Time evolution operator 109for the ... in the Poisson bracket expression.” It can also be written as a differential
operator
iL=∑^3 N
α=1[
∂H
∂pα∂
∂qα−
∂H
∂qα∂
∂pα]
. (3.10.5)
The equation da/dt=iLacan be solved formally fora(xt) as
a(xt) = eiLta(x 0 ). (3.10.6)In eqn. (3.10.6), the derivatives appearing in eqn. (3.10.5) must be taken to act on
the components of the initial phase space vector x 0. The operator exp(iLt) appearing
in eqn. (3.10.6) is known as theclassical propagator. With theiappearing in the
definition ofiL, exp(iLt) strongly resembles the quantum propagator exp(−iHˆt/ ̄h) in
terms of the Hamiltonian operatorHˆ, which is why theiis formally included in eqn.
(3.10.4). Indeed, the operatorLcan be shown to be a Hermitian operator so that the
classical propagator exp(iLt) is a unitary operator on the phase space.
By applying eqn. (3.10.6) to the vector function a(x) = x, we have a formal solution
to Hamilton’s equations
xt= eiLtx 0. (3.10.7)
Although elegant in its compactness, eqn. (3.10.7) amounts to little more than a formal
device since we cannot evaluate the action of the operator exp(iLt) on x 0 exactly. If
we could, then any and every problem in classical mechanics could be solved exactly
analytically and we would not be in the business of developing numericalmethods or
buying expensive computers in the first place! What eqn. (3.10.7) does do is provide us
with a very useful starting point for developing approximate solutions to Hamilton’s
equations. As eqn. (3.10.5) suggests, the Liouville operator can bewritten as a sum of
two contributions
iL=iL 1 +iL 2 , (3.10.8)
where
iL 1 =∑N
α=1∂H
∂pα∂
∂qαiL 2 =−∑N
α=1∂H
∂qα∂
∂pα. (3.10.9)
The operators in eqn. (3.10.9) are examples ofnoncommutingoperators. This means
that, given any functionφ(x) on the phase space,
iL 1 iL 2 φ(x) 6 =iL 2 iL 1 φ(x). (3.10.10)That is, the order in which the operators is applied matters. The operator difference
iL 1 iL 2 −iL 2 iL 1 is an object that arises frequently both in classical and quantum
mechanics and is known as thecommutatorbetween the operators:
iL 1 iL 2 −iL 2 iL 1 ≡[iL 1 ,iL 2 ]. (3.10.11)If [iL 1 ,iL 2 ] = 0, then the operatorsiL 1 andiL 2 are said tocommute.