Heisenberg time dependent operator

Here <5FS signifies the fluctuating solvent force on the coordinate qs, while < qs (t) is the Heisenberg time-dependent operator, with dynamics governed by the full internal anharmonic molecular Hamiltonian, associated with the fluctuation <5qs = qs — (i qs f). Finally, the prefactor yi( is (2)... 

In the Heisenberg picture the operators themselves depend explicitly on the time and the time evolution of the system is determined by a differential equation for the operators. The time-dependent Heisenberg operator AH(t) is obtained from the corresponding Schrodinger operator As by the unitary transformation... 

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. 

The result of using these identities as well as the Heisenberg definition of the time-dependence of the dipole operator... 

In the Heisenberg representation a time-dependent dipole operator p(t) is generated from its value at some previous time t by a unitary transformation with the time-displacement operator exp — t )/h, so that... 

Exercise. For any observable A one defines the time-dependent or Heisenberg operator A(t) by... 

The average value of any operator O can be written as (O) = (t Os t) in the Schrodinger representation or (O) = (0 Off(t) 0) in the Heisenberg representation, where 0) is some initial state. This initial state is in principle arbitrary, but in many-particle problems it is convenient to take this state as an equilibrium state, consequently without time-dependent perturbation we obtain usual equilibrium Green functions. 

The plan of the paper is the following. In section 2 we introduce the elements at the basis of the Heisenberg group representation representation theory [12-14,20] that are needed to understand the alternative group-theoretical formulation of quantum mechanics. In section 3 the Heisenberg representation of quantum mechanics (with the time dependence transferred from the vectors of the Hilbert space to the operators) is used to introduce quantum observables and quantum Lie brackets within the group-theoretical formalism described in the previous section. In section 4 classical mechanics is obtained by taking the formal limit h —> 0 of quantum observables and brackets to obtain their classical counterparts. Section 5 is devoted to the derivation of... 

The group-theoretical formalism we have introduced so far is particularly suited to formulate quantum mechanics in the Heisenberg representation, where the time dependence is shifted from the wavefunctions to the operators. As we shall show in the next section, the formalism allows to show in a straightforward way that the Poisson brackets are obtained as a formal limit of the commutator when h —> 0. 

The form of the expressions in Eqs (5.98) and (5.114) is closely related to the classical expressions for the rate constant given in Section 5.1. The quantum mechanical trace becomes in classical statistical mechanics an integral over phase space [9] and the Heisenberg operators become the corresponding classical (time-dependent) functions of coordinates and momenta [8]. Thus, Eq. (5.78) is the classical version of Eq. (5.114). Furthermore, note that Eq. (5.98) is related to Eq. (5.49), i.e., the relevant classical (one-way) flux through Ro, at a given time, becomes S(R - Ro)(p/p)9(p/p), exactly as in Eq. (5.49). 

Now, use the effective non-Hermitean Hamiltonians (228) to get the time dependence of the transition moment operator through Heisenberg transformation, but ignore non-Hermitean part for the Boltzmann operator. Then, the ACF taking into account both direct and indirect dampings, appears to be given by... 

The operators P and obey the usual equal time anticommutation relations. The time-dependence of the field operators appearing here is due to the Heisenberg representation in the L-space. In view of the foregoing development which parallels the traditional Schrodinger quantum theory we may recast the above Green function in terms of the interaction representation in L-space. This leads to the appearance of the S-matrix defined only for real times. We will now indicate the connection of the above to the closed-time path formulation of Schwinger [27] and Keldysh [28] in H-space. Equation (82) can be explicitly... 

Unlike classical quantum mechanics, the spontaneous processes of the damped oscillator are irreversible, so its quantum mechanical description needs changes to some instruments of classical quantum mechanics. To do this, we use the Heisenberg picture of quantum processes. In this picture, the observables are time-dependent linear Hermitian operators, and the state vector of the system is time independent. Using the terminology introduced in the first part, the infinitesimal time transformation of the Hermitian operator could happen in two ways ... 

In the case of s linearly damped oscillator, the transformation of the Heisenberg picture into the Schrodinger picture by the method applied in classical quantum theory is impossible because the operator has a time-dependent part due to the dissipative process. Thus, a new way must be found to construct the wave equation of the oscillator. Kostin introduced a supplementary dissipation potential into his wave equation and constructed this dissipation potential by an assumption that the energy eigenvalues of the oscillator decay exponentially over time [39]. In Kostin s version of the wave equation, the operators are time independent, but the dissipation potential is nonlinear with respect to the wave function. In our theory, it is assumed that the abstract wave equation of the linearly damped oscillator has the form... 

Obviously, is simply 4 (Z = 0) and is by definition time independent. Equation (2.62) is a unitary transformation on the wavefunctions and the operators at time t. The original representation in which the wavefunctions are time dependent while the operators are not, is transformed to another representation in which the operators depend on time while the wavefunctions do not. The original fonnulation is referred to as the Schrodinger representation, while the one obtained using (2.62) is called the Heisenberg representation. We sometimes use the subscript S to emphasize the Schrodinger representation nature of a wavefunction or an operator, that is. 

As an example for the use of this formulation let us calculate the (in-principle time-dependent) variance, (Ax(Z)2), defined by Eq. (2.149) for a Harmonic oscillator in its ground state. Using the expression for position operator in the Heisenberg representation from Eq. (2.166) and the fact that (0 Ax(Z)2 0) = <0 x(Z) 0 for an oscillator centered at the origin, this can be written in the from... 

The field operators are time dependent and thus Heisenberg operators, i.e., they satisfy... 

In the Schrodinger representation of the latter matrix element in (8.10), the molecular states are regarded as time-dependent basis functions exp( — i fcf/ft) fe> and exp( —i t/ft) w>, and the operator /t is considered to be time-independent. For present purposes, it is more illuminating to use the Heisenberg representation, in which the molecular states are the time-independent basis functions lfc>, m> and the operator is viewed as time-dependent. Since H i> = , i> for each of the molecular states i>> we have... 

Considering the time-dependent position of a crystal electron r(t) as a quantum mechanical operator, we have from the usual formulation in the Heisenberg picture (see Appendix B) ... 

In the general treatment of propagators (see e.g. Zubarev 1960, 1964) it is usual to define all operators in the so-called Heisenberg picture this picture (implicit in the elementary treatment (Section 12.1) of time-dependent perturbations) must therefore be made explicit. To do this, we momentarily add a superscript S to distinguish wavefunctions and operators in the usual Schrodinger representation, and start from the time-dependent Schrodinger equation, which now becomes... 

The resultant wavefunction in the Heisenberg picture is thus reduced to rest , becoming UV)U(t)y (0) = f (0) but all operators (even those, like the Hamiltonian, that do not depend on time in the Schrddinger picture) in general acquire a time dependence. Thus a general Heisenberg operator is... 

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