HW6 - PHY4221 Quantum Mechanics(Fall Term Of 2004 Problem...

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PHY4221 Quantum Mechanics (Fall Term Of 2004) Problem Set 6 (Answers) Question 1: The Spherical Harmonic Functions (SHFs) are the common eigen functions of L 2 and L Z : ( 29 2 2 1 lm lm Z lm lm L Y l l Y L Y m Y = + = h h (1) Or Equation (1) is sometimes written as: ( 29 2 2 , 1 , , , Z L l m l l l m L l m m l m = + = h h (2) The raising angular operator L + , as defined as x y L L iL + = + , has the following property: ( 29 ( 29 , 1 , 1 L l m l m l m l m + = - + + + h (3) So we have: ( 29 ( 29 2,1 2 1 2 1 1 2,2 2 2,2 L + = - + + = h h (4) As a result: 1 2,2 2,1 2 L + = h (5) Or equally: ( 29 ( 29 ( 29 ( 29 ( 29 22 21 21 21 2 2 , 1 , 2 1 , 2 1 sin cot cos cos cot sin , 2 15 sin cot cos cos cot sin sin cos 8 2 15 sin 32 x y i i Y L Y L iL Y i Y i e e φ θ φ θ π + = = +   = + + - ÷  ÷     + + - ÷  ÷   = - = h h h h (6) Question 2:
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This is a typical question of central force field, for which the angular momentum is conserved. This is easy to show:
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HW6 - PHY4221 Quantum Mechanics(Fall Term Of 2004 Problem...

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