
QCM701S Theoretical Exercise 2 This exercise is due first week after the mid-semester break 1. Given the three operators A, B, and C, show that (a) [A, B] =- [B, A] (b) [A?, B] = A[A, B] + [A, B]A (c) (A,BC] = B?A,C]+[A, B]C [2] [5] [5] 21+ 2. The solution of the Schrödinger equation of a plane rigid rotor is of the form Y(6)= Acos($) for 0 < • <21. Determine the normalisation constant, A. 1 (Given: cos' = 1 + cos2%) [6] 3. If , and are real normalized and orthogonal atomic orbitals belonging to A and PA B, respectively, show that the molecular orbital of their linear combination below is also normalized. Y = +0) [5] 4. The normalized wavefunctions for a particle-in-box (length, a = 40 nm) in the X- direction are given by: Tz com Y(x)= () sin nit -X a for 0 SX Sa and n = 1, 2, 3, 4, etc. (a) Using Excel and on the same diagram, plot (n=2), Y (n = 3) and the product Y(n=2). Y (n = 3) and comment on the result. (b) Using Excel, plot diagrams of the variation of the Y(n = 5) and the corresponding probability density function, y(n = 5) [6] (c) At what values of x are the functions (n=2), Y (n = 3), Y (n = 2). Y (n = 3) and Y' (n = 5) above equal to zero in the range 0