# Why do particles have spins such as $1/2$, $3/2$, $5/2$? [duplicate]

What does it mean to have 'half' spin? I have looked on Wikipedia and a few youtube videos on spin but they don't explain what it means to have $1/2$ spin. I am 18 and only starting to learning about quantum mechanics not so long ago, so please keep the vocabulary to a minimal.

• possible duplicate of Why does spin have a discrete spectrum? – user10851 Jan 27 '15 at 6:14
• To be precise, spin(intrinsic or inernal) comes from the fact that we are living in a world of 3 spatial dimension and the Lie algebra describing its rotation. There is a discrete sptrum of spin because they belong to different representations of 3 space-dimensional group of rotation, particularly $SO(3)$ or $SU(2)$ if you like. I may not be 100% precise in words but that's more or less the reasoning. – mastrok Jan 27 '15 at 6:28
• @ChrisWhite the other question is not particularly about the half integer, so not exact duplicate? – anna v Jan 27 '15 at 7:20

Quantum mechanics (QM; also known as quantum physics, or quantum theory) is a fundamental branch of physics which deals with physical phenomena at nanoscopic scales, where the action is on the order of the Planck constant

The Planck constant is a very small number, 6.6*10^-34 Joulesecond

Quantum mechanics was invented because the data showed that at these small dimensions measurable variables were often not continuous, but came in packets eventually called quanta. The necessity for this solution came from the photoelectric effect, the black body radiation, the discrete spectra of excited atoms, and it has been experimentally established that quantum mechanics is the underlying level of nature. For every measurable observable there corresponds a quantum mechanical operator which operating on the quantum mechanical state gives the probability of measuring the specific measurement. In the case of the operator corresponding to the angular momentum, the values are quantized. This theory developed because of the observation of quantization in orbital angular momentum in the solutions describing atoms.

It was then found experimentally that there existed an intrinsic angular momentum (named spin) characterizing particles like protons, neutrons, electrons which make up atoms and molecules. Spin 1/2 is the smallest quantum of angular momentum, conceptually in the same way that charge +/- 1/3 is the smallest quantum assignable to elementary particles..

The spin of the electrons is 1/2*h_bar, where

As elementary particles make up all matter, by algebra the only allowed values for spin are multiples of 1/2 and angular momentum multiples of 1 times h_bar. The smallness of the constant ensures that at macroscopic values angular momentum is to all intents and purposes continuous.

Thus the real answer is "because that is what we have observed to be the case in the microscopic interactions of particles".

• But how exactly is the 1/2 derived? – Ray Kay Jan 27 '15 at 6:07
• @RayKay from the experimental values of the experiments linked above. From measurements it was found that an intrinsic angular momentum should be assigned to electrons, for example, equal to 1/2*h_bar to fit the measurements. – anna v Jan 27 '15 at 6:52
• Lets put it this way: physics is about observations/measurements and the mathematical theories that can fit the observations measurements. Generally to get a new theory come first observations that disagree with the established ones, then new theoretical models are suggested, predicting observations that had not yet been seen, then a new established theory is accepted because it was validated by the data. Intrinsic spins were an observed/measured anomaly for the first models of QM which disappeared with the postulation of intrinsic spin. – anna v Jan 27 '15 at 7:03
• @RayKay If you're looking for an answer to "how is it even a logical possibility in the first place" that takes a bit more math. There are features about the structure of 3D space that let you detect a $360^\circ$ rotation but not a $720^\circ$ one; see the "plate trick" or "Dirac belt trick". Apparently nature decided to make use of this option. – Robert Mastragostino Jan 27 '15 at 9:49