130_notes.dvi

Since the highest m value is 6, we expect to have aj= 6 state which uses up one state for each m
value from -6 to +6. Now the highest m value left is 5, so aj= 5 states uses up a state at each m
value between -5 and +5. Similarly we find aj= 4,j= 3, andj= 2 state. This uses up all the
states, and uses up the states at each value of m. So we find in this case,

|ℓ 1 −ℓ 2 |≤j≤|ℓ 1 +ℓ 2 |

and thatjtakes on every integer value between the limits. This makes sense in the vector model.

21.7.4 Two electrons in an atomic P state

If we have two atomic electrons in a P state with no external fields applied,states of definite
total angular momentum will be the energy eigenstates. We will learn later that closed
shells in atoms (or nuclei) have a total angular momentum of zero, allowing us to treat only the
valence electrons. Examples of atoms like this would be Carbon, Silicon, and Germanium.

Our two electrons each haveell= 1 (P state) ands=^12 (electrons). We need to add four angluar
momenta together to get the total.

J~=L~ 1 +L~ 2 +S~ 1 +S~ 2

We will find it useful to do this addition in two steps. For low Z atoms, it ismost useful to add
L~ 1 +L~ 2 =L~andS~ 1 +S~ 2 =S~then to add these results~L+S~=J~.

Since the electrons are identical particles and they are in the same radial state, the angular momen-
tum part of the wavefunction must be antisymmetric under interchange. This will limit the allowed
states. So let’s do the spinor arithmetic.

|ℓ 1 −ℓ 2 |≤ ℓ ≤ℓ 1 +ℓ 2

ℓ = 0, 1 , 2

s = 0, 1

These states have a definite symmetry under interchange. Before going on to make the total angular
momentum states, lets note the symmetry of each of the above states. The maximum allowed state
will always need to be symmetric in order to achieve the maximum. The symmetry will alternate as
we go down in the quantum number. So, for example, theℓ= 2 andℓ= 0 states are symmetric, while
theℓ= 1 state is antisymmetric. Thes= 1 state is symmetric and thes= 0 state is antisymmetric.
The overall symmetry of a state will be a product of the these two symmetries (since when we addℓ
andsto givejwe are not adding identical things anymore). The overall state must be antisymmetic
so we can use:

ℓ = 1 s= 1 j= 0, 1 , 2 3 P 0 ,^3 P 1 ,^3 P 2

ℓ = 2 s= 0 j= 2^1 D 2

ℓ = 0 s= 0 j= 0^1 S 0

Each atomic state will have the angular momentum quantum numbers

ℓ 1 , ℓ 2 , s 1 , s 2 , ℓ, s, j, m.