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Is psi square just an assumption? Or there is a physical reason why they defined like that? My procedure is:

  1. It is intuitive for me to think possibility is proportional to energy distribution. Because lots of energy means lots of electron. Since energy of the wave is square of wave’s amplitude, it is exotic to define square of psi is proportional to possibility.
  2. Squaring psi means getting squared length of complex number. I know the length of complex number is to multiply the complex conjugate with itself. It is also intuitive to say that the length can represent the possibility. (Just intuition not mathematical reason.) since the length is abs of psi not square of abs psi, it is valid to say that sqrt(psi sqaure) is proportional to possibility. So, if I let the total possibility to be 3, not 1, then the integration of psi square must be 9.

I am struggling with this paradox. I need some mathematical reason or clear evidence that prove square psi is possibility.

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    $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Community
    Commented Jan 30 at 7:49
  • $\begingroup$ Sorry for not including details. I want to know the reason why possibility is proportional to the length square not length. $\endgroup$
    – 김건형
    Commented Jan 30 at 8:16
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/8062/2451 , physics.stackexchange.com/q/116595/2451 and links therein. $\endgroup$
    – Qmechanic ♦
    Commented Jan 30 at 8:49
  • $\begingroup$ physics.stackexchange.com/q/166232 $\endgroup$
    – alanf
    Commented Jan 30 at 8:54
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    $\begingroup$ To reopen this post (v3), consider to explicitly distinguish between $|\psi|^2$ and $\psi^2$. $\endgroup$
    – Qmechanic ♦
    Commented Jan 30 at 9:01

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