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Those particles have discrete values of mass and electric charge. So it comes naturally that their electromagnetic radiation would come in quantified amounts.

Not really.In classical physics, there is nothing to stop r1 or r2 taking any value it likes

Since the charges can only change in multiplication of integer, so does the radius.

In a stable orbital motion, the radius depends on centripetal force.

In a stable orbital motion,

Quote from: hamdani yusufSince the charges can only change in multiplication of integer, so does the radius.For a given atom, the electron can exist in one of a potentially infinite number of orbitals.- The charge on the electron is always -1, regardless of which orbital it is in- The charge on the nucleus is fixed and positive- The product of electron charge and nucleus charge is fixedSo how come you can have several different permitted radii, if the product of the charges is fixed?- That's not it!QuoteIn a stable orbital motion, the radius depends on centripetal force.When a planet is orbiting the Sun in a circular orbit, the attraction between the Sun and planet balances the tendency of the planet to fly off into space in a straight line (hence your comment that "In a stable orbital motion, the radius depends on centripetal force"). - There are an infinite number of combinations of radius and velocity where these forces balance for a circular orbit, so this is not the source of quantization.- It gets more complicated if you try to account for elliptical orbits of planets, as the gravitational force does not balance the centripetal force for most of the orbit.- And most planets have elliptical orbitsIn atoms, only certain orbitals are permitted (quantization)- To calculate these orbitals, you need to solve the wave equation for the electron. - Some of these orbitals are spherical, but others look like a cluster of balloons assembled by a clown. How do you calculate the radius and centripetal motion for these? If you want a simple understanding, have a look at Bohr's model of the atom, where an electron's angular momentum is quantized (classical physics has no equivalent).- Or de Broglie's model where the electron has a wavelength, and that wavelength must have an integer number of wavelengths to be stable (classical physics has no equivalent).- But for a good model, you have to solve the relativistic Schroedinger equation, which gets quite complex for anything bigger than a hydrogen atom. Even a Hydrogen atom is beyond what they are paying me here!See: https://en.wikipedia.org/wiki/Atomic_orbital#Bohr_atom

Neither side "needs" to be quantised.the radius is a continuous variable.

ω².r³ = k.q_{1}.q_{2/}m

I want to know how far we can follow classical physics until it inevitably fails

Classical physics, with no quantisation of orbital energy immediately collapses in an "ultraviolet catastrophe".

The discovery of Planck's constant in the year 1900 was one of the most important discoveries that catalyzed the quantum revolution. What started as a simple idea to resolve one of the greatest physics mysteries of the time, turned out to be the key to unlocking the quantum realm. While Planck assumed that the constant would be 0 when measured the constant had a definite and real value, meaning that there was a lower limit on the universe. Preforming a basic version of this measurement is actually really easy and we explore the process of that measurement in this video using some LEDs and a diffraction grating.

A Simple Method For Measuring Plancks Constant//www.youtube.com/watch?v=q0jLUpBqessQuoteThe discovery of Planck's constant in the year 1900 was one of the most important discoveries that catalyzed the quantum revolution. What started as a simple idea to resolve one of the greatest physics mysteries of the time, turned out to be the key to unlocking the quantum realm. While Planck assumed that the constant would be 0 when measured the constant had a definite and real value, meaning that there was a lower limit on the universe. Preforming a basic version of this measurement is actually really easy and we explore the process of that measurement in this video using some LEDs and a diffraction grating.

We report the high-frequency modulation of individual pixels in 8 × 8 arrays of III-nitride-based micro-pixellated light-emitting diodes, where the pixels within the array range from 14 to 84 μ m in diameter. The peak emission wavelengths of the devices are 370, 405, 450 and 520 nm, respectively.