In following [Prof. Toyer's Computational Quantum Physics lecture notes](http://edu.itp.phys.ethz.ch/fs08/cqp/cqp.pdf), I came across the following:

In computing the Schrödinger equation in real space, one can make a "split operator" Ansatz, for which we rewrite the time evolution operator as

> 
> $$
> e ^ { - i \hbar \Delta \_ { t } H } = e ^ { - i \hbar \Delta \_ { t } \hat { V } / 2 } e ^ { - i \hbar \Delta \_ { t } \hat { T } } e ^ { - i \hbar \Delta \_ { t } \hat { V } / 2 } + \mathcal{O} \left( \Delta \_ { t } ^ { 3 } \right)
> $$
> 
> 
> 

with the Hamiltonian $\hat{H} = \hat { T } + \hat { V }$.

I'm not sure how to get to the $\mathcal{O}(\Delta\_t^3)$ term. 

I know that the [Zassenhaus formula](https://en.wikipedia.org/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula#The_Zassenhaus_formula), which follows from the [Baker-Campbell-Hausdorff formula](https://en.wikipedia.org/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula) states, for two non-commuting operators

$$
e ^ { t ( X + Y ) } = e ^ { t X } e ^ { t Y } e ^ { - \frac { t ^ { 2 } } { 2 } [ X , Y ] } e ^ { \frac { t ^ { 3 } } { 6 } ( 2 [ Y , [ X , Y ] ] + [ X , [ X , Y ] ] ) } \dots
$$

but there, I would pick up a term of $\mathcal{O}(\Delta\_t^2)$?