# HW6 - rigid rod are E n = ¯ h 2 n ( n + 1) ma 2 , for n =...

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PHY4221 Quantum Mechanics I Fall term of 2004 Problem Set No.6 (Due on December 1, 2004) 1. Given Y 21 ( θ,φ ) = - s 15 8 π sin θ cos θ e , apply the raising operator ˆ L + to ﬁnd Y 22 ( 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 . 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
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Unformatted text preview: 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 momentum. (b) What are the normalized eigenfunctions for this system? What is the degeneracy of the n th energy level? 4. (a) Find h r i and h r 2 i for an electron in the ground state of hydrogen. Express your answers in terms of the Bohr radius. (b) Find h x i and h x 2 i for an electron in the ground state of hydrogen. (c) Find h x 2 i in the state n = 2, l = 1, m = 1. —— End ——...
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## This note was uploaded on 12/10/2011 for the course PHYS 4221 taught by Professor Cflo during the Spring '11 term at CUHK.

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