110 Microcanonical ensemble
ThatiL 1 andiL 2 donotgenerally commute can be seen in a simple one-dimensional
example. Consider the Hamiltonian
H=
p^2
2 m+U(x). (3.10.12)According to eqn. (3.10.9),
iL 1 =p
m∂
∂x, iL 2 =F(x)∂
∂p, (3.10.13)
whereF(x) =−dU/dx. The action ofiL 1 iL 2 on a functionφ(x,p) is
p
m∂
∂xF(x)∂
∂pφ(x,p) =p
mF(x)∂^2 φ
∂p∂x+
p
mF′(x)∂φ
∂p, (3.10.14)
whereas the action ofiL 2 iL 1 onφ(x,p) is
F(x)∂
∂pp
m∂
∂xφ(x,p) =F(x)p
m∂^2 φ
∂p∂x+F(x)1
m∂φ
∂x, (3.10.15)
so that [iL 1 ,iL 2 ]φ(x,p) is
[iL 1 ,iL 2 ]φ(x,p) =p
mF′(x)∂φ
∂p−
F(x)
m∂φ
∂x. (3.10.16)
Since the functionφ(x,p) is arbitrary, we can conclude that the operator
[iL 1 ,iL 2 ] =p
mF′(x)∂
∂p−
F(x)
m∂
∂x, (3.10.17)
from which it can be seen that [iL 1 ,iL 2 ] 6 = 0.
SinceiL 1 andiL 2 generally do not commute, the classical propagator exp(iLt) =
exp[(iL 1 +iL 2 )t]cannotbe separated into a simple product exp(iL 1 t) exp(iL 2 t). This is
unfortunate because in many instances, the action of the individual operators exp(iL 1 t)
and exp(iL 2 t) on the phase space vector can be evaluated exactly. Thus, it would be
useful if the propagator could somehow be expressed in terms of these two factors.
In fact, there is a way to do this using an important theorem known as theTrotter
theorem(Trotter, 1959). This theorem states that given two operatorsAandBfor
which [A,B] 6 = 0,
eA+B= lim
P→∞[
eB/^2 PeA/PeB/^2 P]P
, (3.10.18)
wherePis an integer. In fact, eqn. (3.10.18) is commonly referred to as thesymmetric
Trotter theoremorStrang splitting formula(Strang, 1968). The proof of the Trotter