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# Baker Campbell Hausdorff-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

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 $$\hat{H} = \hat { T } + \hat { V }$$.

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

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

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

1

# Baker Campbell Hausdorff Formula for the Time Evolution Operator

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