A Kerr effect in an electromagnetic cavity containing a nonlinear material may be described by the following Hamiltonian 1 = hw + -hg( ) where N is the number operator and g is a coupling constant that is proportional to the third-order susceptibility of the material in the cavity. a) Is the number operator a constant of motion for this system? Justify your answer. b) Find the time evolution of the annihilation operator by the Heisenberg equation of motion. c) What is the time evolution of the creation operator? d) What would be the expectation value of the energy if the cavity contains a coherent state with average photon number, .