Expectation value definition

Probability Density Function, Cumulative Distribution Function, and Expected Value Definitions... 

This definition is based on and proportional to the g-expectation value. However, it is more useful since it is not necessary to evaluate the partition function to compute an average. 

With this definition, the classical entropy per system equals the ensemble average of the expectation value of 8 in occupation number representation. 

By definition, the average or expectation value of x is just the sum over all possible values of x of the product of x and the probability of obtaining that value. Since x is a continuous variable, we replace the probability by the probability density and the sum by an integral to obtain... 

We see that the energy and time obey an uncertainty relation when At is defined as the period of time required for the expectation value of S to change by one standard deviation. This definition depends on the choice of the dynamical variable S so that At is relatively larger or smaller depending on that choice. If d(S)/dt is small so that S changes slowly with time, then the period At will be long and the uncertainty in the energy will be small. 

We remind our readers here that AE, as we have been using it in this derivation is, as you will recall, the difference between AE and AE 0 in equation 41-4 and the expected value in the statistical nomenclature is therefore 21/2 times as large as AE (due to the fact that it is the result of the difference between random variables with equal variance), a difference that should be taken note of when comparing results with the original definition of S/N in equation 41-2. 

Stereospecificity as applied to olefin metathesis may be considered in two ways (a) How does the cis/trans isomer ratio of a product olefin compare with its equilibrium ratio, or (b) how does this cis/trans value differ from 1.0, which is the statistically expected value in terms of probabilities. In the present discussion, the latter definition applies. 

Quantum mechanics applies to a segment of a system, that is, to an open system, if the segment is bounded by a surface of zero flux in the gradient vector field of the density. Thus the quantum mechanical and topological definitions of an atom coincide [1]. The quantum mechanical rules for determining the average value of a property for a molecule, as the expectation value of an associated operator, apply equally to each of its constituent atoms. 

In general, the one-point joint velocity PDF can be used to evaluate the expected value of any arbitrary function / (U) of U using the definition... 

In order to show that the expected-value and derivative operations commute, we begin with the definition of the derivative in terms of a limit 16... 

We start by considering an arbitrary measurable10 one-point11 scalar function of the random fields U and 0 Q U, 0). Note that, based on this definition, Q is also a random field parameterized by x and t. For each realization of a turbulent flow, Q will be different, and we can define its expected value using the probability distribution for the ensemble of realizations.12 Nevertheless, the expected value of the convected derivative of Q can be expressed in terms of partial derivatives of the one-point joint velocity, composition PDF 13... 

Since is deterministic, the expected value in this definition is with respect to the composition PDF. However,... 

Use of Equation (1) in numerical work requires a means of generating x(r, r i(o) as well as the average charge density. Direct variational methods are not applicable to the expression for E itself, due to use of the virial theorem. However, both pc(r) and x(r, r ico) (39-42, 109-112) are computable with density-functional methods, thus permitting individual computations of E from Eq. (1) and investigations of the effects of various approximations for x(r, r ico). Within coupled-cluster theory, x(r, r ico) can be generated directly (53) from the definition in Eq. (3) then Eq. (1) yields the coupled-cluster energy in a new form, as an expectation value. 

To gain an understanding of this mechanism, consider the Hamiltonian operator (H — Egl) with only two-body interactions, where Eg is the lowest energy for an A -particle system with Hamiltonian H and the identity operator I. Because Eg is the lowest (or ground-state) energy, the Hamiltonian operator is positive semi-definite on the A -electron space that is, the expectation values of H with respect to all A -particle functions are nonnegative. Assume that the Hamiltonian may be expanded as a sum of operators G,G,... 

An A-representable RDM is also defined to be S-representable if it derives from an A-particle wavefunction or an ensemble of A-particle wavefunctions with a definite spin quantum number 5 [57]. By definition, an 5-representable two-electron RDM yields the correct expectation value... 

If X is an operator on and if x can be written as x take the matrix X, with entries X,y, to be the matrix representation of x. With this definition, expectation values can be written... 

A -representability conditions [28]. Let us start this description by focusing on the RDM s properties, which may be deduced from their definition as expectation values of density fermion operators. Thus the ROMs are Hermitian, are positive semidefinite, and contract to finite values that depend on the number of electrons, N, and in the case of the HRDMs on the size of the one-electron basis of representation, 2/C. Thus... 

These definitions are easily generalized from a pure state, described by to ensemble states, described by a system density matrix V, for which an expectation value is... 

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