrJxrWzJyKet.dual_classcCrUrXr@rdrHrHrNrY|rWzJyKet.coupled_classcKrr)rPrrHrHrNrrnzJyKet._represent_default_basiscKr`NrrFrHrbrrHrHrNrKrzJyKet._represent_JxOpcKrZr[rrrHrHrNrPrnzJyKet._represent_JyOpcK,|jdttddt dtdd|SNrrF)rrrrHr^rrHrHrNr,zJyKet._represent_JzOpNrcrHrHrHrNr8lrdr8c@re)r9zEigenbra of Jy. See JzKet for the usage of spin eigenstates. See Also ======== JzKet: Usage of spin states cCrUrXr8rdrHrHrNrJrWzJyBra.dual_classcCrUrXrArdrHrHrNrYrWzJyBra.coupled_classNrhrHrHrHrNr9rir9c@rT)r:aN Eigenket of Jz. Spin state which is an eigenstate of the Jz operator. Uncoupled states, that is states representing the interaction of multiple separate spin states, are defined as a tensor product of states. Parameters ========== j : Number, Symbol Total spin angular momentum m : Number, Symbol Eigenvalue of the Jz spin operator Examples ======== *Normal States:* Defining simple spin states, both numerical and symbolic: >>> from sympy.physics.quantum.spin import JzKet, JxKet >>> from sympy import symbols >>> JzKet(1, 0) |1,0> >>> j, m = symbols('j m') >>> JzKet(j, m) |j,m> Rewriting the JzKet in terms of eigenkets of the Jx operator: Note: that the resulting eigenstates are JxKet's >>> JzKet(1,1).rewrite("Jx") |1,-1>/2 - sqrt(2)*|1,0>/2 + |1,1>/2 Get the vector representation of a state in terms of the basis elements of the Jx operator: >>> from sympy.physics.quantum.represent import represent >>> from sympy.physics.quantum.spin import Jx, Jz >>> represent(JzKet(1,-1), basis=Jx) Matrix([ [ 1/2], [sqrt(2)/2], [ 1/2]]) Apply innerproducts between states: >>> from sympy.physics.quantum.innerproduct import InnerProduct >>> from sympy.physics.quantum.spin import JxBra >>> i = InnerProduct(JxBra(1,1), JzKet(1,1)) >>> i <1,1|1,1> >>> i.doit() 1/2 *Uncoupled States:* Define an uncoupled state as a TensorProduct between two Jz eigenkets: >>> from sympy.physics.quantum.tensorproduct import TensorProduct >>> j1,m1,j2,m2 = symbols('j1 m1 j2 m2') >>> TensorProduct(JzKet(1,0), JzKet(1,1)) |1,0>x|1,1> >>> TensorProduct(JzKet(j1,m1), JzKet(j2,m2)) |j1,m1>x|j2,m2> A TensorProduct can be rewritten, in which case the eigenstates that make up the tensor product is rewritten to the new basis: >>> TensorProduct(JzKet(1,1),JxKet(1,1)).rewrite('Jz') |1,1>x|1,-1>/2 + sqrt(2)*|1,1>x|1,0>/2 + |1,1>x|1,1>/2 The represent method for TensorProduct's gives the vector representation of the state. Note that the state in the product basis is the equivalent of the tensor product of the vector representation of the component eigenstates: >>> represent(TensorProduct(JzKet(1,0),JzKet(1,1))) Matrix([ [0], [0], [0], [1], [0], [0], [0], [0], [0]]) >>> represent(TensorProduct(JzKet(1,1),JxKet(1,1)), basis=Jz) Matrix([ [ 1/2], [sqrt(2)/2], [ 1/2], [ 0], [ 0], [ 0], [ 0], [ 0], [ 0]]) See Also ======== JzKetCoupled: