1
$\begingroup$

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

$\endgroup$
2
$\begingroup$

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

$\endgroup$
1
$\begingroup$

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) $.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.