Discussion Fr 24.6.2005  10:30 - 11:15 am lecture hall SN 20.2

Problem Set 9   (angular momentum, rotation, hydrogen atom)
 

Problem 1

Determine the angle J between the z axis and

1. the orbital angular momentum l^→  for the quantum number l = 2 and
2. the spin vector s^→ with s = ½, respectively?
3. Why is it impossible to measure the vector l^→ ?

Problem 2

a) Determine the total angular momentum J for a system with one d electron (i.e. orbital angular momentum l  = 2 and spin s = ½)? What is the degree of degeneracy?
b) What are the possible values of the total angular momentum J if two angular momenta with  j₁ = 2 and j₂ = 2 are combined. What is the degree of degeneracy for each possible total angular momentum J?
 

Problem 3

The complete wave function y(r,J,j) for hydrogen-like atoms is given by the product of  R_n,l(r) and the angular dependence Y_l,m(J,j): y_n,l,m(r,J,j)  =  R_n,l(r) Y_l,m(J,j). Due to proper linear combinations of the +/-m states real wave functions are obtained. For the 3 p_x orbital one gets y_3,1,+/-1(r,J,j)  =  R_n,l(r) S_l with :

At which distance r und for what angles J and j the probability density |y_3,1,+/-1(r,J,j)|² and the probability, |y_3,1,+/-1(r,J,j)|² r²sinJ, of observing the electron within [r,r+dr],[J,dJ+J],[j,j+dj] reaches a maximum?

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