Midterm 5 Solutions

# Calculating the energy eigenvalues printed by

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Unformatted text preview: 0 0 Aê2 0 0 0 0 00 00 0 0 -1 2 0 02 0 0 —w + 20 2 -1 0 0 0 -2 0 0 01 00 0 0 0 0 0 — w0 - HA ê 2L A 0 A -— w0 - HA ê 2L 0 0 0 Aê2 which agrees with our result from Method 1. ü Calculating the energy eigenvalues Printed by Mathematica for Students 0 0 0 1 3 4 MidtermExam5Solutions.nb Calculating the energy eigenvalues ` We see from the “outside” block - which is diagonal - that H has eigenvalue A with multiplicity 2, corresponding to the eigenstates +z, +z\ and -z, -z\. What remains is to diagonalize the “inside” block, as detB A 2 — w0 - -l A -— w0 - A BIl + A2 M 2 A 2 -l F=0 - —2 w2 F - A2 = 0 0 l2 + l A - J 3 A2 4 + —2 w2 N = 0 0 l± = 1 B- A 2 A l± = - 2 ± A2 + 4 J ± 3 A2 4 + —2 w2 N F 0 IA2 + —2 w2 M 0 A Therefore, the four energy eigenvalues are E œ : 2 , A , 2 A l+ = - 2 + A IA2 + —2 w2 M , l- = - 2 0 IA2 + —2 w2 M > 0 ü Part b We have the total spin operator ` ` ` Stot = S1 + S2 Thus ` 2 S1 ` 2 S1 `2 `2 `` `2 Stot = S1 + 2 S1 ÿ S2 + S2 ` `2 `2 `2 ÿ S2 = Stot - S1 - S2 ` `2 3 —2 ` ÿ S2 = Stot - 2 1ཽ ` ` ¬༼ S1 and S2 commute `2 ` ¬༼ Si = —2 ji H ji + 1L 1ཽ = so for w0 = 0, our Hamiltonian is given by ` H= A —2 `2 KStot - 3 —2 2 ` 1ཽO `2 Therefore, the energy eigenstates when w0 = 0 are the same as the eigenstates of Stot . ü Part c ` ü Calculating the energy eigenstates of H There are again two ways of doing this. ü Printed by Mathematica for Students 3 —2 4 ` 1 1ཽ for spin- 2 MidtermExam5Solutions.nb 5 Method 1 ` See lecture note 5. From the matrix representation and eigenvalues of H from (a) for w0 = 0 ` H Hw0 = 0L Ø ` ` Sz Äȩ Sz basis Aê2 0 0 0 0 -A ê 2 A 0 0 A -A ê 2 0 0 0 0 Aê2 A A A , 2, 2 with eigenvalues E œ 9 2 , - 3A = 2 for which we solve for the eigenvectors as Aê2 0 0 0 0 -A ê 2 A 0 0 A -A ê 2 0 0 0 0 Aê2 a b c d Aê2 0 0 0 0 -A ê 2 A 0 0 A -A ê 2 0 0 0 0 Aê2 a b c d = A 2 ﬂ a=a -b + 2 c = b 2b - c = c d=d ﬂ a b c d a = -3 a -b + 2 c = -3 b 2 b - c = -3 c d = -3 d ﬂb=c ﬂb=c and =- 3A 2 a b c d ﬂ ﬂ ﬂ ﬂ a b b d...
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