Addition of Kinetic Moments

If we have any two kinetic moments 1 und 2 (it's an orbital moment and electron spin or orbital moments of two electrons) we will have:

J 1² y = j1(j1+1) h² y and J 2² y = j2(j2+1) h² y

J 1z y = m1 h y and J 2z y = m2 h y

Fig.1.: Vector addition of two kinetic moments j1 and j2

For total kinetic moment = 1 + 2 it should be the following

J ² y = J(J+1) h² y and J z y = m h y

m = −J, −J + 1, ... J

What values of J are possible for given J1 and J2?

After introducing simple vector model the total maximum kinetic moment will be as follows

Jmax = j1 + j2

and the total minimum kinetic moment Jmin :

Jmin = |j1 − j2|

where J takes only integer values.

Fig.2: The addition of two kinetic moments 1 and 2 into the total kinetic moment . The quantum numbers j1, j2 and j obey to the vector addition rule.

The quantum number of kinetic moment can take all possible values in the range from |j1 − j2| to j1 + j2 :

|j1−j2|, |j1−j2|+1, .......... , j1+j2−1, j1+j2

The last possible value (j1 + j2) corresponds to maximum possible parallel alignment between 1 and 2; and the first value corresponds to maximum antiparallel alignment. The total number of alignments is as follows:
For 1 there are 2j1 + 1 possible states (m1). And so the total possible amount of states is (2j1+1)·(2j2+1) since there are (2j2+1) states of 2 for each 1 state.

For any kinetic moment J there are 2J + 1 states:

J=|j1−j2| Σj1+j2 (2J + 1) = (2j1+1)·(2j2+1)

We can describe our system by using total kinetic moment , that is more rational (that's for sure) because only the total kinetic moment keeps the same.

Fig.3: and in the spin-orbital interaction with the total kinetic moment .

The simplest example for p-electron having spin s = ½ :
l = 1 , s = ½ (ml = −1, 0, +1; ms = −½, +½) ; the possible j-values begin from |l-s|=½ and end by l+s=3/2 and that's why the only possible values are j = ½ and j = 3/2.

mj = { | −½, +½ for j = ½
−3/2, −½, +½, +3/2 for j = 3/2

Here we assign the quantum numbers by small letters as we have already seen in the example with electron; the general case when one has a lot of electrons create orbital moment and spin are characterized by big letters which correspond to quantum numbers.

The levels doubling as a consequence of above-mentioned interaction between spin and orbital moment gives rise to orientation of spin ½ relative to orbital moment l (>0). Before we will come to this interaction we firstly must have a look at magnetic field influence on our electron.