Meaning Of The Amplitude

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As a student, I asked why, in some types of waves, the square of the amplitude and the density of energy are directly proportional. I got answers based on general laws of physics.

 

Then I asked why, in the De Broglie - Schrödinger waves, the square of the amplitude is related to probability and not to energy density. I got a response referring to an idea proposed by Max Born, without mention of the general laws of physics.

 

If possible, I would like to understand what is the relationship between that idea of Max Born and the general laws of physics.