Properties of wave functions

The Symmetry Properties of Wave Functions of Li3 Electronically Ground State in [Pg.582]

It is important to distinguish between mmetiy properties of wave functions on one hand and those of density matrices and densities on the other. The symmetry properties of wave functions are derived from those of the Hamiltonian. The "normal" situation is that the Hamiltonian commutes with a set of symmetry operations which form a group. The eigenfunctions of that Hamiltonian must then transform according to the irreducible representations of the group. Approximate wave functions with the same symmetry properties can be constructed, and they make it possible to simplify the calculations. 

The major difference between classical and quantum mechanical ensembles arises from the symmetry properties of wave functions which is not an issue in classical systems. 

Besides the modules of maximum values of ig-function radial part were compared with Po-parameter values, and the line dependence between these values was found. Using some properties of wave function for P-parameter, the wave equation of P-parameter was obtained. 

All terms on the left vanish except that for n = due to orthogonality property of wave functions and from the conditions of normalization,... 

It will be obvious from the content of Chapter 5 why such combinations are desired. First, only such functions can, in themselves, constitute acceptable solutions to the wave equation or be directly combined to form acceptable solutions, as shown in Section 5.1. Second, only when the symmetry properties of wave functions are defined explicitly, in the sense of their being bases for irreducible representations, can we employ the theorems of Section 5.2 in order to determine without numerical computations which integrals or matrix elements in the problem are identically zero. 

Second-quantization formalism was introduced into the theory of many-electron atoms by Judd [12]. This formalism enables one to give a simple and elegant description of both the rotation symmetry of a system and its permutational symmetry the tensorial properties of wave functions are translated to electron creation and annihilation operators, and the Pauli exclusion principle stems automatically from the anticommutation relations between these operators. 

For the basis (18.27) to be used effectively in practical computations an adequate mathematical tool is required that would permit full account to be taken of the tensorial properties of wave functions and operators in their spaces. In particular, matrix elements can now be defined using the Wigner-Eckart theorem (5.15) in all three spaces, so that the submatrix element will be given by... 

To sum up the potentialities of the isospin method are not exhausted by the results stated above. There is a deep connection between orthogonal transformations of radial orbitals and rotations in isospin space (see (18.40) and (18.41)). This shows that the tensorial properties of wave functions and operators in isospin space must be dominant in the Hartree-Fock method. This issue is in need of further consideration. 

The relationships describing the tensorial properties of wave functions, second-quantization operators and matrix elements in the space of total angular momentum J can readily be obtained by the use of the results of Chapters 14 and 15 with the more or less trivial replacement of the ranks of the tensors l and s by j and the corresponding replacement of various factors and 3nj-coefficients. Therefore, we shall only give a sketch of the uses of the quasispin method for jj coupling, following mainly the works [30, 167, 168]. For a subshell of equivalent electrons, the creation and annihilation operators a and a(jf are the components of the same tensor of rank q = 1/2... 

As in the case of LS coupling, the tensorial properties of wave functions and second-quantization operators in quasispin space enable us to separate, using the Wigner-Eckart theorem, the dependence of the submatrix elements on the number of electrons in the subshell into the Clebsch-Gordan coefficient. If then we use the relation of the submatrix element of the creation operator to the CFP... 

Spectroscopists usually use notations based upon the symmetry properties of wave functions to describe excited states. Since many molecules have no special symmetry properties, such devices are not really strictly applicable in general. However, symmetry notation applicable to related molecules of high symmetry is often extended to unsymmetrical systems. The procedure has been treated formally by Platt (7) who introduced the concept of local symmetry. ... 

Bouckaert, L. P., Smoluchowski, R. and Wigner E. (1936) Theory of Brillouin zones and symmetry properties of wave functions in crystals. Phys. Rev. 50, 58-67. 

Properties of Wave Functions in the Continuous Range,—For hyperbolic states the parameter I is imaginary as well as a l = IK, a = i [ a, A and a being real and positive). The question arises, therefore, whether the function x( ) is complex. It was shown (loc. cit.) that our series (1) is a limiting case of the hypergeometric series... 

It is found that the symbols f, g, h, etc. representing such arrays can be manipulated by an algebra closely related to ordinary algebra, differing from it mainly in the process of multiplication. The rules of this algebra can be easily derived from the properties of wave functions, which we already know. 

The above anticommutation relations for second-quantization operators have been derived using the symmetry properties of one-determinant wave functions with relation to the permutation of the coordinates of particles. Since the second-quantization operators are only defined in the space of antisymmetric wave functions, the reverse statement is true -in second-quantization formalism the permutative symmetry properties of wave functions automatically follow from the anticommutation relations for creation and annihilation operators. We shall write these relations together in the form... 

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