Solutions To Stellar Line Formation Problems (3)

  The Problems:

1) Enumerate the possible values of j and m _J for the states in which ℓ = 1 and s = 1/2 and draw the associated vector diagrams.

2) Consider an electron for which n = 4, ℓ = 3, and m _ℓ = 3. Calculate:

a) the numerical value of L, the total orbital angular momentum

b) the z-component of the orbital angular momentum.

3) A two-electron atom for which the orbital angular momentum quantum numbers are ℓ1 =3 and ℓ2 = 2 can have what values for the total orbital angular momentum number L?  Determine the possible values of the total angular momentum quantum number of single f  electron.

Solutions:

1) The vector solutions will show:

j = ℓ + s  = 1 + ½ = 3/2

The diagram for the vectors is shown below:

Then:

m _J  = -3/2, -½, ½, 3/2

2)  a) The numerical value of the total angular momentum is given by:

L = [ℓ (ℓ + 1)]^1/2 (ħ)

Where ℓ = 3, then:

L = [3 (3 + 1)]^1/2 (ħ)  =    [3 (4)]^1/2 (ħ)   =   3Ö2  ( ħ)

b) The z-component of the orbital angular momentum is given by:

L(z) = m _ℓ ħ

For this election,  m _ℓ  = 3, so that:   L(z) = 3 ħ

(3) We have ℓ1 =3 and ℓ2 = 2, then:

Therefore, the possible values of L will be found  from letting ℓ1 =3 and adding each next descending value of m _ℓ  from 2, to 1, to 0, to -1, to -2:

(3) + 1 =   4

(3) + 0 = 3

(3) + (-1) = 2

(3) +  (-2)  = 1

So the total angular momentum L can have the values:

5, 4, 3, 2 and 1.

The f electron has ℓ =3  so that the total angular momentum quantum number possibilities are:

j = ℓ + ½,   ℓ - ½

Then: j = 7/2,  5/2