Atomic Structure Theory

276 Solutions
The S = 1 eigenstates are symmetric and the S = 0 eigenstate is antisymmetric.
The product
1
r1r2
P1s(r1)Y00(ˆr1)P2p(r2)Y1m(ˆr2)
is an eigenstate of L 2 and Lz with eigenvalues L = 1 and ML = m. Thesame
is true of the product with indices 1 and 2 interchanged. The combinations
|L, ML, ±〉 = 1
r1r2
1
√ 2 [P1s(r1)Y00(ˆr1)P2p(r2)Y1m(ˆr2)
±P1s(r2)Y00(ˆr2)P2p(r1)Y1m(ˆr1)]
are symmetric and antisymmetric eigenstates of L 2 and Lz, withL = 1 and
ML = m. The combinations of orbital and spin functions,
| 1 P, m〉 = |1,m,+〉|0, 0〉 ,
| 3 P, ml,ms〉 = |1,ml, −〉|1,ms〉 ,
are antisymmetric eigenstates of spin and orbital angular momentum. There
are 3 + 3 × 3 = 12 magnetic substates.
3.2 A variational calculation, starting from screened Coulomb wave functions,
gives a reasonable approximation to energies of bound states of helium-like
ions. For a doubly excited (nl) 2 state, a variational calculation leads to
2 (Z − σnl)
E (nl) 2 = −
n2 ,
where
σnl Eion >E (1s) 2.
4
Therefore, the (1s) 2 state is bound, but all other (nl) 2 states lie above the
ionization limit.
3.3 Define the radial Hamiltonian operator ˆ h by
where
ˆh = − 1
2
d2 l(l +1)
+
dr2 2r2 Z
− + Vdir,
r