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Probability densities

When providing input for the STOMP calculation a range of values of porosity (and all of the other input parameters) should be provided, based on the measured data and estimates of how the parameters may vary away from the control points. The uncertainty associated with each parameter may be expressed in terms of a probability density function, and these may be combined to create a probability density function for STOMP. [Pg.159]

It is common practice within oil companies to use expectation curves to express ranges of uncertainty. The relationship between probability density functions and expectation curves is a simple one. [Pg.159]

Figure 6.6 The probability density function and the expectation curve... Figure 6.6 The probability density function and the expectation curve...
To extract infomiation from the wavefimction about properties other than the probability density, additional postulates are needed. All of these rely upon the mathematical concepts of operators, eigenvalues and eigenfiinctions. An extensive discussion of these important elements of the fomialism of quantum mechanics is precluded by space limitations. For fiirther details, the reader is referred to the reading list supplied at the end of this chapter. In quantum mechanics, the classical notions of position, momentum, energy etc are replaced by mathematical operators that act upon the wavefunction to provide infomiation about the system. The third postulate relates to certain properties of these operators ... [Pg.7]

Figure Al.1.2. Probability density (v[/ vt/) for the n = 29 state of the hamionic oscillator. The vertical state is chosen as in figure A1.1.1. so that the locations of the turning points comcide with the superimposed potential fiinction. Figure Al.1.2. Probability density (v[/ vt/) for the n = 29 state of the hamionic oscillator. The vertical state is chosen as in figure A1.1.1. so that the locations of the turning points comcide with the superimposed potential fiinction.
The canonical distribution corresponds to the probability density for the system to be in a specific microstate with energy E- H, from it one can also obtain the probability P( ) that the system has an energy between E and E + AE i the density of states D E) is known. This is because, classically. [Pg.398]

I i i(q,01 in configuration space, e.g. as defined by the possible values of the position coordinates q. This motion is given by the time evolution of the wave fiinction i(q,t), defined as die projection ( q r(t)) of the time-dependent quantum state i i(t)) on configuration space. Since the quantum state is a complete description of the system, the wave packet defining the probability density can be viewed as the quantum mechanical counterpart of the classical distribution F(q- i t), p - P t)). The time dependence is obtained by solution of the time-dependent Schrodinger equation... [Pg.1057]

Figure A3.13.6. Time evolution of the probability density of the CH cliromophore in CHF after 50 fs of irradiation with an excitation wave number = 2832.42 at an intensity 7q = 30 TW cm. The contour... Figure A3.13.6. Time evolution of the probability density of the CH cliromophore in CHF after 50 fs of irradiation with an excitation wave number = 2832.42 at an intensity 7q = 30 TW cm. The contour...
Figure A3.13.9. Probability density of a microcanonical distribution of the CH cliromophore in CHF within the multiplet with cliromophore quantum nmnber V= 6 (A. g = V+ 1 = 7). Representations in configuration space of stretching and bending (Q coordinates (see text following (equation (A3.13.62)1 and figure A3.13.10). Left-hand side typical member of the microcanonical ensemble of the multiplet with V= 6... Figure A3.13.9. Probability density of a microcanonical distribution of the CH cliromophore in CHF within the multiplet with cliromophore quantum nmnber V= 6 (A. g = V+ 1 = 7). Representations in configuration space of stretching and bending (Q coordinates (see text following (equation (A3.13.62)1 and figure A3.13.10). Left-hand side typical member of the microcanonical ensemble of the multiplet with V= 6...
Figure A3.13.10. Time-dependent probability density of the isolated CH clnomophore in CHF. Initially, tlie system is in a Fenni mode with six quanta of stretching and zero of bending motion. The evolution occurs within the multiplet with chromophore quantum number A = 6 = A + 1 = 7). Representations are given... Figure A3.13.10. Time-dependent probability density of the isolated CH clnomophore in CHF. Initially, tlie system is in a Fenni mode with six quanta of stretching and zero of bending motion. The evolution occurs within the multiplet with chromophore quantum number A = 6 = A + 1 = 7). Representations are given...
Figure A3.13.il. Illustration of the time evolution of redueed two-dimensional probability densities I I and I I for the exeitation of CHD between 50 and 70 fs (see [154] for further details). The full eurve is a eut of tire potential energy surfaee at the momentary absorbed energy eorresponding to 3000 em during the entire time interval shown here (as6000 em, if zero point energy is ineluded). The dashed eurves show the energy uneertainty of the time-dependent wave paeket, approximately 500 em Left-hand side exeitation along the v-axis (see figure A3.13.5). The vertieal axis in the two-dimensional eontour line representations is... Figure A3.13.il. Illustration of the time evolution of redueed two-dimensional probability densities I I and I I for the exeitation of CHD between 50 and 70 fs (see [154] for further details). The full eurve is a eut of tire potential energy surfaee at the momentary absorbed energy eorresponding to 3000 em during the entire time interval shown here (as6000 em, if zero point energy is ineluded). The dashed eurves show the energy uneertainty of the time-dependent wave paeket, approximately 500 em Left-hand side exeitation along the v-axis (see figure A3.13.5). The vertieal axis in the two-dimensional eontour line representations is...
The wave paeket motion of the CH eliromophore is represented by simultaneous snapshots of two-dimensional representations of the time-dependent probability density distribution... [Pg.1074]

