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 threefold 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.
 Fall '13
 AkimasaMiyake

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