Wigner’s theorem

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. 

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]. 

A fundamental property of the wave function is that it can be used as basis for irreducible representations of the point group of a molecule [13], This property establishes the connection between the symmetry of a molecule and its wave function. The preceding statement follows from Wigner s theorem, which says that all eigenfunctions of a molecular system belong to one of the symmetry species of the group [14],... 

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... 

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, ... 

Selection rules arise on considering how the transition operator f(t) transforms under the double group SF El We note that if f(t) is spin free it transforms the same way under SF El ° a as it does under S SF. Selection rules follow from the Wigner-Eckart theorem, just as for the spin-free case discussed in Section III. Selection rules for operators which contain spin may also be derived on considering22 how such operators transform in S SF El... 

As has been shown, second-quantized operators can be expanded in terms of triple tensors in the spaces of orbital, spin and quasispin angular momenta. The wave functions of a shell of equivalent electrons (15.46) are also classified using the quantum numbers L, S, Q, Ml, Ms, Mq of the three commuting angular momenta. Therefore, we can apply the Wigner-Eckart theorem (5.15) in all three spaces to the matrix elements of any irreducible triple tensorial operator T(JC K) defined relative to wave functions (15.46)... 

To that end, variational transition-state theory has been introduced, which is based on Wigner s variational theorem, Eq. (5.10). When a saddle point exists, it represents a bottleneck between reactant and products. It is the point along the reaction coordinate where we have the smallest rate of transformation from the reactant to products. This can be seen from Eq. (7.50), where it should be noted that only the sum of states G (E ) changes as the reaction proceeds along the reaction coordinate. We have the smallest sum of states of the activated complex G (E ) on top of the barrier because at this point the available energy E is at a minimum see Fig. 7.3.3. 

The tensorial structure of the spin-orbit operators can be exploited to reduce the number of matrix elements that have to be evaluated explicitly. According to the Wigner-Eckart theorem, it is sufficient to determine a single (nonzero) matrix element for each pair of multiplet wave functions the matrix element for any other pair of multiplet components can then be obtained by multiplying the reduced matrix element with a constant. These vector coupling coefficients, products of 3j symbols and a phase factor, depend solely on the symmetry of the problem, not on the particular molecule. Furthermore, selection rules can be derived from the tensorial structure for example, within an LS coupling scheme, electronic states may interact via spin-orbit coupling only if their spin quantum numbers S and S are equal or differ by 1, i.e., S = S or S = S 1. 

In our discussion of the FIR laser magnetic resonance spectrum of CH in its a 4 " state we encountered the reduced matrix element of P(.S. . S. . S ). The result was presented in equation (9.155), which we now derive. First we note that, by the Wigner Eckart theorem, the following result applies in the molecule-fixed coordinate system with... 

The matrix elements of the tensor operator Uq which are diagonal in the spin S can be rewritten using the Wigner-Eckart theorem with the 3-j symbols and the reduced matrix elements... 

It is highly useful to employ symmetry relations and selection rules of angular momentum operators for SOC matrix elements [108, 109], The Wigner-Eckart theorem (WET) allows calculations of just a few matrix elements of manifold S. M. S, M in order to obtain all other matrix elements. The WET states that the dependence of the matrix elements on the M, M quantum numbers can be entirely... 

In principle, the correspondence between the two theories is not complete, because scattering theory is the more general formulation. For our purposes, however, the fact that the applications to atomic physics obtained by both methods are quite consistent with each other is an important and useful conclusion. The same result and connections have been obtained independently by Komninos and Nicolaides [378]. Both [373] and [378] noted that the derivation of MQDT from Wigner s scattering theory establishes its basic structure and theorems without special assumptions about the asymptotic forms of wavefunctions. The approach of Komninos and Nicolaides [378] is designed for applications involving Hartree-Fock and multiconfigurational Hartree-Fock bases. In the present exposition, we follow the approach and notation of Lane [379] and others [380, 381], who exploit the analytic K-matrix formalism and include photon widths explicitly when interferences occur. 

An important theorem which derives from Schur s lemma is the fundamental theorem for irreducible representations (3). This can be formulated as the Wigner-Eckart theorem, Eq. (3). In order to obtain this theorem in the present formulation, each irreducible representation must be chosen in identically the same form each time it occurs, rather than in an equivalent form. Therefore the irreducible representations are conveniently generated in standard form by applying the operators of the symmetry group to a properly cho.sen set of standard bases for the irreducible representations. 

The power of the Wigner-Eckart theorem (Messiah, 1960, p. 489 Edmonds, 1974, p. 75) is that it relates one nonzero matrix element to another, thereby vastly reducing the number of integrals that must either be explicitly evaluated or treated as a variable parameter in a least-squares fit to spectral data. For example, consider S k a tensor operator of rank k that acts exclusively on spin variables. The Wigner-Eckart theorem requires... 

Evaluation of the reduced matrix element (5 T ( S, 5) 5 ) is shghtly more comph-cated. First we use the Wigner Eckart theorem to obtain... 

The symbols a and y are multiplicity labels that are required if A and B occur more than once in the respective decompositions of A, x A2 and B, x B2. The symbol Pi is a multiplicity label of a different kind it is required when the reduction G yields more than one identical irreducible representation B,- in a given A,. Like the proof of the Wigner-Eckart theorem, the derivation of (51) depends crucially on Schur s lemma. The second factor on the right-hand side of eq. (50), for want of a better expression, has become known over the years as an isoscalar factor, in analogy to the situation for SU(3) (Edmonds 1962). 

One of the most remarkable features of the analysis of Racah (1949) for the Coulomb energies in the f shell is that the theory works much better and shows more simpliiications than could possibly have been anticipated in 1949. As was explained in section 4.3.6, Racah s use of the lie groups SO(7) and Gi provides explanations for the vanishing of many matrix elements and for the proportionalities that some matrices bear to others but it also goes beyond a straightforward application of the Wigner-Eckart theorem that such examples represent. The principal surprises are as follows ... 

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