Expectation values of dynamical quantities

The transform A(p, t) is ealled the momentum-space wave function, while (jc, /) is more accurately known as the coordinate-space wave function. When there is no confusion, however, (jc, /) is usually simply referred to as the wave function. 

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. 

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 

More generally, the expectation value (f(x) of any function f(x) of the variable x is given by 

The expectation value p) of the momentum p may be obtained using the momentum-space wave function A p, i) in the same way that (x) was obtained from F(x, i). The appropriate expression is 

More generally, the expectation value variable x is given by 

---

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... 

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]... 

Thus far we have focussed on the dynamics of quantum-classical systems. In practice, we are primarily interested instead in computing observables that can be compared eventually to experimentally obtainable quantities. To this end, consider the general quantum mechanical expression for the expectation value of an observable,... 

With our realization that dynamical variables are represented as operators, we can also raise the question of how to compute observed quantities within the framework of quantum mechanics. The essential rule is that if we are interested in obtaining the expectation value of the operator 0(r, p) when the system is in the state xj/, then this is evaluated as... 

The energy eigenvalues for the confined hydrogen atom, for some states and some values of R are given in Table 1. A large range of eigenvalues is available in the literature [8,23]. The expectation values of some related dynamical quantities can be deduced [25] for these states. 

Whether the inactive region is a true continuum (e.g., photofragmentation) or a quasi-continuum comprised of an enormous density of rigorously bound eigenstates (polyatomic molecule dynamics, Section 9.4.14) is often of no detectable consequence. The dynamical quantities discussed in Section 9.1.4 (probability density, density matrix, autocorrelation function, survival probability, transfer probability, expectation values of coordinates and conjugate momenta) describe the active space dynamics without any reference to the detailed nature of the inactive space. 

As a result, the early time dynamics of acetylene could be affected by numerous near resonant anharmonic interactions. Nine such anharmonic resonances are well characterized and known to be dynamically relevant. One needs a framework to relate the Fourier amplitudes and phases of the time-dependence of any observable quantity to the expectation values of each of the resonance operators in order to establish the relative importance of each of the resonances. 

Thus, although the usefulness of MC simulation as a method for the study of the dynamics of polymeric systems is somewhat restricted, a consideration of the dynamics, nevertheless, always is necessary, in order to judge correctly the accuracy of the desired averages of equilibrium quantities. Let us return to eqn [6], and suppose that M = M-Mq successive observations =A Xf ),/u= 1,..., M have been generated and stored, with M 1. We now consider the expectation value of the square of the statistical error. 

The expectation values of all the observables mentioned above play a role in the prediction and interpretation of the properties of molecules, in a variety of elaborations and combinations depending on the formulation of the theory for the specific theme under examination (theory of the chemical bond, theory of the geometrical equilibrium structure, theories for the various spectroscopic properties, for molecular dynamics, for reactivity, for the chemical reaction mechanisms, etc.). We shall limit our attention to the theories for chemical reactivity a very important subject, but limited with respect to the variety of problems of interest in the molecular sciences shortly summarized above. This limitation of the theme also implies a limitation in the use of electrostatic quantities. The emphasis is placed here on the energy of molecular interactions and on the forces acting on the molecular interacting units. 

As the evolution of the electronic phase is much faster than the dynamics of the nuclei, the time step Af needs to be adjusted relative to the time step At of the nuclear dynamics. The time-dependent expectation value of the nuclear coordinates R(t) = irtot(R,t) R ktot(R,t)) is evaluated using the solution of Eq. (8.3). Subsequent quantum chemical calculations are performed at the nuclear geometries R(t) to obtain the electronic wavefunctions /,o(r R t))- The quantities ai(t) aj(t) xlfi(R,t) xlfj(R,t))R are also obtained from Eq. (8.3). The integrals over the electronic wavefunctions /[Pg.223]

A unique situation is encountered if Fe-M6ssbauer spectroscopy is applied for the study of spin-state transitions in iron complexes. The half-life of the excited state of the Fe nucleus involved in the Mossbauer experiment is tj/2 = 0.977 X 10 s which is related to the decay constant k by tj/2 = ln2/fe. The lifetime t = l//c is therefore = 1.410 x 10 s which value is just at the centre of the range estimated for the spin-state lifetime Tl = I/Zclh- Thus both the situations discussed above are expected to appear under suitable conditions in the Mossbauer spectra. The quantity of importance is here the nuclear Larmor precession frequency co . If the spin-state lifetime Tl = 1/feLH is long relative to the nuclear precession time l/co , i.e. Tl > l/o) , individual and sharp resonance lines for the two spin states are observed. On the other hand, if the spin-state lifetime is short and thus < l/o) , averaged spectra with intermediate values of quadrupole splitting A q and isomer shift 5 are found. For the intermediate case where Tl 1/cl , broadened and asymmetric resonance lines are obtained. These may be the subject of a lineshape analysis that will eventually produce values of rate constants for the dynamic spin-state inter-conversion process. The rate constants extracted from the spectra will be necessarily of the order of 10 -10 s"F... 

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. 

To separate the effects of static and dynamic disorder, and to obtain an assessment of the height of the potential barrier that is involved in a particular mean-square displacement (here abbreviated (x )), it is necessary to find a parameter whose variation is sensitive to these quantities. Temperature is the obvious choice. A static disorder will be temperature independent, whereas a dynamic disorder will have a temperature dependence related to the shape of the potential well in which the atom moves, and to the height of any barriers it must cross (Frauenfelder et ai, 1979). Simple harmonic thermal vibration decreases linearly with temperature until the Debye temperature Td below To the mean-square displacement due to vibration is temperature independent and has a value characteristic of the zero-point vibrational (x ). The high-temperature portion of a curve of (x ) vs T will therefore extrapolate smoothly to 0 at T = 0 K if the sole or dominant contribution to the measured (x ) is simple harmonic vibration ((x )y). In such a plot the low-temperature limb is expected to have values of (x ) equal to about 0.01 A (Willis and Pryor, 1975). Departures from this behavior indicate more complex motion or static disorder. 

For over a century it has been known that two classes of variables have to be distinguished the microscopic variables, which are functions of the points of ClN and thus pertain to the detailed positions and motions of the molecules and the macroscopic variables, observable by operating on matter in bulk, exemplified by the temperature, pressure, density, hydro-dynamic velocity, thermal and viscous coefficients, etc. And it has been known for an equally long time that the latter quantities, which form the subject of phenomenological thermo- and hydrodynamics, are definable either in terms of expected values based on the probability density or as gross parameters in the Hamiltonian. But at once three difficulties of principle have been encountered. 

---