27 - Physics 6210/Spring 2007/Lecture 27 Lecture 27...

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Unformatted text preview: Physics 6210/Spring 2007/Lecture 27 Lecture 27 Relevant sections in text: § 3.6, 3.7 Angular momentum addition in general We can generalize our previous discussion of 2 spin 1/2 systems as follows. Suppose we are given two angular momenta ~ J 1 and ~ J 2 ( e.g., two spins, or a spin and an orbital angular momentum, or a pair of orbital angular momenta). We can discuss both angular momenta at once using the direct product space as before, with a product basis | j 1 ,m 1 i ⊗ | j 2 ,m 2 i . We represent the operators on product vectors as ~ J 1 ( | α i ⊗ | β i ) = ( ~ J | α i ) ⊗ | β i , and ~ J 2 ( | α i ⊗ | β i ) = | α i ⊗ ( ~ J | β i ) , and extend to general vectors by linearity. The product basis | j 1 ,m 1 i ⊗ | j 2 ,m 2 i is the basis corresponding to the commuting observables provided by ( J 2 1 ,J 2 2 ,J 1 z ,J 2 z )). The total angular momentum is defined by ~ J = ~ J 1 + ~ J 2 . A set of commuting observables that includes the total angular momentum is provided by the operators ( J 2 1 ,J 2 2 ,J 2 ,J z ). Note that both bases are eigenvectors of J 2 1 and J 2 2 since these commute with all components of the individual and total angular momentum (exercise). We also note that product eigenvectors | j 1 ,j 2 ,m 1 ,m 2 i are in fact eigenvectors of J z with eigenvalues given by m = m 1 + m 2 since J z | j 1 ,j 2 ,m 1 ,m 2 i = ( J 1 z + J 2 z ) | j 1 ,j 2 ,m 1 ,m 2 i = ( m 1 + m 2 )¯ h | j 1 ,j 2 ,m 1 ,m 2 i . But we will have to take linear combinations of product basis vectors to get total angular momentum vectors – eigenvectors of J 2 ....
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27 - Physics 6210/Spring 2007/Lecture 27 Lecture 27...

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