Figure A3.13.13. Illustration of the time evolution of redueed two-dimensional probability densities I and I I for the isolated CHD2T (left-hand side) and CHDT, (right-hand side) after 800 fs of free evolution. At time 0 fs the wave paekets eorresponded to a loeafized, ehiral moleeular stnieture (from [154]). See also text and figure A3.13.il. Figure A3.13.13. Illustration of the time evolution of redueed two-dimensional probability densities I and I I for the isolated CHD2T (left-hand side) and CHDT, (right-hand side) after 800 fs of free evolution. At time 0 fs the wave paekets eorresponded to a loeafized, ehiral moleeular stnieture (from [154]). See also text and figure A3.13.il.
Figure Bl.2.4. Lowest five hannonic oscillator wavefimctions / and probability densities i if. Figure Bl.2.4. Lowest five hannonic oscillator wavefimctions / and probability densities i if.
Here pyy r ) represents the probability density for finding the 1 electrons at r, and e / mutual Coulomb repulsion between electron density at r and r. ... [Pg.2159]

The magnitude and shape of such a mean-field potential is shown below [21] in figure B3.1.4 for the two 1 s electrons of a beryllium atom. The Be nucleus is at the origin, and one electron is held fixed 0.13 A from the nucleus, the maximum of the Is orbital s radial probability density. The Coulomb potential experienced by the second electron is then a function of the second electron s position along the v-axis (coimecting the Be nucleus and the first electron) and its distance perpendicular to the v-axis. For simplicity, this second electron... [Pg.2159]

Figure B3.1.5. Probability (as a fimction of angle) for finding the seeond eleetron in He when both eleetrons are loeated at the maximum in the Is orbital s probability density. The bottom line is that obtained using a Hylleraas-type fiinetion, and the other related to a highly-eorrelated multieonfigurational wavefLinetion. After [22],... Figure B3.1.5. Probability (as a fimction of angle) for finding the seeond eleetron in He when both eleetrons are loeated at the maximum in the Is orbital s probability density. The bottom line is that obtained using a Hylleraas-type fiinetion, and the other related to a highly-eorrelated multieonfigurational wavefLinetion. After [22],...
By expressing the mean-field interaction of an electron at r with the N- 1 other electrons in temis of a probability density pyy r ) that is independent of the fact that another electron resides at r, the mean-field models ignore spatial correlations among the electrons. In reality, as shown in figure B3.T5 the conditional probability density for finding one ofA - 1 electrons at r, given that one electron is at r depends on r. The absence of a spatial correlation is a direct consequence of the spin-orbital product nature of the mean-field wavefiinctions... [Pg.2163]

Figure B3.3.5. Energy distributions. The probability density is proportional to the product of the density of states and the Boltzmaim factor. Figure B3.3.5. Energy distributions. The probability density is proportional to the product of the density of states and the Boltzmaim factor.
A way of looking at the points raised in the previous section is to compare energy distributions in two systems whose free energies we wish to relate. In particular, consider measuring, in a simulation of system 0, the fiinction Pq(AE), i.e., the probability density per unit AE of configurations for which and differ by the... [Pg.2263]

Bruce A D 1981 Probability density functions for collective coordinates in Ising-like systems J. Phys. O Soiid State Phys. 14 3667-88... [Pg.2285]

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A Probability Density Functions (PDF)

