A particle of mass m is in a region with potential energy operator V = ki. If theparticle is in its ground state, then its normalized wavefunction is1/42mEVo(1,t) =(1)and its energy is E, = th, where w = Vk/m. This state is called the ground state ofthe quantum harmonic oscillator.Recall that the definition of the expectation value of a one-dimensional quantumoperator A is(4) = [_ våvdz.(2)where Å is sandwiched betreen the wavefunction and its complex conjugate, NOT infront of both of them. For example, (2) = 0 and () = 0, because of the symmetry ofthe wavefunction about the vertical axis.(a) Calculate the expectation values () and () of our wavefunction. You may findthe following Gaussian integral identities useful:dr=(3)dr =0(4)dr=(5)(b) Standard deviations are defined from expectation values as o, = V() – (2)²and o, = V) - (. Heisenberg's uncertainty principle can be formalized withstandard deviations representing uncertainty, o, 2 Show that the ground state ofthe quantum harmonic oscillator saturates Heisenberg's uncertainty principle, that is,the inequality becomes an equality for Ve(r,f).(c) Show that our wavefunction satisfies E, = E) + (v).

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