__Question based on Many-particle systems and indistinguishability in Quantum Mechanics__

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A pair of non-interacting particles is in a one-dimensional infinite square well with walls at r =0 and r = L. Throughout the question, one particle is in the ground state with normalized energy eigenfunction IN9 sin WA(I) = for 0 < < < L L elsewhere, and the other particle is in the first excited state with normalized energy eigenfunction 2 sin L L for 0 < < < L elsewhere. (a) Suppose that the particles are identical spin-- fermions and that the two-particle system has total spin quantum number S =0. Explain what this implies about the symmetries of the spin and spatial parts of the total wave function and hence write down an explicit expression for the spatial wave function (11, 12) of the two-particle system at time

t = ﬂ. Your expression should include an appropriate normalization factor, but no proof of normalization is required. [b] For the case discussed in part [a], write down an expression for the probability density that one particle [artiﬁcially labelled 1) is at 1:1 and the other particle [labelled 2) is at :2. Hence calculate the probability that both particles are found in the left-hand side of the box [i.e. with position coordinates U 11'. 1:1 if. L f 2 and U E. 5:32 sf. LIE). You may use the integrals 142 _ 2 m: L sm — d3:— f, (L) 4 I"? 27m: L - 2 sm — d3:— fa (L) 4 L32 , 7m: _ 27m: 2L 1,; sm (I) srn (T) d1: = 3—1r' {c} Suppose that the two identical fermions occupy the same two energ.»r eigenstates as before, but that they are now in a two-particle state with total spin quantum number 3 = 1. Calculate the probability that both particles are found in the left-hand side of the box in this case. You need not include all steps of your working, but should indicate clearly how and why some signs in the working given for part {b} are modiﬁed in this case. [d] Finally, suppose that the particles occupy the same two energy eigenstates, but that they are now identical spinless bosons. Would you expect the spatial distribution of this pair of particles to be similar to the identical fermions in part (b) or to the identical fermions in part (c)? Outline your reasoning.

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