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# ps13 - Homework for Physics 316 Modern Physics...

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Homework for Physics 316, Modern Physics I (Hoffstaetter/Drasco/Thibault) Due Date: Friday, 05/06/03 - 9:55 in 132 Rockefeller Hall Exercise 1: Compute the radial part R nl ( r ) of the Hydrogen wave function for n < 3 for all possible l . Show by direct integration that the hydrogen state | n = 1 , l = 0 , m = 0 i is orthogonal to the state | n = 2 , l = 0 , m = 0 i . Exercise 2: For Hydrogen, an eigenstate of the total angular momentum ˆ ~ J 2 and its z -component ˆ J z is expanded as a linear combination of orbital angular momentum eigenstates: | nljsm j i = A | n, l, m l = m j - 1 2 , s, m s = 1 2 i + B | n, l, m l = m j + 1 2 , s, m s = - 1 2 i . (1) Using ˆ ~ J = ˆ ~ L + ˆ ~ S , argue that the linear combination cannot contain any other states. Exercise 3: (a) Use the Schroedinger Equation for the radial part of the wave function for hydrogen, and verify that a solution of the form R nl = ( c 0 - c 1 r ) r l +1 e - br (2) corresponds to the energy eigenvalue - 1 / ( l + 2) 2 in dimensionless units.
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