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[3] [5] 1. (a) Show that in one dimension, I and commute, where i and are the kinetic energy and momentum operators, respectively. (b) Discuss whether or not the wave functions eike, e-ikr, and Aeikir + Be-ike are eigenfunctions of T and P, and state the corresponding eigenvalues in each case. (c) Given the unnormalised wave function for a hydrogenic atom Unul.mr.0,0) = NR(r)e2i sine, (] [2] [3] where N is a constant, determine whether nem is an eigenfunction of a 1, = -ia and if so determine the corresponding eigenvalue and angular momentum quantum number. (d) Determine a value for the normalisation constant N for the wave function in part (c). You may assume that °\R(r)l® r2dr = 1. If required, you may use: sind ode = sin de S. sin Ꮎ sin de sin de 37 | in 0 - 4 ca tai - tu tap – can ao – 3 ( a 6h - 8 : 165 4 3 511 16 2 (e) Now consider a different wave function, /j 11.0.6) = R(-) (31-10,9) + tz V100,6) + živi. (0. m), = + where Yim are the spherical harmonic functions. Determine the possible [5] results of a measurement of l’ and , and calculate the expectation value of Ly. (1) Consider the spherical harmonic functions Y10, and Y... State, giving reasons, [2] whether or not these two eigenfunctions are degenerate in Î?, and whether or not they are degenerate in i. If required, the first few spherical harmonics are: 70,0 = VE 71,0 3 cose; 47 Y1,1 VW sin Dei; 3л Y1-13 sin de 47 37