Momentum probability distribution

Figure 8. The angular momentum probability distribution of diatomic molecule in absence of an electric field — case a and when electric field is applied along the c axis — case b.

We will refer to this model as to the semiclassical QCMD bundle. Eqs. (7) and (8) would suggest certain initial conditions for /,. However, those would not include any momentum uncertainty, resulting in a wrong disintegration of the probability distribution in g as compared to the full QD. Eor including an initial momentum uncertainty, a Gaussian distribution in position space is used... 

This is the probability of finding particle 1 with coordinate rx and velocity vx (within drx and dVj), particle 2 with coordinate r2 and velocity v2 (within phase space with velocity rather than momentum for convenience since only one type of particle is being considered, this causes no difficulties in Liouville s equation.) The -particle probability distribution function ( < N) is... 

It should be indicated that a probability density function has been derived on the basis of maximum entropy formalism for the prediction of droplet size distribution in a spray resulting from the breakup of a liquid sheet)432 The physics of the breakup process is described by simple conservation constraints for mass, momentum, surface energy, and kinetic energy. The predicted, most probable distribution, i.e., maximum entropy distribution, agrees very well with corresponding empirical distributions, particularly the Rosin-Rammler distribution. Although the maximum entropy distribution is considered as an ideal case, the approach used to derive it provides a framework for studying more complex distributions. 

At this point, one may wonder why there is an interest in the atomic momentum densities and their nature and what sort of information does one derive from them. In a system in which all orientations are equally probable, the full three-dimensional (3D) momentum density is not experimentally measurable, but its spherical average is. The moments of the atomic momentum density distributions are of experimental significance. The moments and the spherically averaged momentum densities are defined in the equations below. 

The information entropy of a probability distribution is defined as S[p(] = — p, In ph where p, forms the set of probabilities of a distribution. For continuous probability distributions such as momentum densities, the information entropy is given by. S yj = - Jy(p) In y(p) d3p, with an analogous definition in position space... 

Theoretical investigations of the problem were carried out on the base of the mathematical model, combining both deterministic and stochastic approaches to turbulent combustion of organic dust-air mixtures modeling. To simulate the gas-phase flow, the k-e model is used with account of mass, momentum, and energy fluxes from the particles phase. The equations of motion for particles take into account random turbulent pulsations in the gas flow. The mean characteristics of those pulsations and the probability distribution functions are determined with the help of solutions obtained within the frame of the k-e model. 

The maximum entropy method (MEM) is an information-theory-based technique that was first developed in the field of radioastronomy to enhance the information obtained from noisy data (Gull and Daniell 1978). The theory is based on the same equations that are the foundation of statistical thermodynamics. Both the statistical entropy and the information entropy deal with the most probable distribution. In the case of statistical thermodynamics, this is the distribution of the particles over position and momentum space ( phase space ), while in the case of information theory, the distribution of numerical quantities over the ensemble of pixels is considered. 

Readers familiar with Fourier transforms may be interested to know that the probability distribution of the momentum of the particle in state is given by ( 2, where (/) denotes the Fourier transform of .)... 

In the general approach to classical statistical mechanics, each particle is considered to occupy a point in phase space, i.e., to have a definite position and momentum, at a given instant. The probability that the point corresponding to a particle will fall in any small volume of the phase space is taken proportional to die volume. The probability of a specific arrangement of points is proportional to the number of ways that the total ensemble of molecules could be permuted to achieve the arrangement. When this is done, and it is further required that the number of molecules and their total energy remain constant, one can obtain a description of the most probable distribution of the molecules in phase space. Tlie Maxwell-Boltzmann distribution law results. 

Use Equation 6.11 to show that changing the definition of the zero point of energy (which is arbitrary, because potential energy is included) by an amount A changes i// (t) by a factor e lAt. Also show that this arbitrary choice has no effect on the probability distribution, or on the expectation values of position, momentum, or kinetic energy. 

It should however be noted that the eigenfunctions of the hamiltonian are not eigenfunctions of the linear momentum operator. Accordingly, a measurement of the momentum does not lead to p = 2mE , but to a probability distribution as shown in Fig. 2.3 (refs. 18 and 19). It can be noted that the most probable momentum is not p , when the particle is in a state except when n becomes large. Then the quantum and classical descriptions are similar. 

Fig. 2.3 Probability distribution of the linear momentum p for a particle in a one-dimensional box. (Adapted with permission from the Journal of Chemical Education, 72, 148 (1995) copyright 1995, EMvision of Chemical Education Inc., ref. 18.)...

Figure 6. The probability distribution of molecular axes for two step laser excitation of diatomic- molecules. For case a we have angular momentum quantum number sequenc e 7-8-9 in molecular transitions, but for case b we have 7-8-8 sequence. Mutual laser polarization for both cases is shown in the figure.

If one is interested in a spatial distribution of angulcu momentum created by laser radiation, then there is a method how to make a transition from (juantum density matrix to the continuous angular momentum spatial distribution probability density. As it is shoAA n in [21], a classical probability density pci 6,if) for angulm momentum spatial distribution can be connected to the density matrix elements Imm At the J —> OC limit these elements can be considered as coefficients of the Fourier expansion of a classical probabihty densitj pd 9, p)... 

At last, we can resolve the paradox between de Broglie waves and classical orbits, which started our discussion of indeterminacy. The indeterminacy principle places a fundamental limit on the precision with which the position and momentum of a particle can be known simultaneously. It has profound significance for how we think about the motion of particles. According to classical physics, the position and momentum are fully known simultaneously indeed, we must know both to describe the classical trajectory of a particle. The indeterminacy principle forces us to abandon the classical concepts of trajectory and orbit. The most detailed information we can possibly know is the statistical spread in position and momentum allowed by the indeterminacy principle. In quantum mechanics, we think not about particle trajectories, but rather about the probability distribution for finding the particle at a specific location. 

As described in Ref. [25], the Hartree approach has been applied to get energies and density probability distributions of Br2(X) 4He clusters. The lowest energies were obtained for the value A = 0 of the projection of the orbital angular momentum onto the molecular axis, and the symmetric /V-boson wavefunction, i.e. the Eg state in which all the He atoms occupy the same orbital (in contrast to the case of fermions). It stressed that both energetics and helium distributions on small clusters (N < 18) showed very good agreement with those obtained in exact DMC computations [24],... 

Figure 1. Joint probability distribution of (a) momenta and (6) position of EPR-pair ensembles. If one measures the position of particle 1, one can predict the position of particle 2 with uncertainty w Ax-, whereas if one measures momentum of particle 1, one can predict the momentum of particle 2 with uncertainty Ap+.

In the example, we choose x = 0 and p 0 and consider only the broadening of the momentum distribution V (p) by varying its width a. The results can be seen in Fig. 2 The transition probability for a = 0.1 is shown in Fig. 2 (a). The distribution is centered around n - 5 and has the Poisson-like shape of a Franck-Condon type of interaction, as is expected for a quasiconstant interaction with small a. With increasing a, the maximum is shifted slightly towards higher values of n. But the most obvious effect is the broadening of the probability distribution. For a — 00, an extremely narrow position wave packet with its center at x = 0 is excited on the upper potential. In this limit, the probability distribution is... 

Thus, the solution (2.144) oscillates with frequency ty in a way that resembles the classical motion First, the expectation values of the position and momentum oscillate, as implied by Eqs (2.145), according to the corresponding classical equations of motion. Second, the wavepacket as a whole executes such oscillations, as can be most clearly seen from the probability distribution... 

---