De Broglie Waves: How Did He Come Up With It?

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How did De Brogie come up with the idea that matter are waves.
 
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He reasoned if a EM radiation can be described as a wave and as a particle (photon). Then perhaps the analog it true and things that were originally described/discovered as particles can behave as waves.

I think then the question was posed well then what is the wave equation describing this behavior.

Schroedinger who had spent most of his life in wave physics then derived this wave equation.
 

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