Bracket poisson

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 ... 

The linear response of a system is detemiined by the lowest order effect of a perturbation on a dynamical system. Fomially, this effect can be computed either classically or quantum mechanically in essentially the same way. The connection is made by converting quantum mechanical conmuitators into classical Poisson brackets, or vice versa. Suppose tliat the system is described by Hamiltonian where denotes an... 

An identical expression holds classically [32] if- //) times the conmuitator is replaced by the classical Poisson bracket. 

It is possible to introduce a generalized Poisson bracket by considering two general differentiable functions/(z,z ) andg(z,z ) and write... 

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

The Liouvillian iLo- = Ho, , where , is the Poisson bracket, describes the evolution governed by the bath Hamiltonian Hq in the held of the fixed Brownian particles. The angular brackets signify an average over a canonical equilibrium distribution of the bath particles with the two Brownian particles fixed at positions Ri and R2, ( -)0 = Z f drNdpNe liW J , where Zo is the partition function. 

Poisson brackets, transition state trajectory, deterministically moving manifolds, 226-228... 

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 ... 

Since for rare gas pairs p(R) = p(R)R, we may calculate the time derivative of p R) from the Poisson bracket... 

By expressing the time derivatives in terms of the Poisson bracket and canonically averaging, we get for the second binary moment... 

The second derivative of the dipole moment, which is needed for the computation of the fourth moment, is obtained by computing the Poisson bracket once more,... 

To do this we bring in another equation of constraint which has the property that its Poisson bracket with Q(x) does not vanish. In Dirac s terminology, this yields a canonical scheme with two second-class constraints. The extra constraint can be ta-... 

Their mutual Poisson brackets calculated using (34) yield a non-singular matrix with elements... 

The Dirac brackets are like the original Poisson brackets, being antisymmetric and satisfying the Jacobi identity, and are the basis for quantization using the usual correspondence [11-13]... 

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,... 

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... 

However, the quantum-classical brackets ( , ) introduced in eq.(7) are not Lie brackets [19], because they do not satisfy properties that are instead satisfied by the commutator and the Poisson brackets (respectively, quantum and classical Lie brackets), e.g., the Jacobi identity. 

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. 

We have thus obtained the expected result classically, the time derivative of a function of the phase-space (that does not depend explicitly on time) is given by the Poisson brackets between the function itself and the hamiltonian. This result has been obtained by taking the formal limit h — 0 on the quantum expression in eq.(30), i.e.,... 

We have thus reconstructed the derivation and interpreted the results of Ref [15], The first two terms, i.e., the commutator and the Poisson brackets, are already present in a theory based on the quantum-classical Liouville representation discussed in section 1. The new term, which appears within the Heisenberg group approach, needs to be explained. In the attempt to provide a physical interpretation to this term we have shown, in Ref. [1], that the new equation of motion is purely classical. This will be illustrated in the following section. 

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