Classical mechanics Poisson bracket

The similarity of these classical mechanical Hamilton s equations of motion to their quantum mechanical Ehrenfest s Theorem counterpart, (see Section 9.1.7), is an expression of the Correspondence Principle equivalence of a quantum mechanical commutator, [A, B], to a classical mechanical Poisson bracket,... 

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... 

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. 

There remains but one important concept to complete our summary of the role of canonical transformations in classical mechanics, that of the Poisson bracket. Let F p,q) and G p,q) denote two mechanical properties of the system. Their Poisson bracket is defined as... 

The analoguous behiaviour of the Poisson bracket and the commutator has been used to establishcorrespondence between classical and quantum mechanics. It is, however/shown in the next section following the derivation of Schwinger s quantum action principle that the correspondence goes deeper and that the analogous behaviour of the Poisson bracket and commutator is a consequence of the properties of infinitesimal canonical transformations which are common to both mechanics. 

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

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 Poisson bracket A, B in classical mechanics is given by the expression... 

The last Eq. (2.84) is called Jacobi s identity it is simple but tedious to prove. We note in passing that the Poisson brackets of classical mechanics presented here will be promoted to the commutators of quantum mechanics (scaled by —i/h) by the transition to quantum theory in chapter 4. 

Note another analogy to classical physics, namely that the commutator relations for conjugate variables resemble the Poisson brackets for these variables introduced in section 2.3.2 in the framework of classical mechanics. 

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