Mixing of wave functions

The 2.56 MeV resonance is not shown in the alpha particle transitions to other low states of Li , including the ground state. This can be explained if the 3.58 MeV state is identified as the first T = 1 state of Li , since then the resonant state of at 8.89 MeV would also be predominantly T = l, and a break-up to T = 0 states, such as Li + a or Be + d would be forbidden, except in so far as mixing of wave functions occurs in the compound state or isotopic spin impurity exists in the wave functions of the T — Q nuclei concerned. The conservation of isotopic spin in the Be ( a) reaction has been especially discussed by Malm and Inglis. The magnitude of the resonant cross section is used by Mackin to predict that / 2 for the 8.89 MeV state of B , and the observed width of the... 

These are produced by autoionization transitions from highly excited atoms with an inner vacancy. In many cases it is the main process of spontaneous de-excitation of atoms with a vacancy. Let us recall that the wave function of the autoionizing state (33.1) is the superposition of wave functions of discrete and continuous spectra. Mixing of discrete state with continuum is conditioned by the matrix element of the Hamiltonian (actually, of electrostatic interaction between electrons) with respect to these functions. One electron fills in the vacancy, whereas the energy (in the form of a virtual photon) of its transition is transferred by the above mentioned interaction to the other electron, which leaves the atom as a free Auger electron. Its energy a equals the difference in the energies of the ion in initial and final states ... 

We have shown that g-factors of excited states in nuclei far from stability can provide important nuclear structure information, such as purity of wave functions, configuration-mixing, number of active protons, dissipation of shell closures, values of g, gv. 

This can give rise to difficulties in tightly coupled spin systems when there may be virtually complete mixing of some pairs of wave functions. (40) Most of the problems can be avoided, however, by an appropriate new choice of wave functions (41) and this approach should always be... 

Resonance hybrids Various possible chemical structures of molecules, each with identical atomic connectivity, but differing in the disposition of electrons. The wave function of the molecule is approximately represented by mixing the wave functions of the contributing structures. The energy calculated for such a mixture is lower, presumably because the representation is more nearly correct than it would be if formally represented by a single structure. 

The transition at 2320 cm-1 in the (8 OH, 7 OH) plane shows four maxima of intensity specific to the combination 000) —> 011) (this transition is practically invisible in the infrared or Raman). Finally, a superposition of the OH stretching, 000) — 1100), and 5 OH overtone, 000) —> 020), is observed at 2560 cm-1. The observed intensity ratio for fundamental and overtone is close to that given by Eq. (6). There is no evidence for Fermi resonance mixing the wave functions [Novak 1963 Lucazeau 1973],... 

For a macroscopic sample it is necessary to define a different set of wave functions because the spins are in a mixed state. The mixed state indicates that the wave functions of a particular nuclear spin are subject to additional molecular contributions that might differ over the whole sample. The expectation value of a mixed state now uses the averaged coefficients and is... 

Hybrid orbitals are derived by mixing atomic wave functions. They are used to explain the geometrical shapes of molecules. Hybridization provides several advantages for bonding. With the larger lobe of the hybrid... 

The mixed-state character of a trajectory outside a non-adiabatic region is a serious weakness of the method. As the time-dependent wave function does not... 

Thus, the neglect of the off-diagonal matrix elements allows the change from mixed states of the nuclear subsystem to pure ones. The motion of the nuclei leads only to the deformation of the electronic distribution and not to transitions between different electronic states. In other words, a stationary distribution of electrons is obtained for each instantaneous position of the nuclei, that is, the elechons follow the motion of the nuclei adiabatically. The distribution of the nuclei is described by the wave function x (R i) in the potential V + Cn , known as the proper adiabatic approximation [41]. The off-diagonal operators C n in the matrix C, which lead to transitions between the states v / and t / are called operators of nonadiabaticity and the potential V = (R) due to the mean field of all the electrons of the system is called the adiabatic potential. 

For the case of intramolecular energy transfer from excited vibrational states, a mixed quantum-classical treatment was given by Gerber et al. already in 1982 [101]. These authors used a time-dependent self-consistent field (TDSCF) approximation. In the classical limit of TDSCF averages over wave functions are replaced by averages over bundles of trajectories, each obtained by SCF methods. 

However, because of the avoided crossing of the potential energy curves the wave functions of Vq and Fi are mixed, very strongly at r = 6.93 A and less strongly on either side. Consequently, when the wave packet reaches the high r limit of the vibrational level there is a chance that the wave function will take on sufficient of the character of Na + 1 that neutral sodium (or iodine) atoms may be detected. 

---