Poisson Brackets

where the function is the system energy at time expressed in terms of the classical coordinates and canonical momenta. This function is usually referred to as the

We are interested in
finding some
construct in classical dynamics that consists of
products of dynamical variables. If such a construct exists then we hope to
generalize it somehow to obtain a
rule describing how dynamical variables
commute with one another in quantum mechanics. There is, indeed,
one well-known construct
in classical dynamics that involves products of dynamical variables. The *classical
Poisson bracket* of two dynamical variables,
and
, is defined [55]

where and are regarded as functions of the coordinates and momenta, and , respectively. It is easily demonstrated that

(See Exercise 1.) The time evolution of a dynamical variable can also be written in terms of a Poisson bracket by noting that

where use has been made of

Let us attempt to construct a quantum mechanical Poisson bracket in which and are non-commuting operators instead of functions. Now, the main properties of the classical Poisson bracket are as follows:

(See Exercise 2.) The final relation is known as the

Actually, we can evaluate the quantum mechanical Poisson bracket, , in two different ways, because we can employ either of the formulae (2.12) or (2.13) first. Thus,

(2.15) |

and

(2.16) |

Note that the order of the various factors has been preserved in the previous expressions, because these factors now represent non-commuting operators. Equating the previous two results yields

(2.17) |

Because this relation must hold for and , quite independent of and , it follows that

(2.18) | ||

(2.19) |

where does not depend on , , , , and also commutes with . Because , et cetera, are general operators, it follows that is just a number. Now, we need the quantum mechanical Poisson bracket of two Hermitian operators to be itself an Hermitian operator, because the classical Poisson bracket of two real dynamical variables is real. This requirement is satisfied if is a real number. Thus, the quantum mechanical Poisson bracket of two dynamical variables and is given by

(2.20) |

where is a new universal constant of nature. Quantum mechanics agrees with experiments provided that takes the value , where

(2.21) |

is

It is easily demonstrated that the quantum mechanical Poisson bracket, as defined in the previous equation, satisfies all of the relations (2.8)-(2.14). (See Exercise 2.)

The strong analogy we have found between the classical Poisson bracket, defined in Equation (2.3), and the quantum mechanical Poisson bracket, defined in Equation (2.22), leads us to assume that the quantum mechanical bracket has the same value as the corresponding classical bracket, at least for the simplest cases. In other words, we are going to assume that Equations (2.4)-(2.6) hold for quantum mechanical, as well as classical, Poisson brackets. This argument yields the fundamental commutation relations

These results provide us with the basis for calculating commutation relations between general dynamical variables. For instance, if two dynamical variables, and , can both be written as a power series in the and then repeated application of Equations (2.8)-(2.14) allows to be expressed in terms of the fundamental commutation relations (2.23)-(2.25).

Equations (2.23)-(2.25) provide the foundation for the analogy between quantum mechanics and classical mechanics. Note that the classical result that dynamical variables commute is obtained in the limit . Thus, classical mechanics can be regarded as the limiting case of quantum mechanics as goes to zero. In classical mechanics, each generalized coordinate and its conjugate momentum, and , respectively, correspond to a different classical degree of freedom of the system. It is clear from Equations (2.23)-(2.25) that, in quantum mechanics, the dynamical variables corresponding to different degrees of freedom commute. In fact, it is only those variables corresponding to the same degree of freedom that may fail to commute.