An extract from the book "Quantum Mechanics and the Path Integrals" by Richard P. Feynman and A. R. Hibbs:

We can state the correct law for $P(x)$ mathematically by saying that $P(x)$ is the absolute square of a certain complex quantity (if electron spin is taken into account, it is a hypercomplex quantity) $\phi(x)$ which we call the probability amplitude of arrival at $x$.

What does hypercomplex quantity mean here?


Usually "hypercomplex" means a quaternion, but he just means that the wavefunction at a point $x$ has two components $\psi_\uparrow(x)$ and $\psi_\downarrow(x)$ with the probability density of being found at $x$ being $$ P(x)= |\psi_\uparrow(x)|^2+ |\psi_\downarrow(x)|^2. $$

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  • $\begingroup$ Thanks for the answer mate! Can you give good ref. to read more about Quaternions? $\endgroup$ – Vidya Sagar V Sep 30 '18 at 14:07
  • $\begingroup$ Thw wikipedia article "Quaternion" seems quite comprehensive. $\endgroup$ – mike stone Oct 1 '18 at 17:51
  • $\begingroup$ I found this video by 3blue1brown helpful. $\endgroup$ – Vidya Sagar V Oct 15 '18 at 10:18

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