Dynamic expectation value

Since each q>i is an eigenfunction of the Hamiltonian operator with eigenvalue Ei, the dynamic expectation value of the total energy reduces to... 

Consider, at t = 0, some non-equilibrium ensemble density P g(P. q°) on the constant energy hypersurface S, such that it is nonnalized to one. By Liouville s theorem, at a later time t the ensemble density becomes ((t) t(p. q)), where q) is die function that takes die current phase coordinates (p, q) to their initial values time (0 ago the fimctioii ( ) is uniquely detemiined by the equations of motion. The expectation value of any dynamical variable ilat time t is therefore... 

The excitation energy and dynamic properties are evaluated from the time-averaged derivatives of the corresponding time-dependent energy functionals [11, 23-25]. However, a more straightforward way to define dynamic properties is through an expectation value of the corresponding properties over a state / ... 

A formulation of electronic rearrangement in quantum molecular dynamics has been based on the Liouville-von Neumann equation for the density matrix. Introducing an eikonal representation, it naturally leads to a general treatment where Hamiltonian equations for nuclear motions are coupled to the electronic density matrix equations, in a formally exact theory. Expectation values of molecular operators can be obtained from integrations over initial conditions. 

The approach to the evaluation of vibrational spectra described above is based on classical simulations for which quantum corrections are possible. The incorporation of quantum effects directly in simulations of large molecular systems is one of the most challenging areas in theoretical chemistry today. The development of quantum simulation methods is particularly important in the area of molecular spectroscopy for which quantum effects can be important and where the goal is to use simulations to help understand the structural and dynamical origins of changes in spectral lineshapes with environmental variables such as the temperature. The direct evaluation of quantum time- correlation functions for anharmonic systems is extremely difficult. Our initial approach to the evaluation of finite temperature anharmonic effects on vibrational lineshapes is derived from the fact that the moments of the vibrational lineshape spectrum can be expressed as functions of expectation values of positional and momentum operators. These expectation values can be evaluated using extremely efficient quantum Monte-Carlo techniques. The main points are summarized below. 

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. 

According to the correspondence principle as stated by N. Bohr (1928), the average behavior of a well-defined wave packet should agree with the classical-mechanical laws of motion for the particle that it represents. Thus, the expectation values of dynamical variables such as position, velocity, momentum, kinetic energy, potential energy, and force as calculated in quantum mechanics should obey the same relationships that the dynamical variables obey in classical theory. This feature of wave mechanics is illustrated by the derivation of two relationships known as Ehrenfest s theorems. 

Equivalently, expectation values of three-dimensional dynamical quantities may be evaluated for each dimension and then combined, if appropriate, into vector notation. For example, the two Ehrenfest theorems in three dimensions are... 

The appearance of the Hamiltonian operator in equation (3.55) as stipulated by postulate 5 gives that operator a special status in quantum mechanics. Knowledge of the eigenfunctions and eigenvalues of the Hamiltonian operator for a given system is sufficient to determine the stationary states of the system and the expectation values of any other dynamical variables. 

The expectation value A) of the dynamical quantity or observable A is, in general, a function of the time t. To determine how A) changes with time, we take the time derivative of equation (3.46)... 

The position, momentum, and energy are all dynamical quantities and consequently possess quantum-mechanical operators from which expectation values at any given time may be determined. Time, on the other hand, has a unique role in non-relativistic quantum theory as an independent variable dynamical quantities are functions of time. Thus, the uncertainty in time cannot be related to a range of expectation values. 

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. 

Here r and v are respectively the electron position and velocity, r = —(e2 /em)(r/r3) is the acceleration in the coulombic field of the positive ion and q = /3kBT/m. The mobility of the quasi-free electron is related to / and the relaxation time T by p = e/m/3 = et/m, so that fi = T l. In the spherically symmetrical situation, a density function n(vr, vt, t) may be defined such that n dr dvr dvt = W dr dv here, vr and vt and are respectively the radical and normal velocities. Expectation values of all dynamical variables are obtained from integration over n. Since the electron experiences only radical force (other than random interactions), it is reasonable to expect that its motion in the v space is basically a free Brownian motion only weakly coupled to r and vr by the centrifugal force. The correlations1, K(r, v,2) and fc(vr, v(2) are then neglected. Another condition, cr(r)2 (r)2, implying that the electron distribution is not too much delocalized on r, is verified a posteriori. Following Chandrasekhar (1943), the density function may now be written as an uncoupled product, n = gh, where... 

In the preceding F = fc(r, r), H = tc(r, vt)G = k(vt, v) and the normalization constant C is fixed by equating the volume integral of n to unity. For further tractability, Sano and Mozumder expand (r v) in a Taylor s series and retain the first two terms only. The validity of this procedure can be established a posteriori in a given situation. At first, the authors obtain equations for the time derivatives of the expectation values and the correlations of dynamical variables. Then, for convenience of closure and computer calculation, these are transformed into a set of six equations, which are solved numerically. The first of these computes lapse time through the relation... 

As in the previous experiments, at each epoch we would like to select a waveform (or really the error covariance matrix associated with a measurement using this waveform) so that the measurement will minimize the uncertainty of the dynamic model of the target. We study two possible measures entropy of the a posteriori pdf of the models and mutual information between the dynamic model pdf and measurement history. Both of these involve making modifications to the LMIPDA-IMM approach that are described in [5]. Since we want to minimize the entropy before taking the measurement, we need to consider the expected value of the cost. To do this we replace the measurement z in the IMM equations by its expected value. In the case of the second measure, for a model we have... 

The program THERMFF solves the same dynamic process model equations as THERM, where it was shown that all the parameters, including the inlet temperature and concentration will influence the steady state. In the case of multiple steady states the values of the steady state parameters cannot be set, because they are not unique. This example should, therefore, be mn under parameter conditions that will guarantee a single steady state for all expected values of the CA0 and T0. These can be selected with the aid of the programs THERMPLOT and THERM. 

To describe the electronic relaxation dynamics of a photoexcited molecular system, it is instructive to consider the time-dependent population of an electronic state, which can be defined in a diabatic or the adiabatic representation [163]. The population probability of the diabatic electronic state /jt) is defined as the expectation value of the diabatic projector... 

To perform a PO analysis of nonadiabatic quantum dynamics, we employ a quasi-classical approximation that expresses time-dependent quantities of a vibronically coupled system in terms of the vibronic POs of the system [123]. Considering the quasi-classical expression (16) for the time-dependent expectation value of an observable A, this approximation assumes that the integrable islands in phase space represent the most significant contributions to the dynamics of the observables considered [236]. As a consequence, the short-time dynamics of the system is determined by its shortest POs and can be approximated by a time average over these orbits. Denoting the A th PO with period 7 by qk t),Pk t) we obtain [123]... 

We have explored whether, on the basis of sound theoretical and physical reasoning, a semiempirical formula can be derived that would directly provide an accurate estimate of the dynamic correlation energy as a whole. Two known rigorous results are suggestive in this context (i) The dynamic correlation energy can be expressed as the expectation value of a perturbing correlation operator (65-67) and (ii) the correlation energy is known to be expressible (10) as sum of contributions of occupied orbital pairs, viz. 

In quantum mechanics, the average (or expectation) value of dynamical variable r is given by... 

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