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Problem 1 - Harmonic Oscillator The eigenfunctions of the harmonic oscillator are )n(ar) = on certain uwo 1/4 Hu(y no one uwo -x)e-twox2 /2h, n = 0,1,2,... with Hermite polynomials H(x) = (-)"ez? (d" /dx)e-z?. (a) Verify that the n = 0 and n = 1 eigenfunctions are orthonormal, i.e., (Un|4m) = Onm. (b) Verify that the n = O and n = 1 eigenenergies En eigenfunctions. (n + 1/2)Hwo correspond to these (c) Calculate (Un|x|4m) for n, m = 0,1. (d) For n = 0 only, find the average value of the kinetic energy T and potential energy V and verify that (T) = (V) in this case. Problem 1 - Harmonic Oscillator The eigenfunctions of the harmonic oscillator are )n(ar) = on certain uwo 1/4 Hu(y no one uwo -x)e-twox2 /2h, n = 0,1,2,... with Hermite polynomials H(x) = (-)"ez? (d" /dx)e-z?. (a) Verify that the n = 0 and n = 1 eigenfunctions are orthonormal, i.e., (Un|4m) = Onm. (b) Verify that the n = O and n = 1 eigenenergies En eigenfunctions. (n + 1/2)Hwo correspond to these (c) Calculate (Un|x|4m) for n, m = 0,1. (d) For n = 0 only, find the average value of the kinetic energy T and potential energy V and verify that (T) = (V) in this case