(91) | |||

(92) |

where the function is the energy of the system 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 of classical dynamics which consists of
*products* of dynamical variables. If such a construct exists 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 which involves products of dynamical variables. The
*Poisson bracket* of two dynamical variables and is defined

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.

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

and

The last relation is known as the

Well, we can evaluate the Poisson bracket
in
two different ways, since we can use either of the formulae (102) or
(103) first. Thus,

(105) | |||

and

(106) | |||

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

(107) |

(108) | |||

(109) |

where does not depend on , , , , and also commutes with . Since ,

(110) |

(111) |

It is easily demonstrated that the quantum mechanical Poisson bracket, as defined above, satisfies all of the relations (98)-(104).

The strong analogy we have found between the classical Poisson bracket, defined
in Eq. (93), and the quantum mechanical
Poisson bracket, defined in Eq. (112), leads
us to make the assumption 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 Eqs. (94)-(96) 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 Eqs. (98)-(103) allows to be expressed in terms of the fundamental commutation relations (113)-(115).

Equations (113)-(115) 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 Eqs. (113)-(115) 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 which
may fail to commute.