4. [25] A particle of mass m moves within a region x [0, 1] outside of which it encounters an infinite potential. The pa

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4 25 A Particle Of Mass M Moves Within A Region X 0 1 Outside Of Which It Encounters An Infinite Potential The Pa 1 (92.59 KiB) Viewed 41 times

4. [25] A particle of mass m moves within a region x [0, 1] outside of which it encounters an infinite potential. The particle is prepared in a state V), such that the initial (normalised) position-picture wave-function reads Ja[L²/4 - (x - L/2)²1 V(x,0) = = for x € (0, L], otherwise. with a ER+ [7] (a) Show that a = 30/L5. Using the representation of the momentum operator in the position picture, calculate the mean energy of the particle prepared in ). (b) Calculate the probability that the particle is found, at t = 0, within the region [0, L/2] and justify your answer using symmetry arguments. (c) Suppose now that the particle is subjected to an external perturbation [8] û û) = eħpx with € a perturbative parameter. Determine the second-order correction to the least- energy eigenstate of the unperturbed Hamiltonian. (Hint: Use without proof the follow- ing results: So sin(ntx/L) cos(Tx/L)dx = nL(1+cos(n)) 7(n2-1) n=0 (An2-1)3 = 72/256]. 7 [10]