Expectation values time dependence

It is now easy to understand the origin of equation (7-31). One sees that it is of the form of the Clausius-Mosotti equation where the complex dielectric constant rather than % or eR values is used. The complex formulation introduces a frequency dependence, which appears in the last term of equation (7-31). One would expect the time-dependent contribution to be related to the difference between instantaneous and long-time behavior and, indeed, this is correct, because the factor multiplying the frequency dependence in equation (7-31) is merely the difference between equations (7-47) and (7-40). In fact, these two expressions may be combined with equation (7-47) to yield... 

It should be noted that the assumption of local defect equilibrium may not hold in the very early stages of the interdiffusion process when steep concentration gradients occur, especially if only the outer crystal surface acts as source or sink for point defects. One would expect then time dependent j6(c)-values and nonplanar equiconcentration surfaces. 

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). 

The quantum degrees of freedom are described by a wave function /) = (x, t). It obeys Schrodinger s equation with a parameterized coupling potential V which depends on the location q = q[t) of the classical particles. This location q t) is the solution of a classical Hamiltonian equation of motion in which the time-dependent potential arises from the expectation value of V with regard to tp. For simplicity of notation, we herein restrict the discussion to the case of only two interacting particles. Nevertheless, all the following considerations can be extended to arbitrary many particles or degrees of freedom. 

Here (r - Rc) (r - Rq) is the dot product times a unit matrix (i.e. (r — Rg) (r — Rg)I) and (r - RG)(r — Rg) is a 3x3 matrix containing the products of the x,y,z components, analogous to the quadrupole moment, eq. (10.4). Note that both the L and P operators are gauge dependent. When field-independent basis functions are used the first-order property, the HF magnetic dipole moment, is given as the expectation value over the unperturbed wave funetion (for a singlet state) eqs. (10.18)/(10.23). 

Since most time-dependent failures have larger life dispersions, we must consider the maximum and minimum ratios of 4 1 and 40 1. Generally, relative life dispersion increases with the absolute value of MTBF. That is, wear items with a relatively short life expectancy such as rider rings on reciprocating compressors will have a comparatively smaller dispersion than components such as gear tooth flanks, which can be expected to remain serviceable for long periods of time. 

While this bound is not a particularly strong one and convergence is generally faster in practice [goles90], it does clearly point out the important fact that transient times are linearly bounded by the lattice size n. Notice that this is not true of more general classes of matrices, even those of the preceding section that are both symmetric and integer-valued. Equation 5.140 shows that the transient time depends on both A and 26 — Al if both A and b are arbitrary (save, perhaps, for A s symmetry), there is of course no particular reason to expect t to be linearly... 

The statistical matrix is then computed via Eq. (7-78). When the expectation value of the energy, Tr (pH), is then calculated in different orders of V, the successive orders of time-dependent perturbation theory emerge. 

This means that any observable not depending explicitly on time and commuting with H has an expectation value that does not depend on time. 

Consider next the rate of change of the expectation value of an operator that does depend explicitly on time, say 22(f). Then in addition to the terms appearing in Eq. (8-244) there is also a term < , f 8Rj8t >0 so that... 

Now let us use the set, <0> to form a matrix representation of some operator Q at time hi assuming that Q is not explicitly a function of time. The expectation value of Q in the various states, changes in time only by virtue of the time-dependence of the state vectors used in the representation. However, because this dependence is equivalent to a unitary transformation, the matrix at time t is derived from the matrix at time t0 by such a unitary transformation, and we know that this cannot change the trace of the matrix. Thus if Q — WXR our result entails that it is not possible to change the ensemble average of R, which is just the trace of Q. 

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 / ... 

The density amplitudes can usually be calculated more efficiently than the density operator because they depend on only one set of variables in a given representation although there are cases, such as shown below for the time-dependent Hartree-Fock density operator, where the advantages disappear and it is convenient to calculate the density operator. Expectation values of operators A t) follow from the trace over the density operator, as... 

In general, A(p, i) depends on the time, so that the expectation values p) and fip)) are also functions of time. 

The first relationship is obtained by considering the time dependence of the expectation value of the position coordinate x. The time derivative of (x) in equation (2.13) is... 

Thus, the energy E of the system, which is equal to the expectation value of the Hamiltonian, is conserved if the Hamiltonian does not depend explicitly on time. 

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. 

Time intervals permitting displacement values in the scaling window a< )tortuous flow as a result of random positions of the obstacles in the percolation model [4]. Hydrodynamic dispersion then becomes effective. For random percolation clusters, an anomalous, i.e., time dependent dispersion coefficient is expected according to... 

The results for the glass crystallization of PET annealed at 80 °C as before are shown in Fig. 8. In the early stage of spinodal decomposition up to 20 min, the characteristic wavelength A remains constant at a value of 15 nm, which agrees with the theoretical expectation that only the amplitude of density fluctuations increases whilst keeping a constant characteristic wavelength. In the late stage from 20 to 100 min it increases up to 21 nm just before crystallization. Such a time dependence of A in nm can be represented by... 

In the Schrodinger picture operators in the case of a closed system do not depend explicitly on the time, but the state vector is time dependent. However, the expectation values are generally functions of the time. The commutator of the Hamiltonian operator H= —(h/2iri)(d/dt) and another operator A, is defined by... 

As a simple model, we confine our attention just to a single mode Ha(t) of the Hamiltonian (23). Note that neither any instantaneous eigenstate of Ha(t) is an exact quantum state nor e-/3ii W is a density operator. To calculate the thermal expectation value of an operator A, one needs either the Heisenberg operator Ah or the density operator pa(t) = UapaUa Now we use the time-dependent creation and annihilation operators (24), invariant operators, to construct the Fock space. 

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