Coupled eigenstates sympy.physics.quantum.tensorproduct.TensorProduct: Used to specify uncoupled states uncouple: Uncouples states given coupling parameters couple: Couples uncoupled states cCrUrXr;rdrHrHrNrJrWzJzKet.dual_classcCrUrXrBrdrHrHrNrYrWzJzKet.coupled_classcKrrrrrHrHrNrrnzJzKet._represent_default_basiscKr\NrrrFrHr^rrHrHrNrKr_zJzKet._represent_JxOpcK*|jdttddtdtdd|Srnr^rrHrHrNrP*zJzKet._represent_JyOpcKrZr[rrrHrHrNr"rnzJzKet._represent_JzOpNrcrHrHrHrNr:sn   r:c@re)r;zEigenbra of Jz. See the JzKet for the usage of spin eigenstates. See Also ======== JzKet: Usage of spin states cCrUrXr:rdrHrHrNrJ2rWzJzBra.dual_classcCrUrXrCrdrHrHrNrY6rWzJzBra.coupled_classNrhrHrHrHrNr;&rir;c Csddt|D}g}g}|D]/\}}}|||||d||dft||d||d}|||dd<q||fS)NcSg|]}|dgqSrGrHrJrrHrHrNrO@z"_build_coupled..rGr)rSrsorted) jcouplinglengthn_list coupled_jn coupled_nn1n2j_newn_sortrHrHrN_build_coupled?s rc@seZdZdZddZddZddZdd Zed d Z ed d Z eddZ eddZ e ddZddZddZddZddZdS)rDz/Base class for coupled angular momentum states.c st||t|dkr5g}tdtD]}|d|tfddt|Dfq|dt|fnt|dkr@|d}n tdt|dtttt fsZtdj j t|ttt fsjtd |j j t d d |Dswtd tdt|kst d tdt|ft dd |Dst dt|}t|}t ddDt dd|D}tdd Drt dtdd |Drt dtfdd |Drt dtdd |Drt dt|t\}}t}t|D]f\}\} } |t| d} |t| d} ||} t| jr[t| jr[t| jr[| | | kr>t d|d| | | ft| | | krSt d|d| | | ft| | r[ | |t| | d<qt|dkrz|dd|krzt dt||||S)NrrFrGcg|]}|qSrHrHrIrrHrNrOVrPz,CoupledSpinState.__new__..z5CoupledSpinState only takes 3 or 4 arguments, got: %srz'jn must be Tuple, list or tuple, got %sz.jcoupling must be Tuple, list or tuple, got %scss |] }t|tttfVqdSrX)rlisttuplerrJtermrHrHrN dz+CoupledSpinState.__new__..z6All elements of jcoupling must be list, tuple or Tuplez(jcoupling must have length of %d, got %dcs|] }t|dkVqdS)rNrrJrrHrHrNrjz,All elements of jcoupling must have length 3cSg|]}t|qSrHrrJjirHrHrNrOorPcSs*g|]\}}}tt|t|t|qSrH)rr)rJrrrrHrHrNrOps css*|]}|jrd|td|kVqdSrFNr rTrrHrHrNrss(z;All elements of jn must be integer or half-integer, got: %scss.|]\}}}|t|kp|t|kVqdSrX)rTrJrr_rHrHrNrus,z%Indices in jcoupling must be integersc3s>|]\}}}|dkp|dkp|tkp|tkVqdS)rGNrrrrHrNrws<z?Indices must be between 1 and the number of coupled spin spacescss0|]\}}}|jrd|td|kVqdSrr)rJrrrHrHrNrys.zGAll coupled j values in coupling scheme must be integer or half-integerzQAll couplings must have j1+j2 >= j3, in coupling number %d got j1,j2,j3: %d,%d,%dzSAll couplings must have |j1+j2| <= j3, in coupling number %d got j1,j2,j3: %d,%d,%dr*z>Last j value coupled together must be the final j of the state)r0rrSrr TypeErrorrrrrrrallrRranyrr3minr r1rr$r)r\rMrrr~rrrjvalsrrj1j2j3rHrrNrNs|  (   $ zCoupledSpinState.__new__c Gs||j||jg}t|jddD]\}}|d|||fqt|jdd|jddD]\}\}}|dd ddt ||D||fq3d |S) NrGr?zj%d=%sr*zj(%s)=%srcs|]}t|VqdSrXrorIrHrHrNrz0CoupledSpinState._print_label..) rrMrr3rrziprrjoinr}) rerlrcr]rKrrrrrHrHrN _print_labels  *" zCoupledSpinState._print_labelc Gs|j|jg}t|jddD]!