Question $5$

$1.$ Compute the classical "orbit time" in attoseconds $\backslash (\backslash$left$(10^\{-18\}\backslash$mathrm$\{$~s$\}\backslash$right$)\backslash )$ for an electron in a hydrogen atom in its ground state. The orbit time is the time it takes for the electron to make one full revolution around the proton. For this problem you can take the orbit length to be the circumference of an orbit at the most probable proton$/$separation$,$ i$.$e$.,$ the Bohr radius $\backslash ($ r$\_\{\backslash$mathrm$\{$MP$\}\}=$a$\_\{0\}=0\mathrm{.}592\backslash$AA $\backslash ).$ For the orbit speed, you need to compute the RMS speed of the electron $\backslash ($ v$\_\{\backslash$text $\{$RMS $\}\}\backslash ).$

$2.$ To put your result in context, if a bond vibration period in a typical molecule is on the order of $10$ femtoseconds, how many times can an electron orbit the proton in the time it takes for one bond vibration? Of course electrons do not move in deterministic orbits like this, but this problem gives a useful measure of atomic time scales.