What are the allowed states when coupling 3 identical bosons?

[malawi_glenn]

Homework Statement

Show that when tree identical quadrupole phonons (boson) are coupled togheter, only states with total angular momentum 0,2,4 and 6 are allowed.

The Attempt at a Solution

I know "how" to do it, but i do not know how to couple three angular momenta.

In the case of two, following is true for the total:

$$\vert j_1 - j_2\vert \leq J \leq \vert j_1 + j_2\vert$$

But how to do a similar thing when there is three? Shall I look up how the triangular inequality works for three vectors?

 

[George Jones]

i do not know how to couple three angular momenta.

In the case of two, following is true for the total:

$$\vert j_1 - j_2\vert \leq J \leq \vert j_1 + j_2\vert$$

But how to do a similar thing when there is three? Shall I look up how the triangular inequality works for three vectors?

You can "iterate" the rule for the coupling of two angular momenta. Couple two of the angular momenta, then, for each possible outcome, couple the resultant angular monenta with the third angular monentum.

I think Wigner 6-j symbols give a more efficient approach, but I haven't looked at them since grad school, so I might be mistaken.

 

[malawi_glenn]

okay, that came up into my head when I just hit the "post" botton. Yes, but I will not have the Wigner 6-j symbols until January I think =P

thanx