Baker-Campbell-Hausdorff (BCH) Formula for the Time Evolution Operator

Question

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 $\mathcal{O}(\Delta_t^2)$?

Answer 1

$$e^{tV/2}e^{tT}e^{tV/2}~=~(e^{tV/2}e^{tT/2})(e^{tT/2}e^{tV/2}) ~\stackrel{\text{BCH}}{=}~e^{tH/2+t^2[V,T]/8+{\cal O}(t^3)}e^{tH/2+t^2[T,V]/8+{\cal O}(t^3)} ~\stackrel{\text{BCH}}{=}~e^{tH+{\cal O}(t^3)},$$ where we have used repeatedly $$ e^{tX}e^{tY}=e^{t(X+Y)+t^2[X,Y]/2+{\cal O}(t^3)}\tag{BCH}.$$

Answer 2

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) $.