33 - Physics 6210/Spring 2007/Lecture 33 Lecture 33...

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Physics 6210/Spring 2007/Lecture 33 Lecture 33 Relevant sections in text: § 5.2, 5.6 Example: Hyperfine structure (cont.) We are evaluating the matrix elements of the perturbation, which now takes the form - μ e · B = - 8 π 3 μ e · μ p δ ( r ) in the degenerate subspace spanned by | 1 , 0 , 0 i ⊗ |±i ⊗ |±i . The “translational part” of the unperturbed degenerate subspace is just the usual ground state of hydrogen and so when computing the matrix elements of the perturbation we get a common factor of: h 1 , 0 , 0 | δ ( r ) | 1 , 0 , 0 i = Z d 3 x | ψ 100 ( r ) | 2 δ ( r ) = | ψ 100 (0) | 2 . Using | ψ 100 (0) | 2 = 1 πa 3 , the 4 × 4 the perturbation takes the form 4 ge 2 3 m p m e a ( S p · S e ) ij , where a is the Bohr radius and the i j refer to the basis | S ( e ) z ,S ( p ) z i = | + + i , | + -i , | - + i , | - -i . So, for example, ( S p · S e ) 12 = h + | S e | + i · h + | S p |-i = 0 . A very straightforward computation of the matrix elements yields ( S p · S e ) ij = ¯ h 2 4 1 0 0 0 0 - 1 2 0 0 2 - 1 0 0 0 0 1 . The eigenvalues and (normalized) eigenvectors of this matrix are (exercise) eigenvalue : ¯ h 2 4 , eigenvectors : 1 0 0 0 , 1 2 0 1 1 0 , 0 0 0 1 , eigenvalue : - h 2 4 , eigenvectors : 1 2 0 1 - 1 0 . 1
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Physics 6210/Spring 2007/Lecture 33 Of course, the eigenvalue ¯ h 2 4 is triply degenerate; any linear combination of its three eigen- vectors is also suitable. You can also get this result by setting S = S p + S e and computing S p · S e = 1 2 ( S 2 - S 2 p - S 2 e ) = 1 2 S 2 - 3 4 ¯ h 2 I. Recall that the singlet and triplet states are eigenvectors of S 2 with eigenvalues of 0 and h 2 respectively. Thus the singlet state is an eigenvector of S p · S e with eigenvalue - 3 4 ¯ h 2 and the triplet states all have the eigenvalue 1 4 ¯ h 2 . The components of the eigenvectors in the product basis which we found above are indeed the components of the singlet and triplet states and the eigenvalues then follow. Our particular choice of basis for the triply
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33 - Physics 6210/Spring 2007/Lecture 33 Lecture 33...

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