Probabilistic descriptions

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. 

Nature In monitoring a moving threadhne, one criterion of quality would be the frequency of broken filaments. These can be identified as they occur through the threadhne by a broken-filament detector mounted adjacent to the threadhne. In this context, the random occurrences of broken filaments can be modeled by the Poisson distribution. This is called a Poisson process and corresponds to a probabilistic description of the frequency of defects or, in general, what are called arrivals at points on a continuous line or in time. Other examples include ... 

Newtonian inochaiiics. Since it is impossible to completely specify the initial state of such a system, kinetic theory (contents itself with describing a smoothed version of the system. The smoothed version is simply one whore all exact state-information below some characteristic length and time is replaced by a probabilistic description. 

Thus, when the attention of the mathematicians of the time turned to the description of overdetermined systems, such as we are dealing with here, it was natural for them to seek the desired solution in terms of probabilistic descriptions. They then defined the best fitting equation for an overdetermined set of data as being the most probable equation, or, in more formal terminology, the maximum likelihood equation. 

MSN.97. B. Misra, 1. Prigogine, and M. Courbage, From deterministic dynamics to probabilistic descriptions, Proc. Natl. Acad. Sci. USA 76, 3607-3611 (1979). 

The carbon nuclei are to be identified with I, the proton nuclei with S, and the carbon-proton internuclear distance with r. The spectral density is the Fourier transform of a correlation function which is usually based on a probabilistic description of the motion modulating the dipole-dipole interactions. The spin-spin relaxation time, T2, is usually written directly as a function of spectral densities ( ). 

A complete probabilistic description of the system is obtained either by specifying all of the joint probability densities, pr, r = 1, 2, 3,..., or by specifying the singlet probability density p2 and all of the conditional... 

One question of interest in setting up a probabilistic description of a physical phenomenon is whether we are free to choose any form for the conditional probabilities which satisfy the necessary conditions (3a-3d). 

The physical state of a system with a varying number of particles is defined uniquely by a set of the population numbers j (f i),..., a(r ) = i/(f ) jv-Assuming the reaction is the Markov process, let us introduce the distribution functions (DF s) P( i/(r ) yv t) yielding a complete probabilistic description of the problem. The recurrent relation... 

The set of functions wTn rn> (m,m = 0,1,...) yields the complete probabilistic description of a system. The variation of a number of particles of both kinds is taken into account by the normalization condition... 

At this point we will, briefly, describe some of the fundamental qualitative differences between a quantum mechanical and a classical mechanical description. First of all, a trajectory R(t) is replaced by a wave packet, which implies that a deterministic description is replaced by a probabilistic description. x(R,t) 2 is a probability density, giving the probability of observing the nuclei at the position R at time t. In... 

The influence of the solvent may be described by the Langevin equation. Since it is a probabilistic description, we want to determine the probability density P(r,v t), where P(r,v t)dvdr is the probability of finding the particle in the position interval (r, r + dr) with velocity in the interval (v, v + dv) at time t. 

The problem of Brownian motion relates to the motion of a heavy colloidal particle immersed in a fluid made up of light particles. In Fig. 11.1.1 the trajectory of a Brownian particle is shown. The coordinates of a particle with diameter 2 /xm moving in water are observed every 30 s for 135 min. At the very first step in the argument one renounces an exact deterministic description of the motion and replaces it with a probabilistic description. 

Assuming the reaction is the Markov process, let us introduce the distribution functions (DF s) t) yielding a complete probabilistic description... 

The key new ideas of qnantnm mechanics include the quantization of energy, a probabilistic description of particle motion, wave-particle duality, and indeterminacy. These ideas appear foreign to ns because they are inconsistent with our experience of the macroscopic world. We have accepted them because they have provided the most comprehensive account of the behavior of matter and radiation and because the agreement between theory and the results of all experiments conducted to date has been astonishingly accurate. 

Brownian motion of particles is the governing phenomenon associated with transitions between states in the above examples as well as in the mathematical derivations in the following [4, p.203]. If we consider a particle as system and the states are various locations in the fluid which the particle occupies versus time, then the transition from one state to the other is treated by the well-known random walk model. In the latter, the particle is moving one step up or down (or, alternatively, right and left) in each time interval. Such an approach gives considerable insight into the continuous process and in many cases we can obtain a complete probabilistic description of the continuous process. 

Two additional conmiciits apply at tliis point. First, there is a conceptual difference between the probabilistic element in quantum and classic statistical physics. For instance, in quantum mechanics, the outcome of a measurement of properties even of a single particle can be known in principle only w ith a certain probability. In classic mechanics, on the other hand, a probabilistic element is usually introduced for many-particle systems where we would in principle bo able to specify the state of the system with absolute certainty however, in practice, this is not possible because we are dealing wdth too many degrees of freedom. Recourse to a probabilistic description within the framework of classic mechanics must therefore be regarded a matter of mere convenience. The reader should appreciate this less fundamental meaning of probabilistic concepts in classic as opposed to quantiun mechanics. 

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