2. It is known that the harmonic oscillator has a Hamiltonian (p²) x² + mw? 2m H, 2 and an electric field of H, = qex With the information known, determine the value of the first-order and second-order energy shifts. (Hint: Use Equation x which contains the Creation operator (a) and the Annihilation operator (a*)

A nucleus traveling at speed vnucleus = c(1 - 10-5) decays with the emission of a particle of speed vdecay particle = c(1 - 2 x 10-4) at an angle of 45°, as shown in the diagram above. i) What is the velocity (speed and direction) of the decay particle on transforming to the stationary frame of reference of the nucleus. Clearly explain the quantities used in your evaluation, and show the directions and signs of quantities. Give the speed of the decay particle in the form c(1 - ?). ii) If the decay particle was an electron, what was its total energy in the rest frame of the nucleus? What was its total energy in the original frame shown in the above diagram?Give your answers in units of MeV.