Chapter4_5

# 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: ∂ ∂ + ∂ ∂ ± = ± = ± ± φ θ θ φ 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 ) ( ) ( ) ( ) ( ) ( φ θ θ θ θ θ 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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