3:1 statistics in QM
I have been impressed for decades with the mysterious 3:1 proportionality of structures that have two forms but do not obtain 50:50 statistics. I've come to QM because my first two examples involve the spin character of QM particles.
Ex. 1. The hydrogen molecule comes in two flavors, traditionally named para-H2 (parallel) and ortho-H2 (usually ortho- means perpendicular, but in reality the spin axes of the protons are mathematically parallel or collinear).
The boiling points differ between the two molecular structures such that the volume of one form is 3 times the volume of the other. I modeled the basic structure of the molecule in more ways than Feynman did and found one that looked the most promising: starting with separate protons being separated vertically at z = ± Δz with the plane of the Pauli type bonding orbit in the xy plane with its center at the origin. Now let the protons’ spin axes be parallel, i.e. horizontal, and oriented in bar-magnet fashion with their north poles pointing in the same direction (repulsive) or in opposite directions (attractive); the latter structure is likely the more frequent one. Since the Pauli-type orbit is non-radiative, i.e. a so-called “standing wave”, it remains midway between the protons and the most stable structure is the one with the protons more closely together.
Ex. 2. In the case of neutral orbits (an electron and a positron called positronium or e+e-) the difference between charges of the leptons satisfies the Pauli Exclusion Principle so that the magnetic orientations of the leptons are dual, and being physically parallel their north poles either point in the same direction or in opposite directions. When the e+e- is in its annihilation mode the so-called triplet is the one with its poles pointing in the same direction thus retarding its lifetime to collapse. When the north poles point in opposite directions the forces are strongly attractive thus hastening the lifetime of the singlet. That the most stable choice is the one whose lifetime is 1000 fold that of the other which means that the triplet is the one which is 3 times more frequent than the singlet.
Ex. 3. Bio-geneticists have discovered that the statistics of human eye color are what I term Dominant/Recessive statistics that divides into the 3:1 proportionality. To savvy this genetic rule, one needs to know the difference between Dominance and recessiveness, phenotype and genotype, “B” for brown and “b” for blue, and hybrid and pure. Let me describe the situation with my own family: my parents were both phenotype (shown color) “B”, and my 3 siblings were also “B”; I ,on the other hand, was blue-eyed which meant that each of my parents had to have one of their gene pair being genotype “b” which meant that my two genes were pure “bb”. It also meant that both my parents were hybrid “Bb” and that 2 of my siblings were statistically “Bb” and that the remaining one could have been pure “BB”. My 3 siblings had the same phenotype and my phenotype was pure recessive. I know for sure that one of my brothers was “Bb” because he had three blue-eyed daughters and his wife was also “Bb”. My other 2 siblings died young so that it could not be determined whether they were “Bb” or “BB”.