4. (20 points). Consider the Hamiltonian H=4Ec2-EJ coso, (6) where the charge operator q describes the number of Cooper pairs on a small superconducting island separated from ground by a Josephson junction and o is the phase of the superconducting 1 wavefunction. (Here, Cooper pairs are the charge carriers in a superconductor, and you can read about Josephson junctions in volume 3 of the Feynman Lectures on Physics. However, for the purposes of solving this problem, no understanding of superconductivity or the Josephson effect is required). The operators and obey the canonical commutation relation [o, q] = i, (7) so we can view as a position-like variable with q as its conjugate momentum. For example, in the o-representation, we have = -i. (Note that in the way I am formulating this problem, the operator q is dimensionless: it corresponds to the number of charged Cooper pairs. Since is a phase, the operator is also dimensionless). a. Expand the cos part of the Hamiltonian to quartic order and rewrite the Hamiltonian as the sum of two parts: Ĥ = Ĥho + Ĥpert (8) where Ĥho describes a harmonic oscillator and Ĥpert describes the lowest order correction to this Hamiltonian. b. Now rewrite the Hamiltonian in terms of the usual raising and lowering operators: 2Ec 3 = (ZEC) 1/4 (at + à), EJ EJ 1/4 (32°c) (ât - à). (9) c. Within the harmonic approximation (e.g., neglecting Apert), what are the energy levels of the system? Approximately how many bound states does the potential support? d. To lowest order in perturbation theory (which we'll study a bit later in the course), the shifts in energy levels induced by a perturbing Hamiltonian are given by the following matrix elements: AEn = (n|Ĥpert|n), (10) where n) are the stationary states of the unperturbed Hamiltonian. For the case of this Hamilto- nian, calculate the lowest order corrections AE, to the energy levels.