Why does e^(-iHt/ℏ) = e^(-iTt/ℏ)e^(-iVt/ℏ) + O(t²)?

[Niles]

Homework Statement

Hi guys

In my book they use the following identity

$$e^{ - i\widehat Ht/\hbar } = e^{ - i\widehat Tt/\hbar } e^{ - i\widehat Vt/\hbar } + O(t^2 )$$

where H = T+V, and the last term means "terms of order t² or higher". I can't quite see how they reach this identity. First, I know that T and V do not commute, so I guess that is where the O(t²) comes from.

Any help will be appreciated. (BTW, this is not an exercise problem -- just something in my book that I cannot understand).Niles.

 

[vela]

Look up the Baker-Hausdorff lemma.