Representation, Heisenberg

The early approaches to this model used perturbative expansion for weak coupling [Silbey and Harris 1983]. Generally speaking, perturbation theory allows one to consider a TLS coupled to an arbitrary bath via the term where / is an operator that acts on the bath variables. The equations of motion in the Heisenberg representation for the a operators, 8c/8t = ih [H, d], have the form... 

The u and v representations are sometimes distinguished as the Schrodinger and the Heisenberg representation. For stationary operators P, then, the Heisenberg equation of motion is... 

Compare this with Eq. (7-59) and observe the order of the factors on the right Equation (7-83) is the equation of motion of the statistical matrix in the Schrodinger representation p is constant, of course, in the Heisenberg representation. 

Here the relative intensities of the components of each branch are determined by the Boltzmann factor Correlation function K (t, J), corresponding to Gq(a>, J), is obviously the correlation function of a transition matrix element in Heisenberg representation... 

Heisenberg representation 267 helium see nitrogen-helium system Hermite polynomials 24, 25 Hubbard dependence 8 Hubbard relation... 

When 3fC is approximated as the sum of one-electron operators, considerable simplifications in the theory can be made. Following the important pioneering work of Bloss and Hone , this has been exploited, most often, by using the Heisenberg representation and solving the equations of motion for the c-operators. In a slightly different approach, Sebastian et wrote the... 

In a mixed quantum-classical calculation the trace operation in the Heisenberg representation is replaced by a quantum-mechanical trace (tTq) over the quantum degrees of freedom and a classical trace (i.e., a phase-space integral over the initial positions xq and momenta Po) over the classical degrees of freedom. This yields... 

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... 

Thus one has formally transferred the time dependence from the probability distribution onto the observed quantity - in analogy with the quantum mechanical transformation from the Schrodinger representation to the Heisenberg representation. Accordingly one may define a time-dependent vector Q(t) by setting... 

Heisenberg representation (matrix mechanics) the position and momentum are represented by matrices which satisfy this commutation relation, and ilr by a constant vector in Hilbert space, the eigenvalues E being the same in two cases,... 

The dynamic form factors can be written in terms of one-sided time-correlation functions. This is accomplished by transformation of Eq. (61) to the Heisenberg representation,... 

Casting this formula into the Heisenberg representation yields the relaxation time... 

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. 

In the following, we indicate the time derivative of a hermitian operator B with the symbol B. In the Heisenberg representation of quantum mechanics, it obeys the Heisenberg equation of motion... 

The derivation of a consistent mixed quantum-classical dynamics discussed in this paper was first proposed in Ref. [15] and commented and clarified in Ref. [1], This derivation is based on a group-theoretical formulation of quantum and classical mechanics, which introduces a very elegant and formally rigorous mathematical apparatus and allows to directly obtain classical mechanics as the limit for h —> 0 of quantum mechanics, in the Heisenberg representation of quantum dynamics. 

An operator in the Heisenberg representation can be expressed in the interaction representation as follows ... 

The preceding discussion has been completely based on the Heisenberg representation. The foundations of DFT, on the other hand, are usually formulated within the framework of the Schrodinger picture, so that one might ask in how far this field theoretical procedure can be useful. It is, however, possible to go over to an appropriately chosen Schrodinger representation as long as one does not try to eliminate the quantised photon fields (compare Sections 7d, lOg of Ref. [34]). The Hamiltonian then reads... 

Moreover, all corresponding counterterms (being expectation values) are independent of the representation, so that the renormalisation scheme remains unchanged and it is just a matter of convenience which representation is used. While the Heisenberg representation (2.20) is more suitable for the derivation of explicit functionals, the Hamiltonian (2.43) (together with Eq. (2.40)) can be utilised for the proof of a relativistic Hohenberg-Kohn theorem. 

Thus /4(tto) contains both the Hamiltonian and its own intrinsic evolution built in, and this may be recognized as defining the Heisenberg representation of any arbitrary operator. A general time correlation function associated with two general TD operators A t) and B t representing two different physical entities at different times t and t, is defined then by the formula... 

Eq. (36) is a version of the Heisenberg representation of an operator in the L-space, which will be useful in later manipulations. Similarly the time correlation functions between two operators given in Eq. (26) in H-space may also be re-expressed as a matrix element of the above form in the L-space. Thus in the... 

The ion-Hamiltonian Hi consists of its kinetic and mutual interaction terms with P(tfK) representing the ion momentum operator conjugate to the ion position operator R Ik), which in the Heisenberg representation is given by M Ri Kt) = P(lKt). The overdot here and elsewhere stands for the time derivative of the operator in the usual way. Here t and k stand for the ion index and species respectively. 

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