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Could someone explain me, what the Hausdorff expansion is?
In my quantum mechanics homework I should use something called the Hausdorff expansion which looks like the following: $$e^ABe^{-A}=B+[A,B]+\frac{1}{2!}[A,[A,B]]+\frac{1}{3!}[A,[A,[A,B]]]+...$$ and I don't really understand what it is or why is it good.

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    $\begingroup$ What part of it do you not "really understand"? $\endgroup$ – the boy who believed Mar 6 at 0:38
  • $\begingroup$ No, what you just wrote down is a simple combinatoric Lemma of Hadamard useful in working out the CBH expansion. If that's all you want, WP proves it just fine. $\endgroup$ – Cosmas Zachos Mar 6 at 2:07
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When A and B do not commute with each other, one can not write $e^A B e^{-A} = e^A e^{-A} B = B$. What you can do is to expand the exponentials and calculate term by term. When you expand them here, you get \begin{equation*} \begin{split} e^A B e^{-A} & = (1 + A + \frac{A^2}{2!} + \cdots) B (1 - A + \frac{A^2}{2!} - \cdots) \\ & = (B + AB + \frac{A^2 B}{2!} + \cdots)(1 - A + \frac{A^2}{2!} - \cdots) \\ & = B + AB + \frac{A^2 B}{2!} - BA - ABA + \frac{B A^2}{2!} + higher~orders \\ & = B + [A, B] + \frac{1}{2!}(A^2 B - 2 ABA + BA^2) + \cdots\\ & = B + [A, B] + \frac{1}{2!} [A, [A, B]] + \cdots\\ \end{split} \end{equation*}

There you arrive at the formula for the Hadamard expansion.

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