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

CHAPTER: 7, 8 AND 9.

Please solve the following problems.

12 42. Consider a system which is in the state described by the state y[x, y, z] = A XZ -2i where A is 2 VT a constant. (a) Express v[x, y, z] in terms of the spherical harmonics then calculate 12 y[x, y, z] and Lz y[x, y, z]. Is y[x, y, z] an eigenstate of L2 or Lz? (b) Calculate L. y[x, y, z] and (4 | L. | v) (c) If a measurement of the z-component of the orbital angular momentum is carried out, find the probabilities corresponding to find the results 0, + h and +2 h 43. The matrix representations of the angular momentum operators for spin NI- are Sx = 2 (10) 8, = 2 (8 8 ) ands= = 2 60-1). The eigenstates of Sz, Is, ms), are spin-up ?' 2/ and spin-down These angular momentum operators obey the same rules as the orbital angular momentum operators. (a) Use the rotation operator Ry, ] to rotate the state - about the y-axis by an angle d. (b) Evaluate for d = 7 / 2 (c) Is the resulting state in (b) an eigenstate of Sx? (d) For what angle will the rotated state return to itself? 44. (a) Calculate the moment of inertia of a diatomic molecule, as depicted in Fig. 7.17. Express the moment two ways: (1) in terms of the individual masses m, and m2 and the coordinates r, and r2, and (2) in terms of the reduced mass m and the atom-atom separation ro. (b) Calculate the rotational constant for the hydrogen iodide (HI) molecule7.6 Motion on a Sphere 237 FIGURE 7.17 A diatomic molecule is the simplest example of a rigid rotor. The two-atom system rotates around an axis perpendicular to the symmetry axis of the molecule. A physical example of this particleonasphere model is the rigid rotor. The simplest rigid rotor is a diatomic molecule, as illustrated in Fig. 7.17. The two atoms with a separation r0 have a moment of inertia about the center of mass of I = org, just as we have assumed in our particleonasphere model. Molecular spectroscopists call the energy z/ 21 the rotational constant of the molecule. For example, consider the diatomic molecule hydrogen chloride HCl. The equilibrium bond length is r0 = 0.127 nm, which gives a rotational constant hz = 1.32 meV = 10.7 cm'l. (7.160) 21 HCI