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Chapter7_3 - ∂ ∂ ∂ ∂ − = ⋅ z z y y x x x z zx z...

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PHY3063 R. D. Field Department of Physics Chapter7_3.doc University of Florida The Angular Momentum Operator (2) The L 2 operator: The square of the angular momentum operator is 2 2 2 2 ) ( ) ( ) ( op z op y op x op L L L L + + = and it is easy to prove that (L x ) op , (L y ) op , and (L z ) op commute with L 2 op : 0 ] , ) [( 2 = op op i L L for i = 1,2, 3. In Cartesian Coordinates: + + + + + = = z z y y x x x z zx z y yz y x xy y x z x z y z y x x y y x x y y x z x x z z x x z y z z y y z z y L op 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 h h h h Also, + + + + + + + + = + +
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Unformatted text preview: + ∂ ∂ + ∂ ∂ − = ⋅ z z y y x x x z zx z y yz y x xy z z y y x x z z y y x x z z y y x x p r op 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ) ( h h r r and hence ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + + − = ⋅ + z z y y x x z y x z y x p r L op op 2 2 2 2 2 2 2 2 2 2 2 2 2 ) ( ) ( h h r r which implies op op op op p r i p r p r L ) ( ) ( 2 2 2 2 r r h r r ⋅ + = ⋅ + , and therefore 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 ) ( ) ( 1 op op op op op L r r r r r r L r p r r i p r r p + ∂ ∂ + ∂ ∂ − = + ⋅ − ⋅ = h r r h r r...
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