I'm reading Quantum Liquids by A.J. Leggett and became confused by the following statement in the first chapter.
Consider now a pair of such identical atoms. In the absence of appreciable coupling between the hyperfine and (atomic center-of-mass) orbital degrees of freedom, the energy eigenfunctions can always be written as a product of a “spin” function (i.e. a function of the hyperfine degrees of freedom) and an orbital function (i.e. a function of the coordinates of the atomic nuclei). The symmetry of the orbital wave function under exchange of the two atoms is $ (−1)^L $ where $L$ is the relative orbital angular momentum. As to the spin wave function, the combination of two atomic (intrinsic) angular momenta each equal to $F$ yields possible values of the total dimer intrinsic angular momentum $K \equiv \left| F_1 + F_2 \right| $ equal to $0, 1, \ldots , 2F$.
$F$ here is the total atomic spin.
I have trouble understanding how the relative phase of the wave function results from the difference between the magnitude of the angular momenta when their $z$-projections ($z$ being the quantization axis) are not known. Does anyone know how $L$ and $K$ are determined?
Ok, so "relative" here had to do with "relative motion" and not "difference".
Now... does anyone know the reasoning behind the factor of $(-1)^L$ ?