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HomeworkSetIII - PHY4221 Quantum Mechanics(Fall Term Of...

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PHY4221 Quantum Mechanics (Fall Term Of 2005) Problem Set 3 (Suggested Solutions) Question 2: The Spherical Harmonic Functions (SHFs) are the common eigen functions of L 2 and L Z : ( ) 2 2 1 lm lm Z lm lm L Y l l Y L Y m Y = + = = = (1) Or Equation (1) is sometimes written as: ( ) 2 2 , 1 , , , Z L l m l l l m L l m m l m = + = = = (2) The raising angular operator L + , as defined as x y L L iL + = + , has the following property: ( )( ) , 1 , 1 L l m l m l m l m + = + + + = (3) So we have: ( )( ) 2,1 2 1 2 1 1 2,2 2 2,2 L + = + + = = = (4) As a result: 1 2,2 2,1 2 L + = = (5) Or equally: ( ) ( ) ( ) ( ) ( ) 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 φ θ φ θ φ θ φ φ θ φ φ θ φ θ φ θ φ θ φ φ θ φ φ θ φ θ θ θ φ θ φ π θ π + = = + = + + + + = − = = = = = (6) Question 3:
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