Wigner theorem

According to the Wigner theorem (see Appendix A), because of time-reversal. symmetry, the functions i ground-state wavefunction does not have a node, we can always make the... 

The optimal choice of the dividing surface S(pj,r) is, according to the Wigner theorem, the surface that gives the smallest rate constant k(T). In principle, it can be determined by a variational calculation of k(T) with respect to the surface such that 6k(T) = 0. 

The approach described above will, in general, not give the exact rate constant, since it is based on a quite arbitrary choice of the dividing surface we do not know if the choice is valid according to the Wigner theorem, namely that the rate constant is at a minimum with respect to variations in the choice of dividing surface. A variational determination of the rate constant with respect to the position of the dividing surface is usually not done directly. 

The determination of the exact rate constant as given above consists of two steps (i) a determination of the surface integral in Eq. (5.44), and (ii) a determination of the factor k in Eq. (5.69) that makes up for the fact that the chosen surface may not be the one leading to a minimum value of the rate constant as required by the Wigner theorem. In addition, a determination of the partition function for the reactants is, of course, required. 

Detailed analysis of the perturbation formulas reveals that the pth order correction to the wavefunction, cptp) jg actually sufficient to calculate all energy corrections up to (7p+ ) which is the content of the Wigner theorem. 

Quantum mechanically, a one-electron property is defined as the expectation value of the corresponding operator expressed with the help of wavefunction or because this is mostly not available with the approximate wavefunction well-defined wavefunction. According to the Wigner theorem (see equation 23), the function is associated with both and (2p+i) However, this does not represent a real disadvantage of MP theory since one-electron properties can be expressed as response properties once analytical energy derivatives are... 

Let us focus on the origin of the principal idea of the J-T effect. Before its final formulation by Jahn and Teller, first the Teller s student Renner [33] was inspired with the von Neumann-Wigner theorem about crossing electronic terms [34] Electronic states of a diatomic molecule do not cross, unless permitted by symmetry . Only if the states have different symmetry, they can cross. 

Wigner s formula is open to criticism also on another point, since he assumes the existence of a stationary electron state where the density is so low that the kinetic energy may be neglected. This is in contradiction to the virial theorem (Eq. 11.15), which tells us that the kinetic energy can never be neglected in comparison to the potential energy and that the latter quantity is compensated by the former to fifty per cent. A reexamination of the low density case would hence definitely be a problem of essential interest. 

Campbell s Theorem, 174 Cartwright, M. L., 388 Caywood, T. E., 313 C-coefficients, 404 formulas for, 406 recursion relations, 406 relation to spherical harmonics, 408 tabulations of, 408 Wigner s formula, 408 Central field Dirac equation in, 629 Central force law... 

In the latter expression the matrix element of operator dq> is transformed according to the Wigner-Eckart theorem and the definition used is... 

The matrix element of operator is written in terms of the Wigner-Eckart theorem, and the integral part is denoted as... 

Equation (4.15) would be extremely onerous to evaluate by explicit treatment of the nucleons as a many-particle system. However, in Mossbauer spectroscopy, we are dealing with eigenstates of the nucleus that are characterized by the total angular momentum with quantum number 7. Fortunately, the electric quadrupole interaction can be readily expressed in terms of this momentum 7, which is called the nuclear spin other properties of the nucleus need not to be considered. This is possible because the transformational properties of the quadrupole moment, which is an irreducible 2nd rank tensor, make it possible to use Clebsch-Gordon coefficients and the Wigner-Eckart theorem to replace the awkward operators 3x,xy—(5,yr (in spatial coordinates) by angular momentum operators of the total... 

Spin projection coefficients as given in (4.84) can be obtained in a general and mathematically proper manner by using the Wigner-Eckhart theorem. A detailed description of this topic is found in the textbook by Bencini and Gatteschi [106]. 

Another important property of the angular momentum is the Wigner-Eckart theorem.2 This theorem states that the matrix elements of any tensor operator can be separated into two parts, one containing the m dependence and one independent of m,... 

The angular part depends only on properties of the angular momentum. Using the Wigner-Eckart theorem, one has... 

At this point, we should invoke the so-called Wigner-Eckart theorem, whose demonstration is beyond the scope of this book (see, e.g., Tsukerblat, 1994). From this theorem, it is possible to establish the following selection rule ... 

By Wigner-Eckart s theorem [6] Eq. (2) can be expressed in terms of a reduced matrix element that is independent of M and M, ... 

Wigner (1930) has shown that if time is reversible in a quantum-mechanical system, then all wavefunctions can be made real. This theorem enables us to use real wavefunctions whenever possible, which are often more convenient than complex ones. Here we present a simplified proof of Wigner s theorem, with some examples of its applications. 

The last two matrix elements in Eq. (8.45) can be evaluated using the Wigner-Eckart theorem [5] ... 

Consideration of the symmetry of the Poincare group also shows that the cyclic theorem is independent of Lorentz boosts in any direction, and also reveals the physical meaning of the E(2) little group of Wigner. This group is unphysical for a photon without mass, but is physical for a photon with mass. This proves that Poincare symmetry leads to a photon with identically nonzero mass. The proof is as follows. Consider in the particle interpretation the PL vector... 

In Section 10.4 we studied projective unitary representations, important because they are symmetries of quantum systems. It is natural to wonder whether projective unitary symmetries are the only symmetries of quantum systems. In this section, we will show that complex conjugation, while not projective unitary, is a physical symmetry, i.e., it preserves all the physically relevant quantities. The good news is that complex conjugation is essentially the only physical symmetry we missed. More precisely, each physical symmetry is either projective unitary or it is the composition of a projective unitary symmetry with complex conjugation. This result (Proposition 10.10) is known as Wigner s theorem on quantum mechanical symmetries. The original proof can be found in the appendix to Chapter 20 in Wigner s book [Wi]. 

From the Wigner-Eckart theorem, these matrix elements may be written in terms of 3-j symbols, as [350]... 

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