Chapter7_2 - ( , [ ] ) ( , ) [( ] ) ( , ) ( [ ] ) ( , ) ( [...

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PHY3063 R. D. Field Department of Physics Chapter7_2.doc University of Florida The Angular Momentum Operator (1) The Momentum Operator in 3 Dimensions: In “position space” with Cartesian coordinates we have z i p y i p x i p op z op y op x = = = h h h ) ( ) ( ) ( and 2 2 2 2 2 2 ) ( ) ( ) ( ) ( op op z op y op x op p p p p = + + = h where 2 2 2 2 2 2 2 z y x op + + = . Angular Momentum: Angular momentum is the vector operator given by op z op y op x op op p p p z y x z y x p r L ) ( ) ( ) ( ˆ ˆ ˆ = × = r r r Hence, op y op z op x p z p y L ) ( ) ( ) ( = op z op x op y p x p z L ) ( ) ( ) ( = op x op y op z p y p x L ) ( ) ( ) ( = and in “position space” with Cartesian coordinates we have = y z z y i L op x h ) ( = z x x z i L op y h ) ( = x y y x i L op z h ) ( Commutation Relations: The commutator of, for example, L x and L y is () op z op x op y op y op z op x op z op y op z op z op y op y op x op x op y op z op z op z op z op z op x op x op z op z op y op x op y op z op z op x op z op z op x op y op z op y op x L i p y p x i p p z x p z p y p p x z p x p z p p z z p z p z p p x y p x p y p p z y p z p y p x p z p z p z p x p y p z p y p x p z p z p y L L ) ( ) ( ) ( ) ]( ) ( , [ ) ]( , ) [( ) ]( ) ( , [ ] ) ( , ) [( ) ]( ) ( , [ ] ) ( , ) [( ) ]( ) ( , [ ] ) ( , ) [( ) ]( )
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Unformatted text preview: ( , [ ] ) ( , ) [( ] ) ( , ) ( [ ] ) ( , ) ( [ ] ) ( , ) ( [ ] ) ( , ) ( [ ] ) ( ) ( , ) ( ) ( [ ] ) ( , ) [( h h = = + = + + + = + = = We see that the commutator of any two of the angular momentum operators gives the third angular momentum operator as follows: op z op y op x L i L L ) ( ] ) ( , ) [( h = op x op z op y L i L L ) ( ] ) ( , ) [( h = op y op x op z L i L L ) ( ] ) ( , ) [( h = This can be summarized by the following: op k ijk op j op i L i L L ) ( ] ) ( , ) [( h = Note: iik = ijj = iji = 0, 123 = 231 = 312 = 1, 213 = 132 = 321 = -1....
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