# What is meant by the spin of a particle? [duplicate]

I have been studying that electrons have quantum number called spin quantum number(s), this number can have either +1/2 or -1/2 value. If s=+1/2, the spin is clockwise and if s=-1/2, the spin is anti clockwise about it's imaginary axis. But, I am facing some problems now with this concept, the problems are, photon have 1-spin, another recently discovered sub-atomic particle having spin 3 (para. 7), how physicists explain these spin?

• Depends on what you mean by "explain". Physics is descriptive. We can observe that quantum mechanical objects have an internal physical property that behaves very similar to angular momentum. We call this spin. On the theoretical level it turns out that spin emerges naturally from relativistic quantum theory. That's however, is not so much an explanation as it is a consistent result: we see that the world is relativistic and nature seems to implement many of the mathematical consequences that come with that. – CuriousOne Oct 11 '14 at 14:49
• Possible duplicates: physics.stackexchange.com/q/1/2451 and links therein. – Qmechanic Oct 11 '14 at 15:41
• In QM angular momentum is a change in the phase of the wave function under rotations, which can come about in not one way (normal angular momentum) but two ways (normal ang mom OR mixing up components of the wave function, spin, which explains the need for all the representation theory) physics.stackexchange.com/q/135885 – bolbteppa Oct 11 '14 at 15:45
• Subquestion: Do we know that anything is actually spinning, or is "spin" just a term applied to a measurable side-effect of the particle. Ie, is spin a physical phenomenon or a mathematical notation? – Hot Licks Oct 11 '14 at 17:32

Spin is best understood as an intrinsic angular momentum. It is probably easier to understand the concept for a charged particle. A classical charged particle moving along a circle has an angular momentum and the "circuit" has a magnetic moment. Further, the two are proportional to each other.

It is experimentally found that a charged particle like an electron has an magnetic moment, the way it has a charge and a mass. We therefore suggest that the electron also has an intrinsic angular momentum $\vec{S}$, proportional to its magnetic moment $\vec{\mu}$.

We also find experimentally that an electron orbital angular momentum $\vec{L}$ is not a conserved quantity but $\vec{J} = \vec{L} + \vec{S}$ is. Therefore, $\vec{S}$ is not just a mathematical convenience but a "real" angular momentum.

Spin arises from the need to represent the rotation group $\mathrm{SO}(3)$ upon our Hilbert space of states. We need such a representation because the rotations (together with space translations) correspond to the non-relativistic changes of reference frames.

Since states are only determined up to rays in the Hilbert space, the true space of states on which we must represent the group is the projective Hilbert space, and the projective representations of a Lie group are (under some conditions) in bijection to the linear representations of its covering group, which is $\mathrm{SU}(2)$.

It turns out that these representations can be labeled by an integer $s \in \mathbb{N}$ or a half-integer $s \in \mathbb{N} + \frac{1}{2}$. This number is what we call spin.

• Apin arises from an experimental need to conserve angular momentum, mathematical explanations for this are an ad hoc addition. – nbubis Oct 11 '14 at 18:31
• +1, sigh... I hope some day I'll finally find the time to learn at least some basic QFT... :( – Phonon Oct 12 '14 at 23:29