\}}d|}t|}t|d}t|||}||q t |j dd|j ddD]+\}\} } d ddt | | D} td | d}t|||}||q>|j||j|g|RS) NrGrj%d=r*rcss |] }td|dVqdS)rr*NrrIrHrHrNrrz7CoupledSpinState._print_label_pretty..rM)rMrr3rrrrrrrrrrr}_print_sequence_prettyrS) rerlrcr]rKrsymbitemrrrrrHrHrN_print_label_prettys"   * z$CoupledSpinState._print_label_prettyc Gs|j|jg|R|j|jg|Rg}t|jddD]\}}|d||j|g|Rfqt|jdd|jddD]$\}\}}d ddt ||D} |d| |j|g|Rfq?|j |S) NrGrz j_{%d}=%sr*rcsrrXrrIrHrHrNrrz6CoupledSpinState._print_label_latex..z j_{%s}=%s) rrMrr3rrrrrrr}rS) rerlrcr]rKrrrrrrHrHrN_print_label_latexs"*" z#CoupledSpinState._print_label_latexcCr`rrrdrHrHrNrrgzCoupledSpinState.jncCr`rrrdrHrHrNrrgzCoupledSpinState.couplingcCt|jdt|jddS)NrrFrGrr]rrdrHrHrNrzCoupledSpinState.coupled_jncCr)NrrFrrrdrHrHrNrrzCoupledSpinState.coupled_ncCsHt|d}|jrtddttd|dddDStd|dS)NrFcSrrHr2rrHrHrNrOrPz8CoupledSpinState._eval_hilbert_space..rGr)rr r(rSrTr')r\r]rMrHrHrNr^s &z$CoupledSpinState._eval_hilbert_spacecKs|}|jjs td|jjjstdt|jjd}|jt|jkr*|jd}nd|jddd|jd}||j|jj di||||d|jddf<|S)Nz1State must not have symbolic j value to representz2State must not have symbolic j values to representrGrFrrrH) uncoupled_classrMr rR hilbert_space dimensionrrTrr)rerzr>r|r?rHrHrN_represent_coupled_bases*    z(CoupledSpinState._represent_coupled_basecOr5rX)rr"r6r0r?r>r7rHrHrNr8r9z$CoupledSpinState._eval_rewrite_as_JxcOr5rX)rr"r6r1rAr@r7rHrHrNr:r9z$CoupledSpinState._eval_rewrite_as_JycOr5rX)rr"r6r2rCrBr7rHrHrNr;r9z$CoupledSpinState._eval_rewrite_as_JzN)rrrrrrrrrrrrrrr^rr8r:r;rHrHrHrNrDKs(B        rDc@rT)r>zCoupled eigenket of Jx. See JzKetCoupled for the usage of coupled spin eigenstates. See Also ======== JzKetCoupled: Usage of coupled spin states cCrUrXrgrdrHrHrNrJrWzJxKetCoupled.dual_classcCrUrXrfrdrHrHrNrrWzJxKetCoupled.uncoupled_classcKrrrrrHrHrNr rnz%JxKetCoupled._represent_default_basiscKrZr[rrrHrHrNrKrnzJxKetCoupled._represent_JxOpcKr\r]rr r rrHrHrNrPr_zJxKetCoupled._represent_JyOpcKr`rarr rrHrHrNrrzJxKetCoupled._represent_JzOpN rrrrrrJrrrKrPrrHrHrHrNr>rdr>c@re)r?zCoupled eigenbra of Jx. See JzKetCoupled for the usage of coupled spin eigenstates. See Also ======== JzKetCoupled: Usage of coupled spin states cCrUrXrXrdrHrHrNrJ$rWzJxBraCoupled.dual_classcCrUrXrVrdrHrHrNr(rWzJxBraCoupled.uncoupled_classNrrrrrrJrrHrHrHrNr?rir?c@rT)r@zCoupled eigenket of Jy. See JzKetCoupled for the usage of coupled spin eigenstates. See Also ======== JzKetCoupled: Usage of coupled spin states cCrUrXrqrdrHrHrNrJ9rWzJyKetCoupled.dual_classcCrUrXrprdrHrHrNr=rWzJyKetCoupled.uncoupled_classcKrrrrrHrHrNrArnz%JyKetCoupled._represent_default_basiscKr`rlrrrHrHrNrKDrzJyKetCoupled._represent_JxOpcKrZr[rrrHrHrNrPGrnzJyKetCoupled._represent_JyOpcKrmrnrrrHrHrNrJrozJyKetCoupled._represent_JzOpNrrHrHrHrNr@-rdr@c@re)rAzCoupled eigenbra of Jy. See JzKetCoupled for the usage