Spin angular momentum operators for 2 and three electrons (Problem 8-22 in McQuarrie). The z component of the total spin angular momentum for an N-electron system is
Show that the following Slater determinants, which are the Hartree Fock wave function:
|1sα(1)1sβ(1)2sα(1) 1,2,3)= 1sα(2)1sβ(2)2sα(2) V3! 1sα(3)1sβ(3)2sα(3)
1[1sα(l)1sβ(l) (1,2) = 2 1sα(2)1sβ(2)
Hint: Do the first part by multiplying out the determinant and operating with S_Total = S_1 + S_2. You can do the second part the same way, there are six terms instead of 2.
To make the second part more obvious, ask yourself first, for example, what is S_Total (1)(2)(3) = (S_1 + S_z + S_2)(1)(2)(3)? You should find out that the result is that the product of spin functions is an eigenfunction of S_Total corresponding to a particular eigenvalue. Now evaluate S_Total (1)(2)(3) where there are the same number of α and β spins, but which belong to different electrons. Do you see the simple pattern? Did the eigenvalue depend on how the electrons were labeled?
