Chapter4_5 - PHY4604 R. D. Field Department of Physics...

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Unformatted text preview: PHY4604 R. D. Field Department of Physics Chapter4_5.doc University of Florida Raising and Lowering Operators Angular Momentum Raising and Lowering operators: Define (L + ) op and (L- ) op as follows: + = = cot ) ( ) ( ) ( i e L i L L i op y op x op h Note that (L + ) op = (L- ) op and ] ) ( , ) [( 2 = op op L L and op op op z L L L ) ( ] ) ( , ) [( = h . Also, ( ) h = + op z op z op op op L L L L L ) ( ) ( ) ( ) ( ) ( 2 ( ) h + = + op z op z op op op L L L L L ) ( ) ( ) ( ) ( ) ( 2 Thus, ( ) + = + = + 2 2 2 2 2 sin 1 sin sin 1 ) ( ) ( ) ( ) ( ) ( h h op z op z op op op L L L L L The (p 2 ) op in Spherical Coordinates: Previously we derived 2 2 2 2 2 2 2 2 1 2 op op op L r r r r p + + = = h h and hence 2 2 2 2 2 2 2 2 sin 1 sin sin...
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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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