of coupled spin eigenstates. See Also ======== JzKetCoupled: Usage of coupled spin states cCrUrXrkrdrHrHrNrJZrWzJyBraCoupled.dual_classcCrUrXrjrdrHrHrNr^rWzJyBraCoupled.uncoupled_classNrrHrHrHrNrANrirAc@rT)rBaCoupled eigenket of Jz Spin state that is an eigenket of Jz which represents the coupling of separate spin spaces. The arguments for creating instances of JzKetCoupled are ``j``, ``m``, ``jn`` and an optional ``jcoupling`` argument. The ``j`` and ``m`` options are the total angular momentum quantum numbers, as used for normal states (e.g. JzKet). The other required parameter in ``jn``, which is a tuple defining the `j_n` angular momentum quantum numbers of the product spaces. So for example, if a state represented the coupling of the product basis state `\left|j_1,m_1\right\rangle\times\left|j_2,m_2\right\rangle`, the ``jn`` for this state would be ``(j1,j2)``. The final option is ``jcoupling``, which is used to define how the spaces specified by ``jn`` are coupled, which includes both the order these spaces are coupled together and the quantum numbers that arise from these couplings. The ``jcoupling`` parameter itself is a list of lists, such that each of the sublists defines a single coupling between the spin spaces. If there are N coupled angular momentum spaces, that is ``jn`` has N elements, then there must be N-1 sublists. Each of these sublists making up the ``jcoupling`` parameter have length 3. The first two elements are the indices of the product spaces that are considered to be coupled together. For example, if we want to couple `j_1` and `j_4`, the indices would be 1 and 4. If a state has already been coupled, it is referenced by the smallest index that is coupled, so if `j_2` and `j_4` has already been coupled to some `j_{24}`, then this value can be coupled by referencing it with index 2. The final element of the sublist is the quantum number of the coupled state. So putting everything together, into a valid sublist for ``jcoupling``, if `j_1` and `j_2` are coupled to an angular momentum space with quantum number `j_{12}` with the value ``j12``, the sublist would be ``(1,2,j12)``, N-1 of these sublists are used in the list for ``jcoupling``. Note the ``jcoupling`` parameter is optional, if it is not specified, the default coupling is taken. This default value is to coupled the spaces in order and take the quantum number of the coupling to be the maximum value. For example, if the spin spaces are `j_1`, `j_2`, `j_3`, `j_4`, then the default coupling couples `j_1` and `j_2` to `j_{12}=j_1+j_2`, then, `j_{12}` and `j_3` are coupled to `j_{123}=j_{12}+j_3`, and finally `j_{123}` and `j_4` to `j=j_{123}+j_4`. The jcoupling value that would correspond to this is: ``((1,2,j1+j2),(1,3,j1+j2+j3))`` Parameters ========== args : tuple The arguments