Up: Position and Momentum
Consider a dynamical system whose state at a particular time
independent classical coordinates
runs from 1 to
Associated with each generalized coordinate
classical canonical momentum
. For instance, a Cartesian coordinate has an associated linear
momentum, an angular coordinate has an associated angular momentum, etc.
As is well-known, the behavior of a classical system can be specified in terms
of Lagrangian or Hamiltonian dynamics. For instance, in Hamiltonian dynamics,
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 Hamiltonian of the system.
We are interested in
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
are regarded as functions of the coordinates
. It is easily demonstrated that
The time evolution of a dynamical variable can also
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
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 Jacobi identity. In the above,
, etc., represent dynamical variables, and
represents a number.
Can we find some combination of non-commuting operators
that satisfies all of the above relations? We shall refer to such a combination as a quantum mechanical Poisson bracket.
Actually, we can evaluate the quantum mechanical Poisson bracket
two different ways, because we can employ either of the formulae (103) or
(104) first. Thus,
Note that the order of the various factors has been preserved, because they
now represent non-commuting operators. Equating the above two results
Since this relation must hold for
quite independent of
, it follows that
does not depend on
, and also
, etc., are general
operators, it follows that
is just a number. We want the quantum
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
is given by
is a new universal constant of nature. Quantum mechanics agrees with
experiments provided that
takes the value
is Planck's constant. The notation
conventionally reserved for the commutator
in quantum mechanics.
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,
can both be written as a power series in the
then repeated application of Equations (99)-(104)
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,
, 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.
Up: Position and Momentum