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:
J1² y
= j1(j1+1) h²
y
and J2²y
= j2(j2+1) h²
y
J1z y
= m1 h y
and J2zy
= m2 hy
![]() |
Fig.1.: Vector addition of two kinetic moments j1 and j2 |
For total kinetic moment
=
1 +
2
it should be the following
J² y
= J(J+1) h² y
and Jzy
= m hy
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
![]() ![]() ![]() |
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:
![]() ![]() ![]() |
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.