Comparision of the distribution of the sum of the squares of quantum numbers related to the energy with the classical distributions

The energy probability distribution of the quantum levels of a particle imprisoned in three dimensions in the manner of Schrödinger's common setup engenders; as contingent upon a fixed energy range, that we determine beforehand, will be the subject of this essay. Such an outlook is interesting from several angles: i) It looks very much like a classical Maxwellian distribution. ii) In the case we have as many free particles in the box as the number of levels depicted by the number of quantum levels in between the predetermined lower bound energy level and the upperbound energy level, all the while assuming that the Pauli Exclusion Principle holds (i.e., "that no pair of particles can occupy the same energy level represented by the corresponding set of quantum numbers"); the distribution we disclose becomes the energy probability distribution of the ensemble of particles imprisoned in the given box. iii) It can even be guessed that, if elastic collisions between the free particles in question, were allowed, just like is the case for molecules in a room, and still assuming quantization and the Pauli Principle, the outcome we disclose should be about the same as that of the energy probability distribution, molecules in a room would display in equilibrium. iv) The quantized energy being proportional to the sum of three squared integers associated with respectively, each of the spatial dimensions; the property we reveal certainly becomes remarkable from the point of view of mathematics of integer numbers – so much so that we further disclose how the approach pertaining to the probability distribution output remains the same for higher dimensions than 3, i.e. the sum of, say, four squared integers yields, be this qualitatively, the same output.