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

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

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 Hamilton's equations, (92)-(93).

Can we construct a quantum mechanical Poisson bracket in which and are non-commuting operators, instead of functions? Well, the main properties of the classical Poisson bracket are as follows:

The last 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 (103) or (104) first. Thus,

(106) |

and

(107) |

Note that the order of the various factors has been preserved, because they now represent

(108) |

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

(109) | ||

(110) |

where does not depend on , , , , and also commutes with . Because , etc., are general operators, it follows that is just a number. We want the quantum mechanical Poisson bracket of two Hermitian operators to be a Hermitian operator itself, 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

(111) |

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

(112) |

is

It is easily demonstrated that the quantum mechanical Poisson bracket, as defined above, satisfies all of the relations (99)-(105).

The strong analogy we have found between the classical Poisson bracket, defined in Equation (94), and the quantum mechanical Poisson bracket, defined in Equation (113), 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 assuming that Equations (95)-(97) 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 (99)-(104) allows to be expressed in terms of the fundamental commutation relations (114)-(116).

Equations (114)-(116) provide the foundation for the analogy between quantum mechanics
and classical mechanics. Note that the classical result (that everything commutes)
is obtained in the limit
. Thus, *classical mechanics
can be regarded as the limiting case of quantum mechanics when
goes to zero*.
In classical mechanics, each
pair of generalized coordinate and its conjugate momentum,
and
, correspond to a different classical degree of freedom of the system.
It is clear from Equations (114)-(116) that in quantum mechanics the *dynamical
variables corresponding to different degrees of freedom all commute*.
It is only those variables corresponding to the same degree of freedom that
may fail to commute.