ħ² d²y 2m dx² 1 • +_mw²x²y = Ey 1. Find the expectation value of the potential and kinetic energy of the nth state of the harmonic oscillator. Hint: Express the operators p and x in terms of the raising and lowering operators. â₁ = (+ip + mwx) 2hmw âââ_¥n = n¥n â_â¥n = (n+1)wn 2. a) 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) ħw, where n is zero or a positive integer. b) 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.

ħ² d²y 2m dx² 1 • +_mw²x²y = Ey 1. Find the expectation value of the potential and kinetic energy of the nth state of the harmonic oscillator. Hint: Express the operators p and x in terms of the raising and lowering operators. â₁ = (+ip + mwx) 2hmw âââ_¥n = n¥n â_â¥n = (n+1)wn 2. a) 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) ħw, where n is zero or a positive integer. b) 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.