Answer Happy

2 B3) Consider a one-dimensional harmonic oscillator of mass Mand angular frequency o. Its Hamiltonian is: A, P21 2M 2 + Mo???. a) Add the time-independent perturbation À, - man??? where i <land is a real constant with the dimension of an angular frequency. i) For each state In) of the unperturbed harmonic oscillator, using time-independent non-degenerate perturbation theory, calculate the energy of the perturbed "state up to second order in a. [18 marks) Show that the energy values obtained in 1) are equal to the expansion of the exact energy eigenvalues of the Hamiltonian À = Ĥ, +À, up to second order in a. [7 marks) ii) b) If the perturbation #1,- M12*8* is only applied for a time interval r, ? a using first-order time-dependent perturbation theory show that, if the harmonic oscillator is initially in its n'h excited state In), the only possible transitions are to the (n+2) excited states in #2). [5 marks]