Lect26_AngMom2 - Lecture 26 Angular Momentum II Phy851 Fall...

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Lecture 26: Angular Momentum II Phy851 Fall 2009
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The Angular Momentum Operator The angular momentum operator is defined as: It is a vector operator: According to the definition of the cross- product, the components are given by: P R L r r r × = z z y y x x e L e L e L L r r r r + + = y z x ZP YP L = z x y XP ZP L = x y z YP XP L =
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Commutation Relations The commutation relations are given by: These are not definitions, they are just a consequence of [ X , P ]= i h Any three operators which obey these relations are considered as ‘generalized angular momentum operators’ Compact notation: z y x L i L L h = ] , [ x z y L i L L h = ] , [ y x z L i L L h = ] , [ l l h L i L L jk k j ε = ] , [ 0 if any two indices are the same 1 cyclic permutations of x,y,z (or 1,2,3) -1 cyclic permutations of z,y,x (or 3,2,1) l jk ε ‘Levi Cevita tensor’ Summation over l is implied
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Simultaneous Eigenstates In the HW will see that: – Where: This means that simultaneous eigenstates of L 2 and L z exist – Let: We want to find the allowed values of a and b . 0 ] , [ 2 = x L L 0 ] , [ 2 = y L L 0 ] , [ 2 = z L L 2 2 2 2 z y x L L L L + + = b a a b a L , , 2 = b a b b a L z , , =
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Algebraic solution to angular momentum eigenvalue problem In a analogy to what we did for the Harmonic Oscillator, we now define raising and lowering operators: Lets consider the action of L + first: y x iL L L + = + y x iL L L =
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