Probability amplitudes

Marcus R A 1970 Extension of the WKB method to wave functions and transition probability amplitudes (S-matrix) for inelastic or reactive collisions Chem. Phys. Lett. 7 525-32... 

Narevicius E, Neuhauser D, Korsch H J and Moiseyev M 1997 Resonances from short time complex-scaled cross- correlation probability amplitudes by the filter-diagonalization method Chem. Phys. Lett. 276 250... 

The END equations are integrated to yield the time evolution of the wave function parameters for reactive processes from an initial state of the system. The solution is propagated until such a time that the system has clearly reached the final products. Then, the evolved state vector may be projected against a number of different possible final product states to yield coiresponding transition probability amplitudes. Details of the END dynamics can be depicted and cross-section cross-sections and rate coefficients calculated. 

The wave function T is a function of the electron and nuclear positions. As the name implies, this is the description of an electron as a wave. This is a probabilistic description of electron behavior. As such, it can describe the probability of electrons being in certain locations, but it cannot predict exactly where electrons are located. The wave function is also called a probability amplitude because it is the square of the wave function that yields probabilities. This is the only rigorously correct meaning of a wave function. In order to obtain a physically relevant solution of the Schrodinger equation, the wave function must be continuous, single-valued, normalizable, and antisymmetric with respect to the interchange of electrons. 

Quantum Cellular Automata (QCA) in order to address the possibly very fundamental role CA-like dynamics may play in the microphysical domain, some form of quantum dynamical generalization to the basic rule structure must be considered. One way to do this is to replace the usual time evolution of what may now be called classical site values ct, by unitary transitions between fe-component complex probability- amplitude states, ct > - defined in sncli a way as to permit superposition of states. As is standard in quantum mechanics, the absolute square of these amplitudes is then interpreted to give the probability of observing the corresponding classical value. Two indepcuidently defined models - both of which exhibit much of the typically quantum behavior observed in real systems are discussed in chapter 8.2,... 

There is a qualitative universality in the quantum behavior of class-3 rules, whose threshold plots typically consist of strong local oscillation patterns. Although clearly a maJiifestation of the fundamental additivity of probability amplitudes, the majority of patterns also possess distinctive local regularities by which evolutions defined by particular rules can be uniquely identified characteristic features of the... 

How might the interaction between two discrete particles be described by a finite-information based physics Unlike classical mechanics, in which a collision redistributes the particles momentum, or quantum mechanics, which effectively distributes their probability amplitudes, finite physics presumably distributes the two particles information content. How can we make sense of the process A scatters J5, if B s momentum information is dispersed halfway across the galaxy [minsky82]. Minsky s answer is that the universe must do some careful bookkeeping, ... 

Figure 4-8. 1NDO/SCI simulation of the wavcfunclion y/(x,xi, = 16, chain I) of the lowest charge transfer-excited stale in a cofacial dimer formed by two five-ring PPV oligomers separated by 4A. Ili/(x,x/, - 16, chain 1) represents the probability amplitude in finding an electron on a given site xt. assuming the hole is centered on site 16 of chain I. The site labeling is the same as that reported on top of Figure 4-7.

The components = are then the probability amplitudes that if a position measurement is made at time t, the particle will be found at x more precisely, 2d3x is the probability that the particle will be found in the volume element d3x about x. In the q representation, Eq. (9-40) becomes... 

According to the usual rules of quantum mechanics the probability amplitude for finding a particle described by the amplitude (f>(x) to be localized at y is then given by the scalar product, ) This quantity is readily computed and one verifies that... 

For a partiole with wave function (k) it is the Fourier transform of 4 k + m (k) which gives the probability amplitude for finding the particle localised at a given point (see Eq. (9-104)). It is therefore natural to inquire as to the properties of the operator UxU 1, x = Vk, where V is the (nonunitary) operator such that if... 

The s-states have spherical symmetry. The wave functions (probability amplitudes) associated with them depend only on the distance, r from the origin (center of the nucleus). They have no angular dependence. Functionally, they consist of a normalization coefficient, Nj times a radial distribution function. The normalization coefficient ensures that the integral of the probability amplitude from 0 to °° equals unity so the probability that the electron of interest is somewhere in the vicinity of the nucleus is unity. 

The d-state probability amplitudes (in two dimensions) are shaped like cloverleafs. For the principal quantum numbers, n = 1 and n = 2, they lie too close to the nuclei of atoms to interact when the atoms are spaced by s-type wave functions. Only for n = 3 and greater, do they extend far enough from... 

Moreover, the extragalactic object 3cl23 was observed to look for the H,C236o. in absorption, and the upper level (3a) for Ti/Tc absorption of 1.5% is achieved. So far, that is rougher than Dr. Matveenko has observed (private communication, 27.04.04, at conference in Pushchino, Russia) the H78a absorption line toward 3c345 quasar (RT100, Bonn) with, probably, amplitude of 0.5%... 

I function which carries maximum information about that system. Definition of the -function itself, depends on a probability aggregate or quantum-mechanical ensemble. The mechanical state of the systems of this ensemble cannot be defined more precisely than by stating the -function. It follows that the same -function and hence the same mechanical state must be assumed for all systems of the quantum-mechanical ensemble. A second major difference between classical and quantum states is that the -function that describes the quantum-mechanical ensemble is not a probability density, but a probability amplitude. By comparison the probability density for coordinates q is... 

Quasienergies from Short Time Cross-Correlation Probability Amplitudes by the Filter-Diagonalization Method. 

---