Symmetric wave functions

Systems containing symmetric wave function components ate called Bose-Einstein systems (129) those having antisymmetric wave functions are called Fermi-Ditac systems (130,131). Systems in which all components are at a single quantum state are called MaxweU-Boltzmaim systems (122). Further, a boson is a particle obeying Bose-Einstein statistics, a fermion is one obeying Eermi-Ditac statistics (132). 

Another aspect of wave function instability concerns symmetry breaking, i.e. the wave function has a lower symmetry than the nuclear framework. It occurs for example for the allyl radical with an ROHF type wave function. The nuclear geometry has C21, symmetry, but the Cay symmetric wave function corresponds to a (first-order) saddle point. The lowest energy ROHF solution has only Cj symmetry, and corresponds to a localized double bond and a localized electron (radical). Relaxing the double occupancy constraint, and allowing the wave function to become UHF, re-establish the correct Cay symmetry. Such symmetry breaking phenomena usually indicate that the type of wave function used is not flexible enough for even a qualitatively correct description. 

In equation (8.32) the operator P is any one of the N operators, including the identity operator, that permute a given order of particles to another order. The summation is taken over all N permutation operators. The quantity dp is always - -1 for the symmetric wave function Ps, but for the antisymmetric wave function Wa, dpis-l-l(—l)if the permutation operator P involves the... 

As pointed out in Section 7.2, electrons, protons, and neutrons have spin f. Therefore, a system of N electrons, or N protons, or N neutrons possesses an antisymmetric wave function. A symmetric wave function is not allowed. Nuclei of " He and atoms of " He have spin 0, while photons and nuclei have spin 1. Accordingly, these particles possess symmetric wave functions, never antisymmetric wave functions. If a system is composed of several kinds of particles, then its wave function must be separately symmetric or antisymmetric with respect to each type of particle. For example, the wave function for... 

The behavior of a multi-particle system with a symmetric wave function differs markedly from the behavior of a system with an antisymmetric wave function. Particles with integral spin and therefore symmetric wave functions satisfy Bose-Einstein statistics and are called bosons, while particles with antisymmetric wave functions satisfy Fermi-Dirac statistics and are called fermions. Systems of " He atoms (helium-4) and of He atoms (helium-3) provide an excellent illustration. The " He atom is a boson with spin 0 because the spins of the two protons and the two neutrons in the nucleus and of the two electrons are paired. The He atom is a fermion with spin because the single neutron in the nucleus is unpaired. Because these two atoms obey different statistics, the thermodynamic and other macroscopic properties of liquid helium-4 and liquid helium-3 are dramatically different. 

The A-particle eigenfunctions I v(l, 2,. .., A) in equation (8.47) are not properly symmetrized. For bosons, the wave function (1, 2,. .., N) must be symmetric with respect to particle interchange and for fermions it must be antisymmetric. Properly symmetrized wave functions may be readily con-... 

Thus, the two wave functions can at most differ by a unimodular complex number e1. It can be shown that the only possibilities occurring in nature are that either the two functions are identical (symmetric wave function, applies to particles called bosons which have inte-... 

There is no theoretical ground for this conclusion, which is a purely empirical result based on a variety of experimental measurements. However, it seems to apply everywhere and to represent a law of Nature, stating that systems consisting of more than one particle of half-integral spin are always represented by anti-symmetric wave functions. It is noted that if the space function is symmetrical, the spin function must be anti-symmetrical to give an anti-symmetrical product. When each of the three symmetrical states is combined with the anti-symmetrical space function this produces what is... 

For an iV-electron system anti-symmetrical wave functions are conveniently represented by determinants. If Wi(x,) represents an electronic wave function with space and spin components, a typical wave function for an A-electron system can be formulated as... 

Any determinant changes sign when any two columns are interchanged. Moreover, no two of the product functions (columns) can be the same since that would cause the determinant to vanish. Thus, in all nonvanishing completely anti-symmetric wave functions, each electron must be in a different quantum state. This result is known as Pauli s exclusion principle, which states that no two electrons in a many-electron system can have all quantum numbers the same. In the case of atoms it is noted that since there are only two quantum states of the spin, no more than two electrons can have the same set of orbital quantum numbers. 

It is noted that if ei = e2 the anti-symmetric wave function vanishes, ipa = 0. Two identical particles with half-spin can hence not be in the same non-degenerate energy state. This conclusion reflects Pauli s principle. Particles with integral spin are not subject to the exclusion principle (ips 0) and two or more particles may occur in the same energy state. 

Three independent anti-symmetric wave functions can be written for these two particles ... 

Particles having half-integral spin and requiring antisymmetric wave functions are called fermions particles having integral spin and requiring symmetric wave functions are called bosons. 

Particles can be classified as fermions or bosons. Fermions obey the Pauli principle and have antisymmetric wave functions and half-integer spins. (Neutrons, protons, and electrons are fermions.) Bosons do not obey the Pauli principle and have symmetric wave functions and integer spins. (Photons are bosons.)... 

The wave function for a system of TV identical particles is either symmetric or antisymmetric with respect to the interchange of any pair of the TV particles. Elementary or composite particles with integral spins (s = 0, 1,2,. ..) possess symmetric wave functions, while those with half-integral spins ( =, . ..)... 

For a spherically symmetrical wave function of an electron in an s state (s orbital, s wave function) there is naturally no preferred direction. This is, however, quite otherwise for a p orbital which, as explained, has as far as the angular dependence is concerned the shape of a dumb bell consisting of two spheres on one another, directed along the x>y or z axis (Fig. 11). The charge distribution, cp 2, has the shape of a figure 8. 

For a pair of magnetically equivalent nuclei (e.g., two protons), symmetrized wave functions must be used, as in Eqs. 6.42 and 6.44, and the resulting calculation gives eigenvalues and transitions as described in Section 6.11, except that the adjacent symmetric energy levels are separated by... 

Use the dipolar Hamiltonian for a homonuclear spin system, Eq. 7.6, together with symmetrized wave functions from Chapter 6, to compute the energy levels and spectrum for a single H20 molecule in a solid. 

These functions comprise the linear combination of the wave functions which we have already discussed, together with terms representing the spin wave functions. It has been shown in the last section that the formation of the chemical bond is described by the symmetric wave function... 

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