4. (i) By expressing the Hamiltonian for a three-dimensional harmonic oscillator as three one- dimensional oscillators, with the same spring constant, show that the energy is equal to (n+3/2) hw, where n is zero or a positive integer. (ii) Show that the degeneracies of the three lowest energy levels are 1, 3 and 6 and that in general the degeneracy of the level n is 1/2(n+1)(n+2). Here degenerate levels have the same energy but different eigenfunctions.

4. (i) By expressing the Hamiltonian for a three-dimensional harmonic oscillator as three one- dimensional oscillators, with the same spring constant, show that the energy is equal to (n+3/2) hw, where n is zero or a positive integer. (ii) Show that the degeneracies of the three lowest energy levels are 1, 3 and 6 and that in general the degeneracy of the level n is 1/2(n+1)(n+2). Here degenerate levels have the same energy but different eigenfunctions.