# How an electron spin 1/2 needs two full rotations instead of one to return to its original state?

In Quantum Mechanics an electron (or any fermion) wave function changes by a negative sign or -1 under a rotation of 360°. And if we rotate the electron in physical space we would need to rotate it twice around or $$2 \times 360° = 720°$$ for it to come back to its original state. Apart from the unphysical? and bizare topology of such an object is there a mathematical proof of this in quantum mechanics?

• Have you studied spinor representations of the rotation group? Commented Oct 7, 2020 at 3:47
• not yet, but I will. thx.
– user276350
Commented Oct 7, 2020 at 4:03

This certainly is a bizarre property of spin-half particles. Remember that angular momentum is generated by the operators of rotation, so that for a state say $$| \psi \rangle$$ if we consider a rotation about, say the z-axis ($$\hbar = 1$$), then

$$J_z | \psi \rangle = -i \frac{\partial}{\partial \theta} | \psi \rangle$$

where $$J_z$$ is the angular momentum operator. Let's say that $$J_z$$ has a definite value so that a measurement of spin along the z-axis gives us a value $$m$$ and $$m$$ is the intrinsic spin ("magnetic") quantum number. That is,

$$-i \frac{\partial}{\partial \theta} | \psi \rangle = m | \psi \rangle$$

and solving this, the state $$| \psi \rangle$$ can be represented by $$e^{i m \theta} | \psi (\theta) \rangle$$. So if $$m = \frac{1}{2}$$ and we rotate by $$2 \pi$$ then $$| \psi \rangle \rightarrow - | \psi \rangle$$. Doing this once again will retain the original state $$| \psi \rangle$$.

• It might be pointed out that multiplication by a phase-factor $$e^{i m \theta}$$ is just that and does not change the physics of the system. This is true, but it should also be noted that the fact that the state changes sign after a rotation (rather rotation of the space around the particle represented by the state $$| \psi \rangle$$) is physically observable.
• thx for that one again @drjh.
– user276350
Commented Oct 7, 2020 at 4:06
• See also the cute " Square Root of 'NOT' " en.wikipedia.org/wiki/Quantum_logic_gate Commented Oct 7, 2020 at 12:39