# Mirror symmetry of spin-1/2 wavefunction: Definition of the reflection operator

I would expect from a reflection operator $$\hat{M}$$ to leave a wavefunction unchanged if two times applied, thus $$\hat{M}^2=1$$. However, for a spin-1/2 particle this is not the case when following the standard definition of $$\hat{M}$$. Is there some scope to define the reflection operator in a different way? Where does the standard definition come from?

By standard definition I mean: The action of a reflection at the yz-plane (with normal in x-direction) $$\hat{M}_x$$ on the spin part is expressed via an inversion (which has no influence on the spin) and a 180° rotation around the x-axis. Rotations $$\hat{R}_{\alpha}(\vec{n})$$ of an angle $$\alpha$$ around the vector $$\vec{n}$$ can be expressed via: $$\begin{equation} \hat{R}_{\alpha}(\vec{n}) =exp\left(-i\frac{\alpha}{2}\vec{\sigma}\cdot\vec{n}\right) = \cos \left(\frac{\alpha}{2} \right) - i \vec{\sigma}\cdot\vec{n} \sin \left(\frac{\alpha}{2} \right) \end{equation}$$ with the Pauli matrices $$\vec{\sigma}$$. For a 180° rotation around e.g. the $$x$$-axis we get $$\begin{equation} \hat{R}_{\pi}(\vec{n}_x) = - i \sigma_x = -i \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}\, . \end{equation}$$ Obviously, $$\hat{M}_x^2=\hat{R}^2_{\pi}(\vec{n}_x)=-1$$.

• The literature and experiment were helpful and helped me to clarify my question: A reflection for spin-1/2 particles can be expressed via a +180° or a -180° rotation, what results in a reflection operator $\hat{M}_x = \mp i \sigma_x$. I think both are equally valid and do not see the need to restrict to one of these operators. If I apply two reflections, I could take two times the same operator and always get a "-"-sign as a result, but if I take one operator for the first reflection and the other for the second, I do not get the "-". Jan 5, 2020 at 13:31