ENPH/PHYS 344, Fall 2012, Assignment # 2 Due Date: Wednesday Nov. (Assignments should be submitted to my oce. Consider a particle beam (x, t) = Ao...

3. This question asks you to examine properties of the 1d SHO using matrix mechanics,specifically the raising and lowering operators discussed in §7.5 of the lecture notes. All answersshould be constructed using matrix mechanics, and not wave mechanics.The so-called number basis corresponds to those vectors which are the eigenvectors of thenumber operator. That is, ˆN = ˆa†ˆa andˆN|qi = q |qi q = 0, 1, 2, . . .(a) Prove thatˆa |qi = √q |q − 1iUsing this information, derive the matrix elements of ˆa†.(b) Suppose that the number basis is truncated at q = 10. Using the matrix representationsof ˆa† and ˆa, obtain the energy spectrum of this system. How do these energies compare withthose of the full 1d SHO?(c) Derive the position and momentum operators in terms of ˆa† and ˆa. Using these operators,obtain that expectation values of the kinetic and potential energies, as a function of time fort ≥ 0, when the state of the system at t = 0 is|