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B3 2M 2M Consider two identical non-interacting spinless particles of mass M confined in an isotropic three-dimensional harmonic oscillator potential of angular frequency o with Hamiltonian in spherical polar coordinates À 12 EV vi+Mor? Men - + where the subscripts 1 and 2 refer to the two particles. a) Consider the trial wavefunction (1:1) = Aexp(-a(r+ +r?)), where A is the normalization constant and a is a parameter that can be varied. i) Show that the expectation value of #, calculated with this trial 3 M2 3a wavefunction is equal to [11 marks] ii) Use the variational method to estimate the ground state energy and wavefunction for the two non-interacting particles. Show that they correspond to those of the exact ground state. [7 marks] b) If the particles have spin write down the exact ground state energy and wavefunction, in terms of the single-particle spatial and spin wavefunctions, for this system. Specify whether the ground state is a singlet or a triplet state. [5 marks] 4 a M c) A spin-dependent perturbing potential , - S, where Sand Sare the components of the spin operator of particle 1 and 2 respectively and a is a small positive constant with the dimension of energy, is added to Ĥ. , describes a weak interaction between the two spin particles. Use time-independent perturbation theory to estimate the first-order energy shift for the ground state energy of the two spin particles due to this perturbation [7 marks]