# How is the spin angular momentum $h/4\pi$ determined?

We can determine the orbital angular momentum by $mvr$, how is the intrinsic spin angular momentum of an electron?

• Did you read this article en.wikipedia.org/wiki/Spin_(physics) or any material related to the early history of spin, as an idea to explain experimental results? – user108787 Dec 1 '16 at 16:17
• You should first take it as an experimental fact. Have a look at the Stern-Gerlach experiment. Later on, Dirac arrived at the spin in a natural way combining relativity with QM. – FGSUZ Feb 6 '19 at 14:09

Spin is a complicated matter that requires a bit of sophisticated mathematics to understand fully. I'll do my best to give a brief overview. The idea of spin is intimately related to the idea of group representations. The Hilbert space of every particle carries a (projective) representation of the rotation group $$SO(3)$$. (Or if you prefer, a standard representation of the double cover SU(2) which is what you will usually see in these discussions. I find it more intuitive to think about the actually 3 dimensional rotation symmetry.) From pure mathematics relating nothing to spin or angular momentum, we find that the representations of this group are defined by either an integer, or a half-integer. In physical language, this means roughly that the angular momentum of a quantum-mechanical particle, either spin or orbital, is either an integer or half-integer multiple of $$\hbar$$.
The representations of the group $$SO(3)$$ are the linear operators on the space of states that we have come to call the "angular momentum operators". One particular representation gives us the operators for spin $$1/2$$ particles $$S_x = \frac{\hbar}{2} \sigma_x, \quad S_y = \frac{\hbar}{2} \sigma_y, \quad S_z = \frac{\hbar}{2} \sigma_z, \quad$$ where the $$\sigma_i$$ are the Pauli spin matrices. These operators act on a state, denoted $$|{s,m_s}\rangle$$ and "return" the spin of the particle. I.e, $$S_z |{s,m_s}\rangle = \hbar m_s |{s,m_s}\rangle$$ In the case of spin 1/2 particles, $$m_s = \pm1/2$$. Hence, we find that the spin of an electron for example is $$\pm \frac{\hbar}{2}$$, or as you wrote it, $$\frac{h}{4\pi}$$. I would be happy to elaborate more if you are interested in seeing more of the mathematics behind this explanation.