Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / Section E / 27 / a / ii / Solution

For an unnormalized representation, divide the first moment by its total mass. Integration by parts givesConsequentlyFor normalized amplitudes the denominator is one. A mode with small weight can still dominate the mean if its lifetime $1/{\nu }_l$ is sufficiently large; this distinction matters when interpreting the later graph.