Two electrons occupy a orbital ( ) and a orbital ( )

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Symmetry, angular momentum and spin

The Structure of Quantum Mechanics, Ch Adding angular momenta, tensor operators and Wigner-Eckart

Problem

Two electrons occupy a

d

orbital (

l_1=2

) and a

p

orbital (

l_2=1

). Consider only their orbital angular momenta, in the uncoupled basis

|m_1\rangle|m_2\rangle

(

m_1=2,1,0,-1,-2

;

m_2=1,0,-1

) and the coupled basis

|J,M\rangle

, together with

\hat J_-|j,m\rangle=\hbar\sqrt{j(j+1)-m(m-1)}\,|j,m-1\rangle

and

\hat J_-=\hat J_{1-}+\hat J_{2-}

. (a) Tabulate

M=m_1+m_2

over all fifteen product states, peel off multiplets from the top, and derive

l_1\otimes l_2=\bigoplus_J J

. Check the dimension count. (b) Lower the unique top state

|J=3,M=3\rangle=|m_1=2\rangle|m_2=1\rangle

once to obtain

|3,2\rangle

. Use orthogonality inside the two-dimensional

M=2

subspace, with the Condon-Shortley sign fixed by requiring the coefficient of

|m_1=2\rangle|m_2=0\rangle

to be positive, to construct the top state

|J=2,M=2\rangle

of the next multiplet.