Chapter4_6 - L | | h and label states by the eigenvalues l and m The angular equation now becomes 1 sin 1 sin sin 1 2 2 2 = ∂ ∂ ∂ ∂ ∂ ∂

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PHY4604 R. D. Field Department of Physics Chapter4_6.doc University of Florida Spherical Harmonics The Angular Equation: The angular equation ) , ( ) , ( sin 1 ) , ( sin sin 1 2 2 2 2 φ θ CY Y Y = + h is simply the L 2 op eigenvalue equation > >= Y C Y L op | | ) ( 2 where + = 2 2 2 2 2 sin 1 sin sin 1 ) ( h op L . Eigenvalues of the Angular Momentum Operators: The angular momentum operators (L x ) op , (L y ) op, and (L z ) op do not commute, op k ijk op j op i L i L L ) ( ] ) ( , ) [( ε h = , and hence it is not possible to simultaneously know all three components of the angular momentum. We choose to label state according to the eigenvalues of the (L 2 ) op and the (L z ) op , which we can do since [L 2 ,L z ] = 0 . The x and y components obey the “uncertainty relation” given by 2 4 2 4 1 2 2 ) ( ) ] , [ ( ) ( ) ( 2 > < = > < z y x y x L L L i L L h . We will write the eigenvalue equations for (L 2 ) op and the (L z ) op as follows, > + >= lm lm op Y l l Y L | ) 1 ( | ) ( 2 2 h > >= lm lm op z Y m Y
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Unformatted text preview: L | | ) ( h and label states by the eigenvalues l and m . The angular equation now becomes ) , ( ) 1 ( ) , ( sin 1 ) , ( sin sin 1 2 2 2 = + + ∂ ∂ + ∂ ∂ ∂ ∂ lm lm lm Y l l Y Y and the functions Y lm ( θ , φ ) are called the “spherical harmonics” . The eigenvalues for the “orbital” angluar momentum operators (L 2 ) op and (L z ) op are: Quantum Number Allowed Values Specifies l 0, 1, 2, 3, . .. L 2 (length of the angular momentum vector) m -l, -l+1,. .., +l L z (component of the angular momentum in the z-direction) y x z h 2 45 o h l = 1 m = 1 2 2 h ≥ ∆ ∆ y x L L We will derive this later using operators!...
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This note was uploaded on 05/29/2011 for the course PHY 4064 taught by Professor Fields during the Spring '07 term at University of Florida.

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