Time dependent Baker-Hausdorf formula

In summary, the conversation discusses the Baker-Hausdorf formula for product of exponentials in the case of time-dependent operators. The formula only works under certain assumptions, such as the commutator commuting with all operators. The speaker initially expresses confusion about why time-dependence would matter, but the other person clarifies that the formula applies to all operators. The conversation also mentions a paper that includes a formula for computing with time-ordered exponentials and double commutators.
  • #1
paweld
255
0
Could anyone show me the Baker-Hausdorf formula for product of exponentials in case of
operators which are time dependent. I know that there is a time-dependent version of this
formula which works under some assumptions are imposed on the operators which appear
in exponentials, like e.g. commutator commutes with all operators.
Thanks.
 
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  • #2
Forgive me, but I don't see why any time-dependence would matter. It's an identity which applies to any and all operators. (I assume you're talking about the full theorem that includes double commutators, triple commutators, etc, not one that has been truncated.)
 
  • #3
Thanks for answer. I looked for the identity which involves time ordered exponentials
and includes only double commutators. Fortunantely I've just found paper
(http://www.sciencedirect.com/science/article/pii/S0167691101001943)
where a formula which enabled me to compute what I wanted was given.
 

1. What is the Time dependent Baker-Hausdorf formula?

The Time dependent Baker-Hausdorf formula is a mathematical formula used in quantum mechanics to calculate the evolution of a system over time. It is also known as the time-dependent Magnus expansion or the time-dependent Campbell-Baker-Hausdorf formula.

2. How does the Time dependent Baker-Hausdorf formula work?

The formula involves calculating a series of nested commutators, which represent the time evolution of an operator. It takes into account the time-dependent Hamiltonian of the system and can be used to calculate the time evolution of any quantum state or operator.

3. What is the significance of the Time dependent Baker-Hausdorf formula?

The formula is important because it allows for the calculation of the time evolution of quantum systems, which is essential in understanding and predicting the behavior of particles and systems in quantum mechanics. It also has applications in fields such as quantum computing and quantum information theory.

4. Can the Time dependent Baker-Hausdorf formula be simplified?

Yes, the formula can be simplified using approximations such as the Magnus expansion, which truncates the infinite series of nested commutators to a finite number of terms. This makes the calculation more manageable and allows for more efficient computation.

5. Are there any limitations to the Time dependent Baker-Hausdorf formula?

Like any mathematical formula, the Time dependent Baker-Hausdorf formula has its limitations. It is most accurate for short time intervals and can become less accurate for longer time intervals or in systems with strong interactions. Additionally, it may not be applicable in certain cases where the Hamiltonian is time-dependent in a non-analytic way.

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