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For an N-electron system, the z component of the total spin angular momentum operator is Sz,total = [Sz.k k If we define the spin eigenstates such that Ŝz,kª(k) = ½ħ a(k) and Ŝz,kß(k) = −¹⁄ħ ß(k) then find the eigenvalues of Ŝz,total for the two spin-orbit eigenstates specified below. Note that k labels the electron, and the spatial orbital in which the electron resides is also indicated in the Slater determinants provided. (a) = = (b) 1|1sa(1) 1sß(1)| √21sa(2) 1s(2)| = 1 √6 Evaluate Ŝz,total. (c) By analogy with orbital angular momentum, Ŝ²4 = s(s + 1)ħ²y, where represents a spin state, and s is the magnitude of spin (like ¤). If Ŝ² = Ŝx² + ₁² + Ŝ₂², evaluate the 2 2 2 Z 2 2 result of ($x² + $₂²) a(k). Is a(k) an eigenfunction of (§×² + §₂²) : Evaluate Ŝz,totalÝ. |1s a(1) 1s ß(1) 2s α(1)| 1s a(2) 1s (2) 2s a (2) 1s a(3) 1s (3) 2s a(3)|

For an N-electron system, the z component of the total spin angular momentum operator is Sz,total = [Sz.k k If we define the spin eigenstates such that Ŝz,kª(k) = ½ħ a(k) and Ŝz,kß(k) = −¹⁄ħ ß(k) then find the eigenvalues of Ŝz,total for the two spin-orbit eigenstates specified below. Note that k labels the electron, and the spatial orbital in which the electron resides is also indicated in the Slater determinants provided. (a) = = (b) 1|1sa(1) 1sß(1)| √21sa(2) 1s(2)| = 1 √6 Evaluate Ŝz,total. (c) By analogy with orbital angular momentum, Ŝ²4 = s(s + 1)ħ²y, where represents a spin state, and s is the magnitude of spin (like ¤). If Ŝ² = Ŝx² + ₁² + Ŝ₂², evaluate the 2 2 2 Z 2 2 result of ($x² + $₂²) a(k). Is a(k) an eigenfunction of (§×² + §₂²) : Evaluate Ŝz,totalÝ. |1s a(1) 1s ß(1) 2s α(1)| 1s a(2) 1s (2) 2s a (2) 1s a(3) 1s (3) 2s a(3)|

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For an N-electron system, the z component of the total spin angular momentum operator is Sz,total = [$₂.k Σ k If we define the spin eigenstates such that Ŝz,ka(k) = ¹ħ a(k) and Ŝz,kß(k) = −¹/ħ ß(k) then find the eigenvalues of Ŝz,total for the two spin-orbit eigenstates specified below. Note that k labels the electron, and the spatial orbital in which the electron resides is also indicated in the Slater determinants provided. (a) (b) 1 |1sa(1) 1sß(1)| √21sa(2) 1sß(2)| 1 √6 Evaluate Ŝz,total. 1s a(1) 1s (1) 2s a(1)| 1s a(2) 1s (2) 2sa(2) 1s a(3) 1s (3) 2s a(3) Evaluate Ŝz,total. (c) By analogy with orbital angular momentum, Ŝ² = s(s + 1)ħ²y, where represents a spin state, and s is the magnitude of spin (like €). If §² = Ŝx² + y² + ŝ₂², evaluate the 2 2 2 S 2 2 2 result of (§² + ŝ₂²) a(k). Is a (k) an eigenfunction of (§₂² + §₂²) ? y 'y
Transcribed Image Text:For an N-electron system, the z component of the total spin angular momentum operator is Sz,total = [$₂.k Σ k If we define the spin eigenstates such that Ŝz,ka(k) = ¹ħ a(k) and Ŝz,kß(k) = −¹/ħ ß(k) then find the eigenvalues of Ŝz,total for the two spin-orbit eigenstates specified below. Note that k labels the electron, and the spatial orbital in which the electron resides is also indicated in the Slater determinants provided. (a) (b) 1 |1sa(1) 1sß(1)| √21sa(2) 1sß(2)| 1 √6 Evaluate Ŝz,total. 1s a(1) 1s (1) 2s a(1)| 1s a(2) 1s (2) 2sa(2) 1s a(3) 1s (3) 2s a(3) Evaluate Ŝz,total. (c) By analogy with orbital angular momentum, Ŝ² = s(s + 1)ħ²y, where represents a spin state, and s is the magnitude of spin (like €). If §² = Ŝx² + y² + ŝ₂², evaluate the 2 2 2 S 2 2 2 result of (§² + ŝ₂²) a(k). Is a (k) an eigenfunction of (§₂² + §₂²) ? y 'y
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