Angular momentums addition in QM

Question

We know that the spatial inversion parity for eigenfunctions of $hat {L}_{z}$ operator (spherical functions) is $(-1)^{l}$, where $l$ refers to angular momentum. So for product of two eigenfunctions with definite summary angular momentum $L = l_{1} + l_{2}$ corresponding wavefunction has parity $(-1)^{L}$.

Also full wavefunction (in $hat {L}_{z}$ representation) of two particles has parity $(-1)^{l_{1} + l_{2}}$ under their interchanging. For example, $l_{1} = l_{2} = l, L = 2l, m_{1} + m_{2} = 2l - 1$ in $hat {L}_{z}$ it $hat {L}_{z}$ representation is given as
$$
langle l_{1}m_{1}, l_{2}m_{2}|L = 2l, M = 2l - 1rangle = frac{1}{sqrt{2}}left(delta_{m_{1}l}delta_{m_{2}(l - 1)} + delta_{m_{1}(l - 1)}delta_{m_{2}l} right).
$$

The question: is it possible to establish a one-to-one correspondence between spatial inverse parity and interchange parity formally? One can imagine a bit classical mental experiment: let's have two particles with definite angular momentums. Let's use coordinate system with center in a middle of imaginary line which connects the particles. So in this case spatial inversion is equivalent to particles interchanging. I want to formalize it in a quantum case if this correspondence is true.

ZeroTheHero · Answer

In order to construct a state $vert L M_Lrangle$ as linear combination of product states $vert ell_1m_1ranglevert ell_2m_2rangle$, one requires Clebsch-Gordan coefficients $(ell_1m_1,ell_2m_2vert L M_L)$.  These coefficients have well-known symmetry properties:

Permuting the  $ell_1$ and $ell_2$ states gives
$$
(ell_2m_2,ell_1m_1vert L M_L)=(-1)^{ell_1+ell_2-L}
(ell_1 m_1,ell_2 m_2vert L M_L)tag{1}
$$
If $ell_1=ell_2$ as per your example, this shows that, for integer $ell_1$ even values of $L$ are symmetric under permutation, while odd values of $L$ are antisymmetric.  For half-integer $ell_1$, even $L$ are antisymmetric, while odd $L$ are symmetric.
Spatial inversion takes $hat zto -hat z$ and thus reverses the sign of the projections $m_1,m_2$ and $M_L$.  This is also a symmetry of the CG:
$$
(ell_1 -m_1, ell_2 -m_2vert L -M_L)=(-1)^{ell_1+ell_2-L}
(ell_1 m_1,ell_2 m_2vert L M_L), .tag{2}
$$

The phase factors in (1) and (2) are the same but this is a bit of a coincidence: there is no formal connection between the permutation group and space inversion.

(1) actually follows by a choice of convention (the Condon-Shortly phase convention) and may not hold is another convention is used.  (CS is by far the most prevalent and people often forget it is just a convention; there is freedom in choosing the sign of the highest weight state $vert L Lrangle$ and other choices will yield perfectly legitimate states.)

Space inversion can be formally implemented using a finite rotation by $pi$ about the $y$ axis i.e. by an element in the rotation group, but the permutation of the first and second state cannot be done by an element of the rotation group.  Indeed, for higher groups like $SU(3)$ the phases you get in the generalizations of (1) and (2) will
not be equal in general.

Angular momentums addition in QM

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