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:

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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

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where

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

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Therefore, we see that tex2html_wrap_inline384 or

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

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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:

eqnarray80

However,

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

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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:

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which can be rearranged to yield

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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

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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:

eqnarray129

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

eqnarray149

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

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

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