Baker-Campbell-Hausdorff (BCH) Formula for the Time Evolution Operator In following Prof. Toyer's Computational Quantum Physics lecture notes, 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, which follows from the Baker-Campbell-Hausdorff 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  $\mathcal{O}(\Delta_t^2)$?
 A: $$e^{tV/2}e^{tT}e^{tV/2}~=~(e^{tV/2}e^{tT/2})(e^{tT/2}e^{tV/2})
~\stackrel{\text{BCH}}{=}~e^{tH/2+t^2[V,T]/8+{\cal O}(t^3)}e^{tH/2+t^2[T,V]/8+{\cal O}(t^3)}
~\stackrel{\text{BCH}}{=}~e^{tH+{\cal O}(t^3)},$$
where we have used repeatedly 
$$ e^{tX}e^{tY}=e^{t(X+Y)+t^2[X,Y]/2+{\cal O}(t^3)}\tag{BCH}.$$
A: You are wasting attention on the Zassenhaus formula, if you appreciate the symmetry between T  and V in your target expression,
$$
e ^ { - i \hbar \Delta _ { t } \hat { V } / 2 } e ^ { - i \hbar \Delta _ { t } \hat { T } } e ^ { - i \hbar \Delta _ { t } \hat { V } / 2 } .
$$
Instead, just heed the lowest nontrivial order CBH expansion, 
$$
\left( e ^ { - i \hbar \Delta _ { t } \hat { V } / 2 } e ^ { - i \hbar \Delta _ { t } \hat { T } }\right )~~ e ^ { - i \hbar \Delta _ { t } \hat { V } / 2 }=e ^ { - i \hbar \Delta _ { t } \hat { V } / 2   - i \hbar \Delta _ { t } \hat { T } -\hbar^2 \Delta _ { t }^2 [\hat V,\hat T]/4   + \mathcal{O} \left( \Delta _ { t } ^ { 3 } \right)}~~ e ^ { - i \hbar \Delta _ { t } \hat { V } / 2 } \\
=e ^ { - i \hbar \Delta _ { t } \hat { V }    - i \hbar \Delta _ { t } \hat { T }    + \mathcal{O} \left( \Delta _ { t } ^ { 3 } \right)}= e ^ { - i \hbar \Delta _ { t } \hat  H } + \mathcal{O} \left( \Delta _ { t } ^ { 3 } \right) .
$$
Perhaps the Zassenhouse formula  might reassure you of the legitimacy of lowering the $ \mathcal{O} \left( \Delta _ { t } ^ { 3 } \right) $.
