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Consider the following defined commutation relations:

$$[\hat a,\hat a^{\dagger}]=1$$

$$[\hat b,\hat b^{\dagger}]=1$$

$$[\hat a,\hat b]=0$$

(where the usual algebra of commutators holds)

Let us now consider the operator:

$$\hat H=\hat a^{\dagger}\hat a+\hat b^{\dagger}\hat b+\hat a^{\dagger}\hat a(b^{\dagger}+b)+(a^{\dagger}+a)(b^{\dagger}+b)$$

I now want to evaluate the following using Mathematica: $$e^{\hat A}\hat He^{−\hat A}=[\hat A,\hat H]+[\hat A,[\hat A,\hat H]]+[\hat A,[\hat A,[\hat A,\hat H]]].....$$

where, $\hat A=(\hat a^{\dagger}-\hat a+\hat b^{\dagger}-\hat b)$

As can be seen, this is a really difficult task to perform by hand and I want to use Mathematica to do it. Since I am new at Mathematica, I don't even know how to begin.

Maybe I would have to somehow define the above relations as rules but I don't then know how to make Mathematica follow the commutator algebra rules.

I am not looking for a complete solution (though it's okay if it comes as an answer since I can always use it to match with my code:), just a guide on how to start this problem in Mathematica.

EDIT: From the link in the comment by evanb I realized that it's rather hard to do this in Mathematica. Has anybody used the SNEG package who can guide me on how to install or use it in Mathematica?

I am hoping that that should make my work easier. Since explicitly writing the code might take a very long time. I even have to change $\hat A$ in the above expression and try for many such $\hat A$'s.

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    $\begingroup$ Related: mathematica.stackexchange.com/q/95616/7936 $\endgroup$
    – evanb
    Aug 5, 2021 at 12:28
  • $\begingroup$ Also: mathematica.stackexchange.com/q/63811/7936 $\endgroup$
    – evanb
    Aug 5, 2021 at 12:33
  • $\begingroup$ The expression you're looking for is $e^\hat{A} \hat{H} e^{-\hat{A}}$ by the Baker-Hausdorff lemma. I assume that you already knew this, but if other non-physicists try to answer the question it might allow them to do an end run around calculating the entire series. (In particular, if you had a finite-dimensional representation of the operators, you could use MatrixExponential; but I suspect from the context that no such representation exists.) $\endgroup$ Aug 5, 2021 at 15:09
  • $\begingroup$ @Michael Seifert Yes. I have directly written the expansion of the above. I will edit and add the expression above. Also, I think the name of the above expansion is Hadamard Lemma. Baker-Hausdorff lemma is for the expansion of $e^{\hat A}e^{\hat B}$ I think. $\endgroup$
    – Lost
    Aug 5, 2021 at 15:12
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    $\begingroup$ It's called the "Baker-Hausdorff lemma" in Sakurai's Modern Quantum Mechanics (not to be confused with the Baker-Campbell-Hausdorff formula, which gives us an expansion for $e^\hat{A} e^\hat{B}$.) But it's entirely possible that there are other names for it. $\endgroup$ Aug 5, 2021 at 15:16

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