It's said in textbooks that electrons won't radiate and fall into nucleus because matter wave of it's form a standing wave but could somebody explain why being a standing wave it doesn't radiate, even standing wave oscillates at a particular position and hence have acceleration and hence it should radiate right but it doesn't happen. could some one explain this . I understand that being standing wave it shouldn't lose energy so that it won't change it's wave property but at the same time it's said that it accelerates over time but acceleration means radiation should happen but it doesn't so what prevents it from radiation
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The standing wave idea doesn't help us with acceleration; as realized by OP, standing waves do indeed have their constituent parts accelerate over time. The idea is that standing waves do not decay: they do not lose energy, they do not change their wave properties over time, they always return to the same initial position, etc. If any part of the wave got lost (if the electron "begins" to radiate), the boundary conditions will no longer hold and we will no longer have a standing wave.
Note that this answer is circular! We just posit that standing waves don't change over time, then use this to explain why electrons don't change over time. This was part of the early quantum theory (https://en.wikipedia.org/wiki/Old_quantum_theory) and was superceded by the Schrödinger equation for explaining where the quantized energy levels come from and why only certain orbits for electrons are allowed.
Quantum Mechanic
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