Question

An atom has 2 electrons in a p(l = 1) sub-shell and the spin- wavefunction of the electrons is in the singlet state. What are the allowed values for L and the possible values for J = L + S? How would this change if the spin-wavefunction was in the triplet state?

Answer 1

When two electron with orbital quantum number with l 1  and l​​​​​​ 2 , then the resultant angular quantum number L are given by

L = ( l ​​​​​​1  + l 2 ) , (l 1 + l​​​​​​ 2 -1).........( l 1 - l 2)

When two or more electron present in a subshell , then their magnetic field couple to produce spin quantum number S. Then the resultant spin quantum number is given by

S = ( s 1 + s 2 ) , ( s 1 +  s 2 - 1 ) , (s 1 - s 2)

In an atom , the magnetic effects of resultant angular quantum number L , the resultant spin moment S couple together give total angular quantum number J

J = ( L+ S ) , ( L + S - 1) , ...........|( L - S)|

For 2 electron (l =1) in the singlet state

​l  1 = 1 , l 2 = 1 and s​​​​​​ 1  = s 2 = 0

L = 2 ,1 ,0

1 .For  L = 2

S = 0

J = L+ S

J = 2

2 .For L = 1 and S = 0

J = 1

3.L =0 and S = 0

J = 0

The J values for singlet states are J = 0, 1, 2

For triplet state S = 1

Therefore

For L = 2 and S= 1

J = L + S

J = 2 + 1

J = 3

For L = 1 and S = 1

J = 1+ 1

J = 2

For L = 0 and S = 1

J = 0+1

J = 1

The J value of triplet state are J = 1, 2, 3