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Ciro's 10 cents:
Why is there a complex number in the equation? Intuitively and mathematically:
Some ideas:
Video 5. Necessity of complex numbers in the Schrödinger equation by Barton Zwiebach (2017) Source.
This useless video doesn't really explain anything, he just says "it's needed because the equation has an in it".
The real explanation is: they are not needed, they just allow us to write the equation in a shorter form, which is always a win:


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