Time dependence of the expectation value

To obtain the Hellmann-Feynman theorem, we differentiate equation (3.67) with respect to X 

Applying the hermitian property of H X) to the third integral on the right-hand side of equation (3.69) and then applying (3.66) to the second and third terms, we obtain 

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) 

Equation (3.55) may be substituted for the time derivatives of the wave funetion to give 

If we set A equal to unity, then the commutator [H, A] vanishes and equation (3.72) becomes 

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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... 

The general concepts of inverse control are simple to grasp, Suppose we specify, a priori, the time dependence of the expectation value of some observable, (O) = y(r). The Schrodinger equation for the system with applied control field is just... 

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... 

Another consequence of the Schrodinger equation is that the time evolution of the expectation value of a physical observable, represented by an operator A (with no explicit time dependence), is given by... 

We will in this section consider the mathematical structure for computational procedures when calculating molecular properties of a quantum mechanical subsystem coupled to a classical subsystem. Molecular properties of the quantum subsystem are obtained when considering the interactions between the externally applied time-dependent electromagnetic field and the molecular subsystem in contact with a structured environment such as an aerosol particle. Therefore, we need to study the time evolution of the expectation value of an operator A and we express that as... 

Thus U(tto) represents the unitary time evolution according to the given system Hamiltonian. The time evolution of the expectation value of an arbitrary Hermitian TD operator A(t) which has an intrinsic time-dependence of its own representing a physical quantity evolves accordingly by the rule... 

The time-dependence of the expected number of survivors is expressed by the monoton-ically decreasing sequence of the binomial expected values Nop, (Nop)p, [(Nop)p]p,..., Nop. Note that this sequence consists of the substitution values of a decreasing exponential function taken at the end of each period. The explanation for this is as follows. Since no atoms are bom in any of the periods, each census simplifies to a survival test. The mathematical equivalent of this is the B(l, p) Bernoulli sampling of the B(N, i, p) distribution of the previous population. Note that the exponential character of the expected value as a function of time is explained by the fact that the Bernoulli sampling has been performed with the assumption that the probability of survival is the same for each period (p = 0.9). This assumption, on the other hand, is equivalent to the assumption of agelessness, which, in turn, implies the exponential character of the lifetime distribution. 

In Section 3.3 we looked at the dependence of an expectation value on a perturbing field P and expanded the expectation value in powers of this perturbation. In this section, we want to study now the time evolution of an expectation value of an arbitrary operator P. Finally, in the section on time-dependent response theory, Section 3.11, we will combine both and study the effects of a time-dependent perturbation Pa. .if)-Let us study the time dependence of an expectation value by deriving an expression for the time derivative of an expectation value, i.e. an equation of motion for the expectation value of the operator P... 

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. 

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

Taking away the whole top layer all at once would yield a value twice as big as (13), but, is still yualitativcly correct.] The dependence of /t on if is rather similar to Ey. (6). Hence we expect the top part of the dome to be facetted after an initial transient, and the time dependence of the bump height to be approximately linear. 

Our starting point involves the determination of the expectation value of a time-independent operator A. In the case of a time-dependent perturbation given by the operator Vit), the expectation value of A is time dependent. We expand the expression for the time-dependent expectation value in orders of the perturbation and find... 

Initially, at f < 0, the isolated system is described by the Hamiltonian H0 that does not depend on time. Thus, any expectation value averaged with the density matrix p0 is time-independent. At t > 0, the system is disturbed by an external time-dependent field. Then, the evolution of the expectation value of operator O can be presented as follows ... 

A further striking feature shown in Fig. 20 is the convergence of the rotational velocities to the value of v( nt,iinm (i.e., lOOkrns ), which is produced by the fact that the time dependence of the critical rotational velocity is almost independent of the mass loss history, in particular its minimum value. Note, however, that although the rotation velocities of the four sequences displayed in Fig. 20 at the end of the main sequence evolution are almost identical, their masses at that time are greatly different, and thus their ensuing post-main sequence evolution is expected to be very different as well. 

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