Why are spectroscopic bands not sharp

We describe the vibrational levels in the lower (ground-state) Morse curve with the quantum number v , the lowermost vibrational level being v = 0. The vibrational states in the upper (excited-state) Morse curve are described by the quantum number v.  

Each electronic state also has vibrational sub-states. 

An electron excites from a vibrational energy level in the lower, ground-state Morse curve, to a vibrational level in the excited-state Morse curve. 

Instead of saying the vibrational level v = 0, we could say that the vibrational quantum number v = 0. 

The simple quantum-mechanical rule governing transitions between vibrational levels is summarized by 

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