Dynamical quantities probability density

Suppose we wish to measure the position of a particle whose wave function is W(jc, i). The Bom interpretation of F(x, as the probability density for finding the associated particle at position x at time t implies that such a measurement will not yield a unique result. If we have a large number of particles, each of which is in state /) and we measure the position of each of these particles in separate experiments all at some time t, then we will obtain a multitude of different results. We may then calculate the average or mean value x) of these measurements. In quantum mechanics, average values of dynamical quantities are called expectation values. This name is somewhat misleading, because in an experimental measurement one does not expect to obtain the expectation value. 

The general relations among the coefficients - and Dy are presented elsewhere [179]. The quantities yj and y2 are the damping constants for the fundamental and second- harmonic modes, respectively. In Eq.(82) we shall restrict ourselves to the case of zero-frequency mismatch between the cavity and the external forces (ff>i — ff> = 0). In this way we exclude the rapidly oscillating terms. Moreover, the time x and the external amplitude have been redefined as follows x = Kf and 8F =. The s ordering in Eq.(80) which is responsible for the operator structure of the Hamiltonian allows us to contrast the classical and quantum dynamics of our system. If the Hamiltonian (77)-(79) is classical (i.e., if it is a c number), then the equation for the probability density has the form of Eq.(80) without the s terms ... 

For over a century it has been known that two classes of variables have to be distinguished the microscopic variables, which are functions of the points of ClN and thus pertain to the detailed positions and motions of the molecules and the macroscopic variables, observable by operating on matter in bulk, exemplified by the temperature, pressure, density, hydro-dynamic velocity, thermal and viscous coefficients, etc. And it has been known for an equally long time that the latter quantities, which form the subject of phenomenological thermo- and hydrodynamics, are definable either in terms of expected values based on the probability density or as gross parameters in the Hamiltonian. But at once three difficulties of principle have been encountered. 

We recall some basic results of quantum dynamics [3], First, the state of the system and the time evolution can be expressed in a generalized (Dirac) notation, which is often very convenient. The state at time t is specified by x(t)) with the representations x(-Rjf) = (R x t)) and x P,t) = (P x(t)) in coordinate and momentum space, respectively. Probability is a concept that is inherent in quantum mechanics. (R x(t)) 2 is the probability density in coordinate space, and (-P x(f) 2 is H e same quantity in momentum space. The time evolution (in the Schrodinger picture) can be expressed as... 

In a refined form of the theory [37] the same quantity features as the well known quantum potential. The notion of a particle emerges in this theory in the form of a highly localized inhomogeneity that moves with the local fluid velocity, v(x, t), thus as a stable dynamic structure that exists in the fluid, for example, as a small stable vortex or a pulse-like distortion. To explain why the causal theory needs probability densities it is argued [37] that the Madelung fluid must experience more or less random fluctuations in its motion to account for irregular turbulence. The turbulence necessitates a wave theory to describe the motion of vortices embedded in the fluid. The particle velocity is therefore not exactly VS/m, nor is the density exactly... 

Whether the inactive region is a true continuum (e.g., photofragmentation) or a quasi-continuum comprised of an enormous density of rigorously bound eigenstates (polyatomic molecule dynamics, Section 9.4.14) is often of no detectable consequence. The dynamical quantities discussed in Section 9.1.4 (probability density, density matrix, autocorrelation function, survival probability, transfer probability, expectation values of coordinates and conjugate momenta) describe the active space dynamics without any reference to the detailed nature of the inactive space. 

The quantum mechanical probability density is compared to an ensemble of classical trajectories in Fig. 32. Both quantities carry the same dynamical features, which, again, supports the intuitive picture evolving from LCT theory. It is seen that a bifurcation occurs, where two wavepackets move out of phase with each other (and likewise do sets of classical trajectories). This then has the consequence that, upon reaching the continuum, the rotational motion is not directional to 100%. Rather, for the present parameters, one finds a ratio of 2.2 in favor of the counterclockwise rotation. 

Simulations using BOMD or CPMD give as result a set of snapshots of the system, as coordinates, velocities, and forces. Exploitation of this information allows to know statistical quantities as well as dynamic quantities. As an example, the radial distribution function gives the probability to find a pair of atoms a distance r apart, relative to the probability for a random distribution at the same density [27]... 

As in the Langevin description, the dynamic description of noninteractmg Brownian particles moving in a fluid in stationary flow, demands a mesoscopic treatment in terms of the probability density /(r,u, f). The evolution in time of this quantity is governed by the continuity equation... 

In Equation (5), we can first notice (i) the factor 1/r6 which makes the spectral density very sensitive to the interatomic distance, and (ii) the dynamical part which is the Fourier transform of a correlation function involving the Legendre polynomial. We shall denote this Fourier transform by (co) (we shall dub this quantity "normalized spectral density"). For calculating the relevant longitudinal relaxation rate, one has to take into account the transition probabilities in the energy diagram of a two-spin system. In the expression below, the first term corresponds to the double quantum (DQ) transition, the second term to single quantum (IQ) transitions and the third term to the zero quantum (ZQ) transition. 

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