Symmetry properties parity

The experiments show that in the ground states of nuclei the electric moments of 2 order with odd I and all 2-pole magnetic moments with even I are missing. Only electric monopole, magnetic dipole, electric quadrupole, magnetic octupole, etc., moments are observed in experiments. This means that the 2-pole character unambiguously determines whether the moment is electric or magnetic, and these latter attributes may be omitted. This rule can be viewed as a consequence of a symmetry property (parity) of the nuclear states. 

Table 11.3 General classification of nuclides significance of parity is related to symmetry properties of nnclear wave fnnctions. A nuclide is said to have odd or even parity if the sign of the wave fnnction of the system respectively changes or not with changing sign in all spatial coordinates (see Friedlander et ah, 1981 for more detailed treatment). The value assigned to dxA is appropriate for A > 80. For < 60, a value of 65 is more appropriate.

The coefficients A defined in Eq. 4.18 satisfy certain symmetry relationships [323, 391], From parity considerations, it follows that X +2.2 + L must be odd. Moreover, because the dipole operator is Hermitian, the expansion coefficients A are all real. The symmetry property... 

Even though (6.105) is only an approximation, its symmetry properties (e.g., its parity) are the same as those of the true molecular wave function, since the symmetry properties of the wave function follow rigorously from the symmetry of the true Hamiltonian H the perturbation H cannot change the overall symmetry of p. 

Parity, as used in nuclear science, refers to the symmetry properties of the wave function for a particle or a system of particles. If the wave function that specifies the state of the system is Tir,. v) where r represents the position coordinates of the system (x, y, z) and s represents the spin orientation, then (r,. v) is said to have positive or even parity when... 

The parity of a system is related to the symmetry properties of the spatial portion of the wave function. Another important quantum mechanical property of a system of two or more identical particles is the effect on the wave function of exchanging the coordinates of two particles. If no change in the wave function occurs when the spatial and spin coordinates are exchanged, we say the wave function is symmetric... 

This expression for /-electrons can be derived using the phase relations established for isoscalar parts of factorized CFP with different parities of the seniority number [24]. It turned out [91] that phase (16.55) provides sign relations between the CFP in the tables for d- and /-electrons, but it is unsuitable for the p-electrons. In this connection, in what follows all the relationships derived using the symmetry properties under transposition of the quantum numbers of spin and quasispin are provided up to the sign. 

Then, because of the symmetry properties given by Eqs. (247) and (248), it appears that the parity operator exchanges the two Hamiltonians ... 

For group-theoretical selection of nonzero matrix elements of the hyperpolarizability components, one has to know the symmetry properties of the operator / < ( w). Owing to the definition (153), /3y (w) is symmetric with respect to the permutation of the indices./ and k, but it has no definite symmetry with respect to the permutation of all the indices and has no definite parity with respect to the operation of time reversal ... 

The parity (see Box 1.9) of a rr-orbital is m, and that of a it -orbital is g. These labels are the reverse of those for a and a -orbitals, respectively (Figure 1.20). The overlap between two Py atomic orbitals generates an MO which has the same symmetry properties as that derived from the combination of the two p,. atomic orbitals, but the ttuipy) MO lies in a plane perpendicular to that of the TtuiPx) MO. The tr ipx) and TTuipy) MOs lie at the same energy they are degenerate. The Ttg py) and Ttg (j>x) MOs are similarly related. 

This last equation shows that if e + ev, then C ft) = 0, thus proving the theorem. This means that in a system such that the Hamiltonian has inversion symmetry, properties of different parity are totally uncorrelated for all time. The initial value of Eq. 

To illustrate the general applicability of the relaxation equations of Section 11.4 let us study the simple case of a single conserved variable A(q, t) which has the form given by Eq. (11.5.32). The property aj of the jth molecule is presumed to have definite time-reversal symmetry and parity. 

In addition to the orbital and spin rotations, one can also consider other symmetry properties. In the case of Russell-Saunders coupling, considerations of parity further limits the possible I values such that only even (odd) values are allowed when 7 and 7 have equal (different) parity quantum numbers. 

Nuclear states are described in quantum mechanics by eigenvalues and eigenfunctions of the Hamilton operator, to which definite quantum numbers belong. The parity of the wave function is its symmetry property under inversion through the origin of the coordinate system, i.e., when the coordinates x, y, zare transformed into —x, —y, —z. This transformation turns a right-handed coordinate system into a left-handed one or vice versa. 

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