that must be passed are ``j``, ``m``, ``jn``, and ``jcoupling``. The ``j`` value is the total angular momentum. The ``m`` value is the eigenvalue of the Jz spin operator. The ``jn`` list are the j values of argular momentum spaces coupled together. The ``jcoupling`` parameter is an optional parameter defining how the spaces are coupled together. See the above description for how these coupling parameters are defined. Examples ======== Defining simple spin states, both numerical and symbolic: >>> from sympy.physics.quantum.spin import JzKetCoupled >>> from sympy import symbols >>> JzKetCoupled(1, 0, (1, 1)) |1,0,j1=1,j2=1> >>> j, m, j1, j2 = symbols('j m j1 j2') >>> JzKetCoupled(j, m, (j1, j2)) |j,m,j1=j1,j2=j2> Defining coupled spin states for more than 2 coupled spaces with various coupling parameters: >>> JzKetCoupled(2, 1, (1, 1, 1)) |2,1,j1=1,j2=1,j3=1,j(1,2)=2> >>> JzKetCoupled(2, 1, (1, 1, 1), ((1,2,2),(1,3,2)) ) |2,1,j1=1,j2=1,j3=1,j(1,2)=2> >>> JzKetCoupled(2, 1, (1, 1, 1), ((2,3,1),(1,2,2)) ) |2,1,j1=1,j2=1,j3=1,j(2,3)=1> Rewriting the JzKetCoupled in terms of eigenkets of the Jx operator: Note: that the resulting eigenstates are JxKetCoupled >>> JzKetCoupled(1,1,(1,1)).rewrite("Jx") |1,-1,j1=1,j2=1>/2 - sqrt(2)*|1,0,j1=1,j2=1>/2 + |1,1,j1=1,j2=1>/2 The rewrite method can be used to convert a coupled state to an uncoupled state. This is done by passing coupled=False to the rewrite function: >>> JzKetCoupled(1, 0, (1, 1)).rewrite('Jz', coupled=False) -sqrt(2)*|1,-1>x|1,1>/2 + sqrt(2)*|1,1>x|1,-1>/2 Get the vector representation of a state in terms of the basis elements of the Jx operator: >>> from sympy.physics.quantum.represent import represent >>> from sympy.physics.quantum.spin import Jx >>> from sympy import S >>> represent(JzKetCoupled(1,-1,(S(1)/2,S(1)/2)), basis=Jx) Matrix([ [ 0], [ 1/2], [sqrt(2)/2], [ 1/2]]) See Also ======== JzKet: Normal spin eigenstates uncouple: Uncoupling of coupling spin states couple: Coupling of uncoupled spin states cCrUrXrxrdrHrHrNrJrWzJzKetCoupled.dual_classcCrUrXrwrdrHrHrNrrWzJzKetCoupled.uncoupled_classcKrrrrrHrHrNrrnz%JzKetCoupled._represent_default_basiscKr\rtrrrHrHrNrKr_zJzKetCoupled._represent_JxOpcKrurnrrrHrHrNrPrvzJzKetCoupled._represent_JyOpcKrZr[rrrHrHrNrrnzJzKetCoupled._represent_JzOpNrrHrHrHrNrBcst   rBc@re)rCzCoupled eigenbra of Jz. See the JzKetCoupled for the usage of coupled spin eigenstates. See Also ======== JzKetCoupled: Usage of coupled spin states cCrUrXrsrdrHrHrNrJrWzJzBraCoupled.dual_classcCrUrXrrrdrHrHrNrrWzJzBraCoupled.uncoupled_classNrrHrHrHrNrCrirCNcs`|t}|D]&tddjDsqtfddjDs$td|t|}q|S)a Couple a tensor product of spin states This function can be used to couple an uncoupled tensor product of spin states. All of the eigenstates to be coupled must be of the same class. It will return a linear