Preparing for a presentation at university (I'm a Bachelor physics student) I have come across the formula below containg the time-ordering operator $T$. Although i have now understood the action of the time ordering operator (which is admittedly fairly easy once you get it), i couldn't prove the second equality in

$F(t)=e^{i H t / \hbar} e^{-i\left(H+\Delta_{0}\right) t / \hbar} = \exp _{\mathrm{T}}\left[\frac{-i}{\hbar} \int_{0}^{t} \mathrm{d} \tau \Delta(\tau)\right]$ with $\Delta(\tau)=e^{i H \tau / \hbar} \Delta_{0} e^{-i H \tau / \hbar}$ (please note that the $\tau$ is no subscript, but rather is multiplied onto $H$)

Starting from the right I got

$\exp _{\mathrm{T}}\left[\frac{-i}{\hbar} \int_{0}^{t} \mathrm{d} \tau \Delta(\tau)\right] = \sum_{n=0}^{inf} \left( -\frac{i}{\hbar} \right)^{n}\frac{1}{n!} \int_{0}^{t}dt_1 \int_{0}^{t}dt_2 ... \int_{0}^{t}dt_n T[\Delta(t_1)\Delta(t_2)...\Delta(t_n)] = \sum_{n=0}^{inf} \left( -\frac{i}{\hbar} \right)^{n} \int_{0}^{t}dt_1 \int_{0}^{t_1}dt_2 ... \int_{0}^{t_{n-1}}dt_n [\Delta(t_1)\Delta(t_2)...\Delta(t_n)] $

However, from that point I don't quite see where to go

Edit: An important information: $H$ and $\Delta_0$ are both time-indipendent.

Update: In order to convince myself of the equation to be true I expanded both sides into a taylor series and ordered them in orders of t. Up to the third order ($t^3$) the terms are equal, thus I am quite sure now, that the equation is true. However, I still lack a proper proof.


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