PHYS4221(Fall2010)_ProblemSet_6

# PHYS4221(Fall2010)_ProblemSet_6 - PHYS4221 Quantum...

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PHYS4221 Quantum Mechanics I Fall term of 2010 Problem Set No.6 (Due on December 3, 2010) 1. Given Y 21 ( θ,φ ) = - s 15 8 π sin θ cos θ e , apply the raising operator ˆ L + and lowering operator L - to ﬁnd Y 22 ( θ,φ ), Y 20 ( θ,φ ), Y 2 , - 1 ( θ,φ ) and Y 2 , - 2 ( θ,φ ). 2. Determine graphically the allowed energies for the inﬁnite spherical well when l = 1. Show that for large n , E nl π 2 ¯ h 2 2 ma 2 ± n + 1 2 ² 2 . Hint : First show that j l ( x ) = 0 = x = tan x . Plot x and tan x on the same graph and locate the points of intersection. 3. Two particles of mass m are attached to the ends of a massless rigid rod of length a . The system is free to rotate in three dimensions about the centre (but the centre point itself is ﬁxed). (a) Show that the allowed energies of this rigid rod are E n = ¯ h 2 n ( n + 1) ma 2 , for n = 0 , 1 , 2 ,..... Hint : First express the (classical) energy in terms of the total angular momen- tum. (b) What are the normalized eigenfunctions for this system? What is the degen-

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PHYS4221(Fall2010)_ProblemSet_6 - PHYS4221 Quantum...

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