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Provide answers to the following questions on separate papers and attach to this problem sheet. Please write your answers clearly. 1. (6 pts) Prove that the following wave function is an eigenfunction of i2 and find the eigenvalue. Determine whether this is an eigenfunction of L2 = to as well. = 1/2 - 1 2. (14 pts) The Hamiltonian of the electron in hydrogen atom (in atomic units) is H = 1 1 a 1 .2 2 - - T 1 , airl trl where meel a sin20362 . The electron is in a state represented by the following wave function: Answer the following questions. (a) (4/14) How many radial nodes and (angular) nodal planes are in 4(0,0)? Justify your answer. (b) (6/14) Prove that (6)0,0) is an eigenfunction of it and find the corresponding eigenvalue. (c) (4/14) Calculate the most probable value of r.

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