Consider an electron in a orbital ( ) and a second electron in a orbital ( )
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Atomic, Molecular & Optical Physics · Many-electron atoms and the periodic table
Deriving the Spectrum, Ch Many-electron terms, levels and Hund's rules
Problem
Consider an electron in a 2p orbital (l1=1) and a second electron in a 3p orbital (l2=1). Because they occupy different radial orbitals, the Pauli principle places no restriction on how their orbital angular momenta combine, so the coupled orbital state can be built exactly as for two independent vectors with l1=l2=1. Write ∣m1,m2⟩ for the product state with electron 1 having ml=m1 and electron 2 having ml=m2.
((a)) Write the state of maximum ML, ∣L=2,ML=2⟩, as a product ∣m1,m2⟩.
((b)) Apply the lowering operator L−=l1−+l2− to both sides, using l−∣1,m⟩=2−m(m−1)ℏ∣1,m−1⟩, to obtain the normalised state ∣L=2,ML=1⟩ as a combination of ∣1,0⟩ and ∣0,1⟩.
((c)) The states ∣1,0⟩ and ∣0,1⟩ span the whole ML=1 subspace. Construct the combination orthogonal to the one found in (b) and identify its L.
((d)) State which of the two ML=1 combinations changes sign when m1 and m2 are exchanged. Then, at ML=0, count how many independent combinations of ∣1,−1⟩, ∣0,0⟩ and ∣−1,1⟩ are symmetric and how many antisymmetric under the exchange, and use the count to decide the symmetry of the L=0 state.