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5. The Quantum Harmonic Oscillator Given 1 a (mwi - ip) and a 2hmw (a) Obtain the expressions 1 (mwl + ip); V2hmw (3) [2] h î= -(+ + a_) and mwh (â+ -â-) 2 p = iv الخيال 27 (4) (b) Using your result in (a) together with the fact that I vmdx = Oman compute (2) and (w) for the nth state, m. [8] Recall: +Vm= Vn + lunt1 and à Un = nen-1 6. Given (-Vofor || <a V(x) = 0 for 121 > a, (a) What is the condition for bound states? (b) Let k = V2mE /h and K = ik. Rewrite the Schrödinger equation in terms of the second derivative of and K. (c) What is the general solution? (d) Write down the physical solution for < <a and r > . (e) Which process is indicated by these solutions? (f) Consider scattering states. The transmission coefficient is given by T-1-1+ sin? 4E(E + V) Find the energies at which complete transmission occurs and comment on this. 7. Delta-Function Potential The wave function of the bound states for the potential described by V() = -a8(2) is given by a (V2m{D + 10) bar) = { Ae- kr > 0 I <0 KE Aekt, -2mb h (5) By integrating the Time Independent Schrodinger Equation (2) show that the wavefunc- tion (5) admits only one energy eigenstate given by [8] 8 E- ma? 212 (6)