When expressing [itex]exp(\frac{i\sigma\cdot \widehat{n}\phi}{2})[/itex] as a series expansion, why can we make the assumption that [itex](\sigma\cdot \widehat{n})^{2}=I[/itex]?(adsbygoogle = window.adsbygoogle || []).push({});

Someone I asked on campus showed me this and I don't quite understand its implications (partly because I don't quite understand permutation tensors):

[itex]\sigma_{i}n_{i}\sigma_{j}n_{j}=n_{i}n_{j}(\delta_{ij}I+i\epsilon_{ijk}\sigma_{k})=(\widehat{n}\cdot \widehat{n})I+0=I[/itex]

The middle two steps are really foreign to me, but I get the component parts in front and the last part. Could someone explain in further detail why the middle two parts exist?

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# I need an explanation behind this calculation

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