Poisson bracket operator

The operator 4 = Vp -Vr —Vr -Vp is the negative of the Poisson bracket operator, and the subscript W indicates the partial Wigner transform. The partial Wigner transform of the total Hamiltonian is written as,... 

We see that the correlation functions have a rather complex form when expressed in terms of Wigner transformed variables, involving exponential operators of the Poisson bracket operator. 

The Wigner form of the quantum evolution operator iLw X ) in (62) for the equation of motion for W X, X2,t) can be rewritten in a form that is convenient for the passage to the quantum-classical limit. Recalling that the system may be partitioned into S and S subspaces, the Poisson bracket operator A can be written as the sum of Poisson bracket operators acting in each of these subspaces as A Xi) = A xi) + A Xi). Thus, we may write... 

Poisson bracket operator in classical mechanics [...] square matrix... 

The arrows over the gradient operators indicate their direction of operation when the Poisson bracket operator A p, q ) is inserted inEq. (209). 

The Liouville and the Poisson bracket operators are related as follows... 

For the quantum mechanical case, p and Ware operators (or matrices in appropriate representation) and the Poisson bracket is replaced by the connnutator [W, p] If the distribution is stationary, as for the systems in equilibrium, then Bp/dt = 0, which implies... 

A stationary ensemble density distribution is constrained to be a functional of the constants of motion (globally conserved quantities). In particular, a simple choice is pip, q) = p (W (p, q)), where p (W) is some fiinctional (fiinction of a fiinction) of W. Any such fiinctional has a vanishing Poisson bracket (or a connnutator) with Wand is thus a stationary distribution. Its dependence on (p, q) through Hip, q) = E is expected to be reasonably smooth. Quanttun mechanically, p (W) is die density operator which has some fiinctional dependence on the Hamiltonian Wdepending on the ensemble. It is also nonnalized Trp = 1. The density matrix is the matrix representation of the density operator in some chosen representation of a complete orthononnal set of states. If the complete orthononnal set of eigenstates of die Hamiltonian is known ... 

Liouville operator LnZi = using the common Poisson brackets... 

Equation (31) is known as Heisenberg s equation of motion and is the quantum-mechanical analogue of the classical equation (17). The commutator of two quantum-mechanical operators multiplied by 2mfh) is the analogue of the classical Poisson bracket. In quantum mechanics a dynamical quantity whose operator commutes with the Hamiltonian, [A, H] = 0, is a constant of the motion. 

The Liouvillan operator is defined in terms of the Poisson bracket with the Hamiltonian ... 

The first term in the evolution operator has the form of a Poisson bracket and evolution under this part of the operator can be expressed in terms of characteristics. The corresponding set of ordinary differential equations is... 

The operator A is given by A =VpVp — VrVp= — , , where , are the Poisson brackets with respect to the classical phase-space variables... 

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 many applications, the operator L is, besides being skew symmetric, also a Poisson operator, which means that (AX,LBX) is a Poisson bracket denoted hereafter by the symbol A,B A and B are real-valued functions (sufficiently regular) of x. We recall that the Poisson bracket, in addition to satisfying the skew symmetry A, B = — B,A, satisfies also the Jacobi identity A, B,C + B, C,A ... 

The Poisson bracket is an operator in Hamiltonian mechanics which has convenient inherent properties considering the time evolution of dynamic variables [61] [73]. The most important property of the Poisson bracket is that it is invariant under any canonical transformation. 

The second equality in Eq. (1.99) defines the Poisson brackets and C is called the (classical) Lionville operator. Consider next the ensemble average 4(Z) = A t of the dynamical variable A. This average, a time-dependent observable, can be expressed in two ways that bring out two different, though equivalent, roles played by the function -4(r, p ). First, it is a function in phase space that gets a distinct numerical value at each phase point. Its average at time t is therefore given by... 

Earlier in this section it was commented on how the minimal-coupling QED Hamiltonian is obtained from fhe classical Lagrangian function. A few words are in order regarding the derivation of the multipolar Hamiltonian (6). One method involves the application of a canonical transformation to the minimal-coupling Hamiltonian [32]. In classical mechanics, such a transformation renders the Poisson bracket and Hamilton s canonical equations of motion invariant. In quantum mechanics, a canonical transformation preserves both the commutator and Heisenberg s operator equation of motion. The appropriate generating function that converts H uit is propor-... 

In classical mechanics the abstract Hermitian operator L, defined as operating on the Hilbert space of distributions, is Lc = -i, H), whereas it is Lq = h l [, H] in quantum mechanics. Here, denotes a Poisson bracket and [, ] denotes a commutator. 

The classical expression, Eq. (5), corresponds to the quantum mechanical one of Eq. (4). In fact, starting from the classical expression and, as suggested by Dirac [77], replacing the Poisson bracket of two classical quantities F and G by the commutator of the respective operators as... 

The essential point is that by virtue of the quantum nature of the problem, the operators x, and p, satisfy certain commutation relations (which in turn come, through proper use of the correspondence principle, from the Poisson brackets). These relations, which contain within themselves the specific aspects of the physical interaction between particles (nuclei, nucleons, etc.), lead to a set of precise commutation relations of the operators a, and a], defined in (2.23). The case of a three-dimensional harmonic oscillator is such that starting from... 

In this notation the arrows indicate whether the partial differentiation acts to the left (on A) or to the right (on B). With the Poisson bracket we associate the adjoint operator... 

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