Of all the physical quantities the acquaintance as porcupine usually thinks like more "Meccano quantum". The word porcupine comes from English "spin", that means draft or to turn, and refers to a physical property of the particles (1) subatomic, as which any elementary particle has an intrinsic angular moment of fixed value. It is a proper characteristic of the particle as it it is the mass or the electrical load, and a magnitude that survives as the energy does it or the linear moment.
In contrast to what it happens with the angular moment of the macroscopic objects, to which we are accustomed, that can take very varied values depending on the actions to which they turn out to be subject, the magnitude of the porcupine of a particle is always the same one for this concrete type of particle. It is only the direction of the axis of draft the one that can change, although in a very strange way.
For an electron, proton or neutron the porcupine quantity is always 1/2 of the minimal value at the moment authorized (ħ). Precisely that's why this quantity at the moment angular would not be allowed for a compound object certainly number of particles orbiting without none of them being turn on itself. The porcupine can only appear because it is an intrinsic property of the proper particle, that is to say, that does not arise from the orbital movement of his parts concerning his center.
A particle that, like the electron, has an odd multiple porcupine of ħ/2 (ħ/2, 3 ħ/3, 5 ħ/2, etc) is called fermión, and presents a curious oddness: a finished rotation of 360th transforms his vector of the state not into himself but into the negative value of himself; he would need therefore a draft of 720th to remain just as before the draft. Most of the particles of the Nature are fermiones, the remaining particles for which the porcupine is a multiple point of ħ (ħ, 2 ħ, 3 ħ, 4 ħ, etc) they are called bosones. Under a rotation of 360th the vector of the state of a bosón returns to itself, and not to his negative.
If we take a porcupine particle 1/2, for example the electron, the space of the states Meccano quantum possible turns out to be two-dimensional, so that we can take a base of only two states that we can represent like [it arrives> and [below>, for the first one the porcupine turns to rights about the vertical direction up and for the second one it does it of the same way down. Likewise in an Euclidean plane any vector is a linear superposition of two bases ortonormales considered, in this case it happens equally, any possible state of porcupine of the electron is a linear superposition, for example:
w [it arrives> + z [below>, being w, z two complex numbers. Since the represented physical state remains unaltered if we multiply the component two by a complex number different from zero, the reason z/q will be the significant complex number that the state of the particle represents.
This complex number is represented on a Riemann of called sphere, as it appears in the figure. In the equator of the same one the singular points 1,-1 are, i and-i. 
The Riemann sphere plays a fundamental role in any quantum system of two states, describing the set of the possible quantum states. For a porcupine particle 1/2, his geometric role is a particularly clear position that the points of the sphere correspond to the possible spatial directions for the draft axis. In other situations the role of the sphere of possibilities of Riemann is more secret enough, with a much less clear relation with the spatial geometry.
The strange draft of 720th of the electron to remain equal is the whole paradox. Frequently it seems to us that the quantum mechanics presents phenomena completely out of any logic, but on having analyzed infinity of completely normal situations for us in view of this amazing theory, we observe that without her they have no explanation. The proper cohesion of the matter, as we know it, or the existence of four fundamental forces they would not have sense. In the latter case in his essentials, paradoxically, is the proper beginning of suspense. A "bothersome" beginning that seems that only it serves to prevent us from measuring with infinite accuracy.
(1) It is admitted that a "particle" can possess individual parts with such that could be treated mecanocuánticamente like a quite simple one, with a well definite entire angular moment.
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