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# p3 - Problem Set 3 Physics 341 Due October 29 Some...

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Problem Set 3 Physics 341 Due October 29 Some abbreviations: S - Shankar. 1 . Let’s start with a straightforward warm-up exercise. Consider four operators A, B, C, D . Show that [ AB, CD ] = A [ B, C ] D + AC [ B, D ] + [ A, C ] DB + C [ A, D ] B. Now consider e λA e λB = e C where A, B, C are operators and λ is a small parameter. Find an expression for C in terms of A and B including terms of O ( λ 4 ). Evaluate this expression for A = x and B = p . 2 . For a density matrix ρ , please show that ρ = ρ, Tr ρ = 1 , Tr ρ 2 1 . For a pure ensemble, show that ρ 2 = ρ, Tr ρ 2 = 1 . 3 . A particle of mass m moves in one dimension in an infinite square well potential extending from - a to + a . At time t = 0, the system is described by the wavefunction Ψ( x, 0) = c 0 ( x ) + 2Ψ 1 ( x )) where Ψ 0 and Ψ 1 are the normalized eigenfunctions for the ground state and first excited state, respectively. (i) Find the value of c for which the wavefunction is normalized. (ii) Compute the probability that the particle is found in the interval - a < x < 0 at time t .

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p3 - Problem Set 3 Physics 341 Due October 29 Some...

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