Midterm 5 Solutions

2 it will thus suffice to pick any orthonormal

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Unformatted text preview: = = = = 0 -c -c 0 The latter set of equations uniquely identifies the eigenstate E = -3 A ê 2\ = 1 2 B + z, - z\ - -z, +z\F We see that the former set of equations however is degenerate, which reflects the three-fold degeneracy in the energy eigenvalue A . 2 It will thus suffice to pick any orthonormal spanning set of eigenvectors for this subspace. We choose E1 = A ê 2\ = + z, + z\ E2 = A ê 2\ = 1 E3 = A ê 2\ = - z, - z\ 2 B + z, - z\ + Ha = 1; b = c = d = 0L -z, +z\F Hb = c = 1; a = d = 0L Ha = b = c = 0; d = 1L Method 2 `2 From (b), we know that the energy eigenstates for w0 = 0 are the eigenstates of Stot and that `2 `` Stot = 2 S1 ÿ S2 + 3 —2 2 ` 1ཽ `2 `` `` `` Stot = JS1+ S2- + S1- S2+ N + 2 S1 z S2 z + `` `` `` `` ¬༼ 2 S1 ÿ S2 = JS1+ S2- + S1- S2+ N + 2 S1 z S2 z 3 —2 2 ` 1ཽ ` ` ` ` ` ` `` `` as in (a). Because S1 z Äȩ S2 z commute with JS1 x Äȩ S2 x + S1 y Äȩ S2 y N = J S1+ S2- + S1- S2+ N, we have that the eigenstates of `2 `` `` `` Stot are mutual eigenstates of S1 z S2 z and J S1+ S2- + S1- S2+ N. These are linear combinations of product states ± z, ± z\ such ` ` that each component has the same product of S1 z and S2 z eigenvalues and such that the linear combination is an eigenstate of `` `` JS1+ S2- + S1- S2+ N. From Printed by Mathematica for Students `` `` `` `` JS1+ S2- + S1- S2+ N +z, +z^ = 0 JS1+ S2- + S1- S2+ N -z, -z^ = 0 `` `` `` `` JS1+ S2- + S1- S2+ N +z, -z^ = —2 -z, +z^ JS1+ S2- + S1- S2+ N -z, +z^ = —2 +z, -z^ 3— 2 Stot = JS1+ S2- + S1- S2+ N + 2 S1 z S2 z + 1ཽ ` ` ` ` ` ` `` `` M as in idtermExam5Solutions.nb commute with JS1 x Äȩ S2 x + S1 y Äȩ S2 y N = J S1+ S2- + S1- S2+ N, we have that the eigenstates of (a). Because S1 z Äȩ S2 z 6 `2 `` `` `` Stot are mutual eigenstates of S1 z S2 z and J S1+ S2- + S1- S2+ N. These are linear combinations of product states ± z, ± z\ such ` ` that each component has the same product of S1 z and S2 z eigenvalues and such that the linear combination is an eigenstate of `` `` JS1+ S2- + S1- S2+ N. From `` `` `` `` JS1+ S2- + S1- S2+ N +z, +z^ = 0 JS...
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This note was uploaded on 02/01/2014 for the course PHYC 491/496 taught by Professor Akimasamiyake during the Fall '13 term at New Mexico.

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