## Thursday, July 30, 2020

### Solutions To Atomic Physics Problems

1)Find the Lande g-factor for an atom in each of the following states:

3 F 3 ,  3 F 2  and   3  F 4

Solutions:

The Lande g-factor is:

g  = 1 +  [J (J + 1) + S (S + 1) - L (L + 1)/ 2 J(J+1)]

For 3 F 3:   J = 3, S = 1,  L = 3

g  = 1 +  [3 (3 + 1) + 1 (1 + 1) - 3 (3 + 1)/ 2 x3(3+1)]

g  = 1 +  [12 + 2 -  12 / 24] = 1 +  1/12 = 1.08

For 3 F 2:   J = 2, S = 1,  L = 3

g  = 1 +  [2 (2 + 1) + 1 (1 + 1) - 3 (3 + 1)/ 2 x2(2+1)]

g  = 1 +  [6 + 2 - 12/ 12] = 1 +  (-4/12) = 8/12= 2/3

For  3 F 4:   J = 4, S = 1,  L = 3

g  = 1 +  [4 (4 + 1) + 1 (1 + 1) - 3 (3 + 1)/ 2 x3(3+1)]

g  = 1 +  [20 + 2 – 12 / 24]  =  1 +  5/12 = 1.41

2(a) Assuming the L· interaction to be much stronger than the interaction with an external magnetic field, calculate the anomalous Zeeman splitting of the lowest energy states:

2 S 1/2 ,  2 P 1/2  and   P 3/2

In the hydrogen atom for a field of 0.05T

Present a table with the results of the calculations showing the energy states in the extreme left side column under ‘State’, with the headers of the other columns:
L, S, J,  g  ,  M JD E (in eV x 10 -5  )

Solution:

From the example problem :

m J =   m J    J / | J |  =   g  (-e ħ /2m) Ö J (J + 1) [  J/ Ö J (J + 1) ħ =
- e ħ /2m  (g J)

The energy splitting is then given by:

D E =  -  m J  B = e ħ /2m  g J·B  e ħ /2m  g B Je ħ /2m  g B M J

And we know already from quantum mechanics that:

M J   = J, J – 1,……, -J + 1, - J

So that for a given field intensity B each energy level will split into 2J + 1 sublevels with the amount of splitting determined by the  g –factor.

Then:   D E =   e ħ /2m  g B M J

= (5.79  x 10-5 eV/T ) g  (0.05T) M J

So, calculate the value of g for each energy state, i.e.

g  = 1 +  [J (J + 1) + S (S + 1) - L (L + 1)/ 2 J(J+1)]

Then substitute into the equation for  D E   with the correct value of  M J

We obtain for the results table:

2(b)  Given that: m s  = - e (2S)/ 2m and:

m  =    (-e (L)/ 2m)

Show in a vector diagram that m  and J  are not parallel.