combination of eigenstates that are subclasses of CoupledSpinState determined by Clebsch-Gordan angular momentum coupling coefficients. Parameters ========== expr : Expr An expression involving TensorProducts of spin states to be coupled. Each state must be a subclass of SpinState and they all must be the same class. jcoupling_list : list or tuple Elements of this list are sub-lists of length 2 specifying the order of the coupling of the spin spaces. The length of this must be N-1, where N is the number of states in the tensor product to be coupled. The elements of this sublist are the same as the first two elements of each sublist in the ``jcoupling`` parameter defined for JzKetCoupled. If this parameter is not specified, the default value is taken, which couples the first and second product basis spaces, then couples this new coupled space to the third product space, etc Examples ======== Couple a tensor product of numerical states for two spaces: >>> from sympy.physics.quantum.spin import JzKet, couple >>> from sympy.physics.quantum.tensorproduct import TensorProduct >>> couple(TensorProduct(JzKet(1,0), JzKet(1,1))) -sqrt(2)*|1,1,j1=1,j2=1>/2 + sqrt(2)*|2,1,j1=1,j2=1>/2 Numerical coupling of three spaces using the default coupling method, i.e. first and second spaces couple, then this couples to the third space: >>> couple(TensorProduct(JzKet(1,1), JzKet(1,1), JzKet(1,0))) sqrt(6)*|2,2,j1=1,j2=1,j3=1,j(1,2)=2>/3 + sqrt(3)*|3,2,j1=1,j2=1,j3=1,j(1,2)=2>/3 Perform this same coupling, but we define the coupling to first couple the first and third spaces: >>> couple(TensorProduct(JzKet(1,1), JzKet(1,1), JzKet(1,0)), ((1,3),(1,2)) ) sqrt(2)*|2,2,j1=1,j2=1,j3=1,j(1,3)=1>/2 - sqrt(6)*|2,2,j1=1,j2=1,j3=1,j(1,3)=2>/6 + sqrt(3)*|3,2,j1=1,j2=1,j3=1,j(1,3)=2>/3 Couple a tensor product of symbolic states: >>> from sympy import symbols >>> j1,m1,j2,m2 = symbols('j1 m1 j2 m2') >>> couple(TensorProduct(JzKet(j1,m1), JzKet(j2,m2))) Sum(CG(j1, m1, j2, m2, j, m1 + m2)*|j,m1 + m2,j1=j1,j2=j2>, (j, m1 + m2, j1 + j2)) css|]}t|tVqdSrX)rr0rJrrHrHrNrCszcouple..c3s"|] }|jjdjuVqdS)rN)rrcrrrHrNrFs z!All states must be the same basis)atomsr)rrcrrE_couple)exprjcoupling_listrrrHrrNrDs :rDc! s|j}|d}|dur"g}tdt|D] }|d|dfqt|t|dks:tdt|dt|ftdd|DsGtdtdd|DrTtdtd d|Drdgt|}|D]!\}}||dd ksz||dd kr~td d |t ||d<qfd d |Ddd |Dg}dd tt|D} |D]*} | \}}t | |d} t | |d} || | ft | | | t ||d<qtdd|Drfdd |D} g}t| d dD]}t|}t ||d|}t|D]}t|||}tddt|| Drqg}t }g}t||D]X\\} } }|t | d}|t | d}|||}||t | | d<tfdd | D}tfdd | D}||}|||||||f|t | t | |fq%tdd|Drqtdd|Drqtdd|Drqtdd |D}||||}|||qqt|Sg}g}g}t }|D]|\} } |t | d}|t | d}t| | t|krtd}ndddd | | D} t| }||t | | d<tfdd | D}tfdd | D}||}|||||||f|t | t | |f|||||fqtdd |D}||||}t||g|RS)NrrGz(jcoupling_list must be length %d, got %dcsrrrrJrrHrHrNrZrz_couple..z"Each coupling must define 2 