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B3) # a2 + + )+3M03 **+*+2). 2 A three-dimensional isotropic harmonic oscillator of mass M and angular frequency is described by the Hamiltonian aa? MW 2M Oray ?? a) Calculate the energy of the first excited state. Determine the corresponding eigenstates in terms of the eigenstates ) (n=0,1,2,...) of one-dimensional harmonic oscillators with the same mass M and angular frequency o. Show that the first excited state is triply- degenerate [5 marks) b) The three-dimensional isotropic harmonic oscillator of part a) is now subject to the perturbation À, = 2.132 where is a small constant Calculate the matrix elements of Ả, between the eigenstates corresponding to the first excited energy level. Hence, using first-order degenerate perturbation theory, show that, for the first excited state, the energy shifts due to , are 0 and +- 212 4M*? [15 marks) c) Using first-order perturbation theory, calculate the energy of the first excited state and the energy shift due to the perturbation À for the anisotropic harmonic oscillators of mass M described by the Hamiltonian † (? 2 o? i) À + + 2 & * ) +5M 0% (7++ 32 +93) ii) na??? + 2Max? @y? Oz + + Mc (982 +9j2 +2) * 2° (10 marks]