BOOK: Quantum Mechanics, A Paradigms Approach, David H. McIntyre

CHAPTER: 7, 8 AND 9.

Please solve the following problems.

45. The following 3D Hamiltonian H = p- 2m + k(x2 + y?) can be rewritten in cylindrical coordinates as H = -. h2 a 1 22 22 2m p ap ap p2 242 + az2 + kp where p = (x2 + 12)1/2 (a) For large p what is the essential dependence of glp]? (b) For small p what is the essential dependence of gip]?(a) From the form and symmetry of the Hamiltonian with which of the following operators commutes with the Hamiltonian, Lx, Ly, Lz, Px, Py, Pz? 46. (a) From the form and symmetry of the Hamiltonian in Prob, 45 with which of the following operators does the Hamiltonian commute? Lx, Ly, Lz, Px, Py, Pz (b) Rewrite the Hamiltonian using these operators. (c) Write down the form of the most general wavefunction which is a simultaneous eigenfunction of H and the operators found in (a). Your answer should be in the form g[p] x f[y] x h[z]. 47. Consider an infinite spherical well potential V[r] = 10 for r s a " for r > a (a) Determine the functional forms of the ground state and the first two excited states. (b) What are the energy eigenvalues of these states? (c) What is the degeneracy of these states