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Unformatted text preview: Physics 225/315 February 21, 2008 Adding Angular Momentum Two Particle States Form the direct product of spin states of two spin 1/2 particles. These form a basis. Label each state with a 1 or 2 (the number of the particle). There will be four states. | + 1 | + 2 | + 1 |- 2 |- 1 | + 2 |- 1 |- 2 The Operators are also labeled with particle number and operate on the corresponding state. S 1 z | + 1 |- 2 = + ¯ h 2 | + 1 |- 2 S 2 z | + 1 |- 2 =- ¯ h 2 | + 1 |- 2 ( S 1 z + S 2 z ) | + 1 |- 2 = + ¯ h 2 | + 1 |- 2- ¯ h 2 | + 1 |- 2 S 2 1 | + 1 = 3 4 ¯ h 2 | + 1 = 1 2 ( 1 2 + 1)¯ h 2 | + 1 where S 2 1 = S 2 1 x + S 2 1 y + S 2 1 z . Total Angular Momentum The operator for the square of the total angular momentum of the two particle system is S 2 = ( S 1 + S 2 ) 2 = S 2 1 + S 2 2 + 2 S 1 · S 2 S 1 · S 2 = S 1 x S 2 x + S 1 y S 2 y + S 1 z S 2 z Define the “raising” and “lowering” operators: S + ≡ S x + iS y and S- ≡ S x- iS y Write S + and S- in matrix form and see how they transform the basis states.in matrix form and see how they transform the basis states....
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