Time-independent operator

Now consider die case where Ais itself a time-independent operator, such as that for the position, momenPiin or angidar momenPiin of a particle or even the energy of the benzene molecule. In these cases, the time-dependent expansion coefficients are unaffected by application of the operator, and one obtains... 

Since div ( ) mid div 3 (x) commute with 8(x ) and 3 t (x ) for x0 —x, they have vanishing commutators with the hamiltonian and hence, they are time-independent operators. In fact, their constancy in tame implies that they commute with 3 (x) and S(x) at all times and hence they must be c-number multiples of the unit operator. If these c-numbers are set equal to zero initially, they will remain zero for all times. With this initial choice for div 8(x) and div 3tf(x), the operators S and satisfy all of the Maxwell equations (these now are operator equations ) ... 

The system is prepared at t=0 in the quantum state Pik> and the question is how to calculate the probability that at a later time t the system is in the state Fjn>. By construction, these quantum states are solutions of molecular Hamiltonian in absence of the radiation field, Hc->Ho Ho ik> = e k Fik> and H0 Pjn> = Sjn xPJn>. The states are orthogonal. The perturbation driving the jumps between these two states is taken to be H2(p,A)= D exp(icot), where co is the frequency of the incoherent radiation field and D will be a time independent operator. From standard quantum mechanics, the time dependent quantum state is given by ... 

Here, the integration over X was performed in Eq. (63) to define W%a (X, ) which is the integrated value of the combination of the spectral density function with the time independent operator. This spectral density function contains the quantum equilibrium structure of the system. (X, t) is the time evolved matrix element of the number operator for the product state B. Thus, to calculate the rate, one samples initial configurations from the quantum equilibrium distribution, and then computes the evolution of the number operator for product state B. The QCL evolution of the species operator is accomplished using one of the algorithms discussed in Sec. 3.2. Alternative approaches to the dynamics may also be used such as the further approximations to the QCLE discussed in Sec. 4. 

The dependent expectation value for an arbitrary time-independent operator A is written as... 

When quantizing the theory, one can decompose the (time dependent) operator k into a sum °f time independent operators and time dependent functions ... 

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... 

We start the derivations by considering the expectation value of a time-independent operator A which is expanded in orders of a time-dependent perturbation... 

We now return to time-independent Hamiltonians and describe another method for solving the time-dependent Schrodinger equation. Linear initial value problems described by time-independent operators are conveniently solved using Laplace transforms (Section 1.1.7). In Section 1.1.7 we have seen an example where the equation... 

In order to express Eq. (50) in a more compact form, it is useful to introduce the super-operator formalism (Pickup and Goscinski, 1973 Goscinski and Lukman, 1970). The time-independent operators are construed as elements in a super-operator space with a binary product... 

Consider a time-independent operator A whose matrix elements, yf a, /3 d (both expectation values and transition moments), in the space fl we wish to compute. This goal is to be achieved by transforming the calculation from 0 into one in O, resulting in an effective operator a whose matrix elements, taken between appropriate model eigenfunctions of an effective Hamiltonian h, are the desired As we now discuss, numerous possible definitions of a arise depending on the type of mapping operators that are used to produce h and on the choice of model eigenfunctions. 

Effective Hamiltonians and Effective Operator Definitions Corresponding to a Time-Independent Operator A... 

In the above discussion we have not considered dissipative processes. To include these processes we can, in the framework of the above applied perturbation theory, aside from the perturbation operator U, include a time-independent operator which induces transitions between states tE o- In this case the tensor ej, ij is again defined by the expression (7.52), where in the resonant denominators the energy Huj must be replaced by a complex quantity tuv + ih ylu , k), 7 = 7 + iy", with I7I -C uj. Knowledge of the function 7(0 , k) is important, for example, in the analysis of a lineshape. Below we take into account the exciton-phonon interaction and, for simplicity, consider only the first order of perturbation theory. 

If the external perturbation is turned on with a time dependent function F(t) and takes the form AF(t) where is a time independent operator (or is the sum of such terms), then... 

Let A be a linear, time-independent operator. Consider the integral... 

Their identity can be proved when system properties are time-independent operators. (This may not be the unique possibility.) The condition to fulfill, independency of system properties with time, is expressed by the requirement of commutativity of these operators with the time derivation, which can be written using Poisson s brackets ... 

The required irreversibility of the dissipation results from the presence of a tonporal derivation in only one path, the other being featured by a time-independent operator. This latter is called a kinetic operator, which in most systems reduces to a scalar, called kinetic constant or relaxation frequency. Its inverse is the relaxation time or time constant. 

This postulate pertains to ideal measurements, i.e. such that no error is introduced through imperfections in the measurement apparatus. We assume the measurement of the physical quantity A, represented by its time-independent operator A and, for the sake of simplicity, that the system is composed of a single particle (with one variable only). 

Let A be an arbitrary time-independent operator, referring to the system alone. Denoting a trace over the system variables q by Tr. we find from Eq. (206)... 

The semi-classical Hamiltonian, H, composed of a time-independent operator Ha, describing the molecule, combined with the time-dependent term describing the interaction of the electric field, e(r), with the molecular dipole moment, /r(r), is... 

Multiplication by (f) i) can be done after Ho operates on xp r) because is just a constant as far as the time-independent operator H is concerned.) Making the same substitution for W on the right side gives... 

In these equations, m (Eq. 2.1) and mj (Eq. 2.2) indicate the circular frequencies associated with the perturbing field, is a small positive number that ensures that the perturbation vanishes for t —> —oo, e icoj) is the field strength parameter and B the associated perturbation operator. The expectation value of the time-independent operator A is written as [1]... 

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