The Trotter Theorem and the classical propagator next up previous
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The Trotter Theorem and the classical propagator

The evolution of the phase space vector tex2html_wrap_inline374 that we have been examining via the classical propagator:


might appear to be of only formal significance because we cannot evaluate the action of tex2html_wrap_inline376 for a general Liouville operator.

The difficulty in evaluating the action of tex2html_wrap_inline376 can be seen by noting that iL is of the form




Thus, for any phase space function, f(x,p), it follows that


Therefore, we see that tex2html_wrap_inline384 or


The term in parentheses is known as the commutator between tex2html_wrap_inline386 and tex2html_wrap_inline388 and is denoted


Thus, tex2html_wrap_inline390 , so that the two operators do not commute with each other.

Unfortunately, this makes classical mechanics difficult. If we could split the classical propagator according to tex2html_wrap_inline392 , everything would be simple because we can evaluate the action of the two operators on the right on a phase space vector tex2html_wrap_inline374 . However, because tex2html_wrap_inline386 and tex2html_wrap_inline388 do not commute, the classical propagator cannot be factorized in this simple way. This can be easily verified by Taylor expanding both sides:




Fortunately, there is a theorem that we can use to factorize the classical propagator. This is the Trotter theorem, which states that


Although the proof of the theorem is somewhat beyond the scope of these lectures, it can be found in several places, in particular, see the appendix to Chapter 1 of Techniques and Applications of Path Integration by L. S. Schulman.

How does the Trotter theorem help us? Consider the approximation to tex2html_wrap_inline400 obtained by choosing M large but finite:


which can be rearranged to yield


The first of these expressions looks like approximate propagation of the system up to a time t by M applications of the operator in brackets. The second, therefore, looks like an approximation to the propagator for a small time interval t/M. If we interpret this as a single time step, tex2html_wrap_inline410 , then we have a single-time-step approximation to the propagator of the form


The left side is the exact propagator, tex2html_wrap_inline412 , for a time step, tex2html_wrap_inline410 . The right side represents an approximate propagator tex2html_wrap_inline416 .

There are several things to note about tex2html_wrap_inline416 . First, tex2html_wrap_inline420 . Thus, tex2html_wrap_inline416 is unitary and, therefore, preserves the time-reversibility of the dynamics:


Second, by Taylor expansion, it can be shown that tex2html_wrap_inline416 is accurate to order tex2html_wrap_inline426 :


where the last line indicates equivalence up to second order in tex2html_wrap_inline410 . Thus,


next up previous
Next: Action of the factorized Up: No Title Previous: No Title

Mark Tuckerman
Thu Oct 17 21:36:23 EDT 2002