, ▁even ▁at ▁the ▁absolute ▁zero ▁of ▁temperature . ▁At ▁absolute ▁zero ▁all ▁atoms ▁are ▁in ▁their ▁v ibr ational ▁ground ▁state ▁and ▁show ▁zero ▁point ▁quantum ▁mechanical ▁motion , ▁so ▁that ▁the ▁wave function ▁of ▁a ▁single ▁v ibr ational ▁mode ▁is ▁not ▁a ▁sharp ▁peak , ▁but ▁an ▁exponential ▁of ▁finite ▁width ▁( the ▁wave function ▁for ▁n   =   0 ▁dep icted ▁in ▁the ▁article ▁on ▁the ▁quantum ▁harm onic ▁oscill ator ). ▁ ▁At ▁higher ▁temper atures ▁the ▁v ibr ational ▁modes ▁may ▁be ▁therm ally ▁excited ▁( in ▁a ▁classical ▁interpretation ▁one ▁express es ▁this ▁by ▁stating ▁that ▁" the ▁mole cules ▁will ▁v ibr ate ▁faster "), ▁but ▁they ▁oscill ate ▁still ▁around ▁the ▁recogn izable ▁geometry ▁of ▁the ▁mole c ule . ▁ ▁To ▁get ▁a ▁feeling ▁for ▁the ▁probability ▁that ▁the ▁v ibration ▁of ▁mole c ule ▁may ▁be ▁therm ally ▁excited , ▁we ▁inspect ▁the ▁Bol tz mann ▁factor ▁, ▁where ▁ Δ E ▁is ▁the ▁exc itation ▁energy ▁of ▁the ▁v ibr ational ▁mode , ▁k ▁the ▁Bol tz mann ▁constant ▁and ▁T ▁the ▁absolute ▁temperature . ▁At ▁ 2 9 8   K ▁( 2 5   ° C ), ▁typical ▁values ▁for ▁the ▁Bol tz mann ▁factor ▁ β ▁are : ▁▁ β ▁= ▁ 0 . 0 8 9 ▁for ▁ Δ E ▁= ▁ 5 0 0   cm − 1 ▁▁ β ▁= ▁ 0 . 0 0 8 ▁for ▁ Δ E ▁= ▁ 1 0