Re: Varshalovich Quantum Theory Of Angular Momentum Pdf 30
Laverne Levenstein
The arguments for creating instances of JzKetCoupled are j, m,jn and an optional jcoupling argument. The j and m optionsare the total angular momentum quantum numbers, as used for normal states(e.g. JzKet).
The final option is jcoupling, which is used to define how the spacesspecified by jn are coupled, which includes both the order these spacesare coupled together and the quantum numbers that arise from thesecouplings. The jcoupling parameter itself is a list of lists, such thateach of the sublists defines a single coupling between the spin spaces. Ifthere 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 thejcoupling parameter have length 3. The first two elements are theindices 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 1and 4. If a state has already been coupled, it is referenced by thesmallest index that is coupled, so if \(j_2\) and \(j_4\) has already beencoupled to some \(j_24\), then this value can be coupled by referencing itwith index 2. The final element of the sublist is the quantum number of thecoupled state. So putting everything together, into a valid sublist forjcoupling, if \(j_1\) and \(j_2\) are coupled to an angular momentum spacewith 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 forjcoupling.
The arguments that must be passed are j, m, jn, andjcoupling. The j value is the total angular momentum. The mvalue is the eigenvalue of the Jz spin operator. The jn list arethe j values of argular momentum spaces coupled together. Thejcoupling parameter is an optional parameter defining how the spacesare coupled together. See the above description for how these couplingparameters are defined.
This function can be used to couple an uncoupled tensor product of spinstates. All of the eigenstates to be coupled must be of the same class. Itwill return a linear combination of eigenstates that are subclasses ofCoupledSpinState determined by Clebsch-Gordan angular momentum couplingcoefficients.
Elements of this list are sub-lists of length 2 specifying the order ofthe coupling of the spin spaces. The length of this must be N-1, where Nis the number of states in the tensor product to be coupled. Theelements of this sublist are the same as the first two elements of eachsublist in the jcoupling parameter defined for JzKetCoupled. If thisparameter is not specified, the default value is taken, which couplesthe first and second product basis spaces, then couples this new coupledspace to the third product space, etc
Gives the uncoupled representation of a coupled spin state. Arguments mustbe either a spin state that is a subclass of CoupledSpinState or a spinstate that is a subclass of SpinState and an array giving the j valuesof the spaces that are to be coupled
The expression containing states that are to be coupled. If the statesare a subclass of SpinState, the jn and jcoupling parametersmust be defined. If the states are a subclass of CoupledSpinState,jn and jcoupling will be taken from the state.
The list of the j-values that are coupled. If state is aCoupledSpinState, this parameter is ignored. This must be defined ifstate is not a subclass of CoupledSpinState. The syntax of thisparameter is the same as the jn parameter of JzKetCoupled.
The list defining how the j-values are coupled together. If state is aCoupledSpinState, this parameter is ignored. This must be defined ifstate is not a subclass of CoupledSpinState. The syntax of thisparameter is the same as the jcoupling parameter of JzKetCoupled.
This is the most complete handbook on the quantum theory of angular momentum. Containing basic definitions and theorems as well as relations, tables of formula and numerical tables which are essential for applications to many physical problems, the book is useful for specialists in nuclear and particle physics, atomic and molecular spectroscopy, plasma physics, collision and reaction theory, quantum chemistry, etc. The authors take pains to write many formulae in different coordinate systems thus providing users with added ease in consulting this book. Each chapter opens with a comprehensive list of its contents to ease the search for any information needed later. New results relating to different aspects of the angular momentum thoery are also included. Containing close to 500 pages this book also gathers together many useful formulae besides those related to angular momentum. The book also compares different notations used by previous authors.
The Gaunt coefficient is one of the important coefficients to be known for calculating molecular integrals in quantum theory of coupling of three angular momenta. Generally, these coefficients are calculated analytically by using the properties of the associated Legendre polynomials. In this study, Gaunt coefficients were calculated algebraically by using the recurrence relations and orthogonality conditions of spherical harmonics and different mathematical expressions were obtained from known analytical expressions for Gaunt coefficients in terms of factorial functions or binomial coefficients. By using the program written in the Mathematica programming language, both the analytical expressions and the algebraic expressions were calculated, and the numerical results obtained were compared. Numerical results are in quite agreement with the literature and each other.
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