spacescss|] \}}||kVqdSrXrHrJrrrHrHrNr\rz'Spin spaces cannot couple to themselvescss(|]\}}t|jot|jVqdSrX)rr rrHrHrNr^&r*z8Spaces coupling j_n's are referenced by smallest n valuecSg|]}|jqSrHrLrrHrHrNrOfz_couple..cSrrH)rrrHrHrNrOgrcSryrzrHrIrHrHrNrOlr|css |] }|jjo |jjVqdSrX)rMr rrrHrHrNrwrcs0g|]}tfdd|d|dDqS)cs$g|]}|d|dqSrzrHr{rmnrHrNrOz$z&_couple...rrGrrrrHrNrOzs cs|] \}}||kVqdSrXrH)rJrrrHrHrNrrcg|]}|dqSrzrHrrrHrNrOcrrzrHrrrHrNrOrcss$|] }t|d|dkVqdS)rrNr1rrHrHrNrs"css(|]}|d|d|dkVqdSrrFrNrHrrHrHrNrrcss,|]}t|d|d|dkVqdSrrrrHrHrNrs*cSg|]}t|qSrHr+r rrHrHrNrOrrMcSsg|]}d|qS)z%srHr{rHrHrNrOrPcrrzrHrrrHrNrOrcrrzrHrrrHrNrOrcSg|]}t|qSrHr*rrHrHrNrOrP)rcrYrSrrrrrRrr rr}rr_confignum_to_difflistrrrrrr)!rrstates coupled_evectrj_testrr coupling_listr j_couplingj1_nj2_ndiff_maxr|difftot config_num diff_listcg_terms coupled_jr~ coupling_diffrrrm1m2m3coeffr sum_termsj3_namerHrrNrLs          rcCs,|t}|D] }||t|||}q|S)a Uncouple a coupled spin state Gives the uncoupled representation of a coupled spin state. Arguments must be either a spin state that is a subclass of CoupledSpinState or a spin state that is a subclass of SpinState and an array giving the j values of the spaces that are to be coupled Parameters ========== expr : Expr The expression containing states that are to be coupled. If the states are a subclass of SpinState, the ``jn`` and ``jcoupling`` parameters must be defined. If the states are a subclass of CoupledSpinState, ``jn`` and ``jcoupling`` will be taken from the state. jn : list or tuple The list of the j-values that are coupled. If state is a CoupledSpinState, this parameter is ignored. This must be defined if state is not a subclass of CoupledSpinState. The syntax of this parameter is the same as the ``jn`` parameter of JzKetCoupled. jcoupling_list : list or tuple The list defining how the j-values are coupled together. If state is a CoupledSpinState, this parameter is ignored. This must be defined if state is not a subclass of CoupledSpinState. The syntax of this parameter is the same as the ``jcoupling`` parameter of JzKetCoupled. Examples ======== Uncouple a numerical state using a CoupledSpinState state: >>> from sympy.physics.quantum.spin import JzKetCoupled, uncouple >>> from sympy import S >>> uncouple(JzKetCoupled(1, 0, (S(1)/2, S(1)/2))) sqrt(2)*|1/2,-1/2>x|1/2,1/2>/2 + sqrt(2)*|1/2,1/2>x|1/2,-1/2>/2 Perform the same calculation using a SpinState state: >>> from sympy.physics.quantum.spin import JzKet >>> uncouple(JzKet(1, 0), (S(1)/2, S(1)/2)) sqrt(2)*|1/2,-1/2>x|1/2,1/2>/2 + sqrt(2)*|1/2,1/2>x|1/2,-1/2>/2 Uncouple a numerical state of three coupled spaces using a CoupledSpinState state: >>> uncouple(JzKetCoupled(1, 1, (1, 1, 1), ((1,3,1),(1,2,1)) )) |1,-1>x|1,1>x|1,1>/2 - |1,0>x|1,0>x|1,1>/2 + |1,1>x|1,0>x|1,0>/2 - |1,1>x|1,1>x|1,-1>/2 Perform the same calculation