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1. Show that explicit application of the lowering operator in terms of x and d/dx operators on the
n = 3 harmonic oscillator wavefunction y(x) state leads to the n = 2 wavefunction.

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1.The odd parity eigenstates of the infinte square well , with potential V = 0 in the range −L/2 ≤ ? ≤ L/2, are given by : (see figure) and have Ψn(x, t) = 0 elsewhere , for n=2 , 4 , 6 etc a) Sketch the potential of this system , including in your sketch the positions of the lowest three energy levels . Indicate in your sketch the form of the wavefunction for a particle in each of these energy levels , and state which of the wavefunctions you have drawn could be decirbed by the Ψn written above (see figure) . b) Calculate the expectation value of momentum , ⟨p⟩ for a particle with n=2 c) Calculate the expectation value of momentum squared ⟨p 2⟩ , for a particle with n = 2 . Hint : you may use the mathematical identiy sin2 x = 1/2 (1 − cos 2x) without proof .