Consider the following quantum mechanical Hamiltonian for a particle of mass m in a harmonic oscillator with frequency w, p2 mw 2x 2 H = + 2m 2 where p and x are the momentum and position operators, respectively. Let us define the following operators, 1 a = V2mhow (mwx tip ) , and 1 1 = V2mhw (mwx - ip) . 1. Consider a normalized state |0) such that a |0) = 0. Calculate the energy eigenvalue of |0) . 2. Derive an expression for any given eigenstate In) of H starting from the ground state |0). 3. Show that the eigenstate In), obtained above, is also the eigenstate of the number operator, N = ala, with eigenvalue n i.e., alan) = n). Calculate the energy eigenvalues of the Hamiltonian H