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1. (10 pts) For a harmonic oscillator with unit mass and angular frequency in the atomic units, m = w = h = 1, the eigenfunction is given by where An is the normalization constant and Hn(x) is the Hermite polynomial. (a) Calculate expectation values for ² and i22 given that n = 0. (b) Prove that is an eigenfunction and find out its eigenvalue. 2. (10 pts). CH3F belongs to C3v group. Determine the symmetry of all the normal modes using group theory. Find out the modes that are IR active. Find out the modes that are Raman active.

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