Solve all three problems. For full credit, you must show ALL your work or, when appropriate, justify your answer with a short sentence. The use of any known relationship between partial derivatives must be clearly indicated.

WULSLOe OULUPrCe AL WOLADOraALOrs Solve all three problems. For full credit, you must show ALL your work or, when appropriate, justify your answer with a short sentence. The use of any known rela- tionship between partial derivatives must be clearly indicated. 1. The 1s orbital of the hydrogen atom is Wy,(r, 0, ) = Ae"! where r is the distance of the electron from the proton, @ and @ are, respectively, the azimuthal and polar angle, ap is Bohr's radius, and A is a constant. a) Calculate A such that / Wi.(r,0,6)2aV = 1 / all space b) What are the dimensions of V,,? Integrate |"1,(r,6,@)|? over the solid angle and use the resulting function of r to answer the following questions: c) Find the most likely distance between the electron and the proton in the Is orbital? d) What is the mean distance between the electron and the proton in the 1s orbital? e) What is the probability that the electron can be found a distances greater than 2ag from the proton in the |s orbital