13 Addition of angular momenta

TFY4250/FY2045 Tillegg 13 - Addition of angular momenta 9
J 2 2 = ¯h 2 j 2 (j 2 + 1). It is customary to state that j 1 and j 2 are “good quantum numbers”
in the new states, in addition to j and m.
We should also note that the “coupling term” Ĵ1·Ĵ2 has sharp values in the “new” states.
Using the relation Ĵ1·Ĵ2 = 1 2 (Ĵ2 − Ĵ2 1 − Ĵ2 2), we find that this coupling term commutes
withall the operators Ĵ2 1, Ĵ2 2, Ĵ2 and Ĵz:
[Ĵ1·Ĵ2, Ĵ2 1] = [Ĵ1·Ĵ2, Ĵ2 2] = [Ĵ1·Ĵ2, Ĵ2 ] = [Ĵ1·Ĵ2, Ĵz] = 0.
(T13.18)
And for a state with quantum numbers j, m, j 1 and j 2 we then have that
J 1·J 2 = 1 2 (J2 − J 2 1 − J 2 2) = 1 2¯h2 [j(j + 1) − j 1 (j 1 + 1) − j 2 (j 2 + 1)]
(T13.19)
is sharp.
On page 4 we stressed that Ŝ2 does not commute with Ŝ1z and Ŝ2z. This is shown as
follows:
This can be generalized to
[Ĵ2 , Ĵ1z] = [Ĵ2 1 + Ĵ2 2 + 2(Ĵ1xĴ2x + Ĵ1yĴ2y + Ĵ1zĴ2z) , Ĵ1z]
= 0 + 0 + 2[Ĵ1x, Ĵ1z]Ĵ2x + 2[Ĵ1y, Ĵ1z]Ĵ2y + 0
and then you will probably be able to argue that
From Ĵ2 = Ĵ2 1 + Ĵ2 2 + 2Ĵ1·Ĵ2 it then follows that
so that
= −2i¯hĴ1yĴ2x + 2i¯hĴ1xĴ2y = 2i¯h(Ĵ1 × Ĵ2) z . (T13.20)
[Ĵ2 , Ĵ1] = 2i¯h(Ĵ1 × Ĵ2), (T13.21)
[Ĵ2 , Ĵ2] = −2i¯h(Ĵ1 × Ĵ2). (T13.22)
[Ĵ1·Ĵ2, Ĵ1] = i¯h(Ĵ1 × Ĵ2) and [Ĵ1·Ĵ2, Ĵ2] = −i¯h(Ĵ1 × Ĵ2), (T13.23)
[Ĵ1·Ĵ2, Ĵ] = 0.
(T13.24)
Well, this was quite a few formulae, but it may be practical to have them collected this way,
for possible future reference.
13.5 Addition of orbital angular momentum and spin
An important example (e.g. if we want to study the hydrogen atom more closely) is the
addition of the orbital angular momentum L and the spin S of the electron:
J = L + S.
(T13.25)
The triangular inequality (T13.17) then gives half-integral values for the quantum number
j for the total angular momentum J = L + S:
|l − 1 2 | ≤ j ≤ l + 1 2 .
For l = 0, we have J = S and j = s = 1, so that we have two states, with J 2 z =
1
¯hm = ± 2¯h. (Here, the magnetic quantum number m denotes the z-component of the total
angular momentum.)