- #1
Siupa
- 30
- 5
Consider a system of two identical spin zero particles on a sphere. Let ##\vec{L} = \vec{L}_1 + \vec{L}_2## be the total orbital angular momentum of the two particles, and ##l_1, l_2## be the orbital angular momentum quantum numbers corresponding to particle 1 and particle 2.
Consider the simultaneous eigenstates of ##L^2, L_z## with ##l_1 + l_2 = 1##.
Are these states symmetric or anti-symmetric under particle exchange?
Now, the combined angular momentum quantum number for these states is ##L = 1##.
On one hand, ##(-1)^L## would tell me that they're antisimmetric for ##L = 1## (although this actually refers to parity, not particle exchange. But often these are related)
On the other hand, the symmetry properties of Clebsch-Gordan coefficients would tell me that any state in the highest weight tuple is always symmetric under particles exchange, by ##(-1)^{j_1 + j_2 - J}## with ##j_1 + j_2 = l_1 + l_2 = 1## and ##J = L = 1##.
Which one is the correct approach?
There's a similar question here, but it only deals with the case ##l_1 = l_2##, where the argument from parity and the argument from Clebsch-Gordan agree and give the same symmetry properties. However, in this case we must have ##l_1 \neq l_2##, and this seems to make the two approaches disagree.
Consider the simultaneous eigenstates of ##L^2, L_z## with ##l_1 + l_2 = 1##.
Are these states symmetric or anti-symmetric under particle exchange?
Now, the combined angular momentum quantum number for these states is ##L = 1##.
On one hand, ##(-1)^L## would tell me that they're antisimmetric for ##L = 1## (although this actually refers to parity, not particle exchange. But often these are related)
On the other hand, the symmetry properties of Clebsch-Gordan coefficients would tell me that any state in the highest weight tuple is always symmetric under particles exchange, by ##(-1)^{j_1 + j_2 - J}## with ##j_1 + j_2 = l_1 + l_2 = 1## and ##J = L = 1##.
Which one is the correct approach?
There's a similar question here, but it only deals with the case ##l_1 = l_2##, where the argument from parity and the argument from Clebsch-Gordan agree and give the same symmetry properties. However, in this case we must have ##l_1 \neq l_2##, and this seems to make the two approaches disagree.