Expected Value Operator

Definition 4.1 Suppose is the probability density function of a f-dimen-sional random variable then the expected value (or average) of random variable is [7-9]  

E that is shown in the above formula is called the expected value operator. 

Suppose / is a real function defined in 91, then/( ) is a random function. According to Definition 4.1 the expected value E f i)] of [/ (I)] can be defined as  

If 2 are n random vectors and independent of each other, and there 

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The term 5wl (Eq. [23a]) is the expected value of completely filled pores. Because saturation for completely filled pores equals one, the expected value operation amounts to integration of the gamma distribution between the smallest central pore dimension Lmin (the lower limit of integration depicted in Fig. 1-8), and a certain pore size denoted by Lt. The upper limit Lx is determined from the radius of curvature at the onset of drainage in the central pore, rd (Mason Morrow, 1991 Tuller et al., 1999). It is often referred to as the radius of curvature at air-entry , and is given by ... 

Stochastic (fuzzy) variables Stochastic (fuzzy) vectors Probability measure Possibility measure Necessity measure Credibility measure Expected value operator Probability space Possibility space Empty set Set of real numbers n-dimensional real vector set Mini operator Max operator... 

Uncertainty theory is also referred to as probability theory, credibility theory, or reliability theory and includes fuzzy random theory, random fuzzy theory, double stochastic theory, double fiizzy theory, the dual rough theory, fiizzy rough theory, random rough theory, and rough stochastic theory. This section focuses on the probability theory and fiizzy set theory, including probability spaces, random variables, probability spaces, credibility measurement, fuzzy variable and its expected value operator, and so on. 

Liu YK, Liu B (2003) Fuzzy random variables a scalar expected value operator. Fuzzy Optim Decis Making 2(2) 143-160... 

According to the definition of the expected value operator it follows ... 

OxTi is the standard deviation over Xxy, and E is the expectation value operator. 

Close inspection of equation (A 1.1.45) reveals that, under very special circumstances, the expectation value does not change with time for any system properties that correspond to fixed (static) operator representations. Specifically, if tlie spatial part of the time-dependent wavefiinction is the exact eigenfiinction ). of the Hamiltonian, then Cj(0) = 1 (the zero of time can be chosen arbitrarily) and all other (O) = 0. The second tenn clearly vanishes in these cases, which are known as stationary states. As the name implies, all observable properties of these states do not vary with time. In a stationary state, the energy of the system has a precise value (the corresponding eigenvalue of //) as do observables that are associated with operators that connmite with ft. For all other properties (such as the position and momentum). 

One can show that the expectation value of the Hamiltonian operator for the wavepacket in equation (A3.11.71 is ... 

The difference compared to equation B 1.13.2 or equation B 1.13.3 is the occurrence of the expectation value of the operator (the two-spin order), characterized by its own decay rate pjg and coupled to the one-spin longitudinal operators by the tenus 8j aud 5. We shall come back to the physical origin of these tenus below. 

In NMR, the magnetization in the xy plane is detected, so it is the expectation value of the operator that is measured. This is just the unweighted sum of all the operators for the individual spins . It may be a fimction of several time variables (multi-dimensional experiments), including the tune during the acquisition. 

It is now required for observable quantities that the expectation value of any operator O taken with respect to tl[Pg.616]

Many physical properties of a molecule can be calculated as expectation values of a corresponding quantum mechanical operator. The evaluation of other properties can be formulated in terms of the "response" (i.e., derivative) of the electronic energy with respect to the application of an external field perturbation. 

Once a wave function has been determined, any property of the individual molecule can be determined. This is done by taking the expectation value of the operator for that property, denoted with angled brackets < >. For example, the energy is the expectation value of the Hamiltonian operator given by... 

Properties can be computed by finding the expectation value of the property operator with the natural orbitals weighted by the occupation number of each orbital. This is a much faster way to compute properties than trying to use the expectation value of a multiple-determinant wave function. Natural orbitals are not equivalent to HF or Kohn-Sham orbitals, although the same symmetry properties are present. 

It is sometimes useful to recast the equation as the expectation value of a sum of one-electron and pseudo one-electron operators... 

The operator hi is a one-electron operator, representing the kinetic energy of an electron and the nuclear attraction. The operators J and K are called the Coulomb and exchange operators. They can be defined through their expectation values as follows. 

Boys and Cook refer to these properties as primary properties because their electronic contributions can be obtained directly from the electronic wavefunction As a matter of interest, they also classified the electronic energy as a primary property. It can t be calculated as the expectation value of a sum of true one-electron operators, but the Hartree-Fock operator is sometimes written as a sum of pseudo one-electron operators, which include the average effects of the other electrons. 

The first term in the brackets is the expectation value of the square of the dipole moment operator (i.e. the second moment) and the second term is the square of the expectation value of the dipole moment operator. This expression defines the sum over states model. A subjective choice of the average excitation energy As has to be made. 

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