A compendium of some common probability density functions

Activated dynamics probability density

Angular probability density function

Asymptotic probability density

Atomic orbitals probability densities

Atomic orbitals radial probability density plots

Brownian motion probability density

Canonical equilibrium probability density

Circulation time probability density

Classical equilibrium probability density

Combined probability density

Conditional density probability

Current Density Probability Conservation Theorem

Data Analysis Using Absorption Probability Density (Example Guanidinium Nitroprusside)

Data interpretation probability density function

Density probability, resonantly coupled

Density radial probability

Distribution function and probability density in phase space

Dynamical quantities probability density

Electron probability density function

Electron probability density maps

Electron probability density maxima

Electronic wavefunction and probability density function

Energy eigenfunctions probability density

Energy probability density

Energy spectrum probability density

Energy spectrum probability density distribution

Energy transfer probability density

Ensemble probability density

Estimation of Probability Densities

Excitation probability densities

Expansion of probability density

Exponential probability density

Figures FIGURE 5.7 Bayesian posterior probability density of the fraction affected at median log (HC5) for cadmium

Flame probability density functions

Flux probability density

Function reaction probability density

Gaussian distribution probability density function

Gaussian probability density

Gaussian probability density, molecular

Harmonic oscillator probability density

Helium atom probability density

Hydrogen atom electron probability density

Hydrogen probability density

Initial probability density

Joint probability densities

Likelihood or probability density

Limiting behavior of the probability density

Lognormal distribution probability density function

Marginal probability density

Mathematical concept probability density function

Micro-Probability Density Function Methods

Multivariable joint probability density

Multivariable joint probability density function

Normal distribution probability density function

Normalized probability density function

One-dimensional probability density

One-point probability density function

Orientational probability density

Particle size probability density

Particle size probability density distribution

Phase space probability density

Phase-space conditional probability density

Phase-space conditional probability density function

Polymer probability density

Posterior probability density

Posterior probability density function

Primary Probability density function

Probability (or Failure) Density Function

Probability Density Function Modeling

Probability current density

Probability densities and quantum-mechanical analogy

Probability densities quantum-mechanical tunneling

Probability densities, permutational symmetry

Probability density classical

Probability density curve, table

Probability density curves

Probability density defining wavefunction

Probability density definition

Probability density description

Probability density distribution

Probability density distribution Normal

Probability density distribution bubble

Probability density distribution crystal

Probability density distribution function for the maximum information entropy

Probability density distribution function for velocity fluctuation

Probability density distribution particle size function

Probability density distribution residence time

Probability density function

Probability density function (PDF

Probability density function (PDF method

Probability density function , equilibrium

Probability density function Brownian motion

Probability density function Gaussian

Probability density function Monte Carlo simulation

Probability density function chaotic flow

Probability density function combustion models

Probability density function configurational

Probability density function correlation functions

Probability density function equation

Probability density function example

Probability density function explained

Probability density function interpretation

Probability density function method

Probability density function model

Probability density function presumed

Probability density function reaction rate calculation

Probability density function theories

Probability density function transported

Probability density function turbulent diffusion flame

Probability density function valid

Probability density function value

Probability density function, local control

Probability density function, range

Probability density function, single molecule

Probability density functions and expectation curves

Probability density functions components

Probability density in phase space

Probability density moments

Probability density of angular momenta distribution

Probability density probabilities

Probability density quantum mechanical

Probability density radial distribution function

Probability density single-mode

Probability density symmetry

Probability density terms

Probability density volume-weighted

Probability density, electronic

Probability density, long time trajectories

Probability density, of electron

Probability electron density

Probability proton density

Probability, cumulative density

Probability-density functionals

Probability-density functionals, turbulence

Probability-density functions conditioned

Probability-density functions joint

Probability-density functions marginal

Probability-density functions properties

Probability-density-function analysis

Properties of probability-density functions

Reduced probability density

Residence time probability density

Schrodingers Conditional Probability Density

Size-composition probability density

Size-composition probability density function

Special Probability Densities

Suspensions conditional probability density

The Gibbs-Boltzmann Probability Density

The probability density function

The wavefunctions, probability densities and shape of

Transition probability density

Transition probability density calculation

Turbulence probability density functions

Use of Probability Density Curves

Velocity probability density function

Vibrational probability density

Waiting-time probability density function

Wave packet probability density

Wavefunction probability density

Wavefunctions probability densities and

Weibull probability density function

Weight basis probability density function

Zero-order approximation probability density

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