using a SpinState state: >>> uncouple(JzKet(1, 1), (1, 1, 1), ((1,3,1),(1,2,1)) ) |1,-1>x|1,1>x|1,1>/2 - |1,0>x|1,0>x|1,1>/2 + |1,1>x|1,0>x|1,0>/2 - |1,1>x|1,1>x|1,-1>/2 Uncouple a symbolic state using a CoupledSpinState state: >>> from sympy import symbols >>> j,m,j1,j2 = symbols('j m j1 j2') >>> uncouple(JzKetCoupled(j, m, (j1, j2))) Sum(CG(j1, m1, j2, m2, j, m)*|j1,m1>x|j2,m2>, (m1, -j1, j1), (m2, -j2, j2)) Perform the same calculation using a SpinState state >>> uncouple(JzKet(j, m), (j1, j2)) Sum(CG(j1, m1, j2, m2, j, m)*|j1,m1>x|j2,m2>, (m1, -j1, j1), (m2, -j2, j2)) )rr0rE _uncouple)rrrrrrrHrHrNrEs DrEc sDt|tr|j|j}|j}|nit|trxdur tdttt fs+t d|durRg}t dt D]}| dd|tfddt |dDfq8t|tt fs]t dt |t dksktdt|t \}}|jnt d|j|jg}t}t||D]*\}\} } || d d} || d d} | | | | | |f||t| | d<qjrMjrMd dD} t}t }t||d|}g}t |D]h}t|||td d t| Drqg}|D]0}|\}}} } }tfd d|D}tfdd|D}||}| | || |||fqtdd|D}tfddtD}| ||qt|Sdt d}t|fdd|ddD}|j fdd|dfDtdd|D}ddtD}tfddtD}t||g|RS)Nz'Must specify j-values for coupled statezjn must be list or tuplerGcrrHrH)rJrMrrHrNrOrPz_uncouple..z!jcoupling must be a list or tuplez?Must specify 2 fewer coupling terms than the number of j valueszstate must be a spin statercSsg|]}d|qS)rFrHrrHrHrNrO0rPcsrrXrH)rJrr}rHrHrNr9rz_uncouple..c$g|]}|d|dqSrzrHrrrrHrNrO?rcrrzrHrrrHrNrO@rcSrrHrrrHrHrNrOCrcsg|] \}}|||qSrHrH)rJrMrr>rHrNrOEszm1:%dc s`g|],\}}}}}|tfdd|D|tfdd|D|tfdd||DfqS)crrzrHr{r{rHrNrOLr(_uncouple...crrzrHr{rrHrNrOMrcrrzrHr{rrHrNrONrrrJrrrrrrrHrNrOLs  r*c sJg|]!\}}}}}|tfdd|D|tfdd|DfqS)crrzrHr{rrHrNrOOrrcrrzrHr{rrHrNrOPrrr)rMrr{rHrNrOOs  cSrrHr*)rJcg_termrHrHrNrORrPcSsg|] \}}|| |fqSrHrHrJrMrrHrHrNrOSscsg|] \}}||qSrHrHrrrHrNrOTs)rrDrrrrr0rRrrrrSrrrrrrMrrrr rrrrr)rr)rrrrrrKrj_listrrrrrrrrrr|rrrrrrrrrm_strcg_coeffrrH)rr>rMrrr{rNrs   &     rcCsxg}t|D]3}|}||d}t||d|}||kr2||8}|d8}t||d|}||ks|||q|Sr)rSrr)rrlist_lenrr prev_diff rem_spots rem_configsrHrHrNrXs  rrX)NN)grsympy.concrete.summationsrsympy.core.addrsympy.core.containersrsympy.core.exprrsympy.core.numbersrsympy.core.mulrr r r r sympy.core.singletonr sympy.core.symbolrrsympy.core.sympifyr(sympy.functions.combinatorial.factorialsrr&sympy.functions.elementary.exponentialr(sympy.functions.elementary.miscellaneousr(sympy.functions.elementary.trigonometricrrsympy.simplify.simplifyrsympy.matricesr sympy.printing.pretty.stringpictrr&sympy.printing.pretty.pretty_symbologyrsympy.physics.quantum.qexprrsympy.physics.quantum.operatorrr r!sympy.physics.quantum.stater"r#r$(sympy.functions.special.tensor_functionsr%sympy.physics.quantum.constantsr&sympy.physics.quantum.hilbertr'r(#sympy.physics.quantum.tensorproductr)sympy.physics.quantum.cgr+sympy.physics.quantum.qapplyr,__all__r-rVrrrrr<r=r4r5r0r1r2r3r.r/r0r6r7r8r9r:r;rrDr>r?r@rArBrCrDrrErrrHrHrHrNs                   _,)## I Q!! -!! 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