[hamdani yusuf (OP)]

According to Planck's Law, energy of radiation is quantized.
E = n.h.f
n is integer.
h is Planck's constant
f is frequency

Dimensional analysis tells us that energy has mass and length in it. In the equation above, they reside in Planck's constant. So the quantification of energy must come from some things that determine the value of Planck's constant.

We have learned that electromagnetic radiation comes from moving electrically charged particles. They have kinetic energy according to the formula
Ek=½.m.v²
m is the mass and v is the speed
The particles also have potential energy which depends on their electric charge.

Electrostatic potential energy of q due to Q1 and Q2 charge system:

https://en.wikipedia.org/wiki/Electric_potential_energy#One_point_charge_q_in_the_presence_of_n_point_charges_Qi
Those particles have discrete values of mass and electric charge. So it comes naturally that their electromagnetic radiation would come in quantified amounts.

[puppypower]

If you compare quantized energy levels, to the hypothetical energy levels that would be expected using continuous mathematical functions, quantized places a narrower limit on what is possible. Not all states are possible in the quantum world.

 From a practical POV, quantum saves time; lowers time potential. In other words, if we needed a given value to appear, so we can go from step 1 to step 2, by placing a quantum restriction, we will reach the needed state much sooner, throwing dice, compared to if all all states were continuous and possible. This allows the universe to evolve faster; saves time or speeds up evolution.

The affect is similar to loading dice. If we place a load on a dice, the odds will increase for certain outcomes, based on where the load has been placed. Other outcomes become unlikely, since they are in the gaps between the quanta This makes a purely random universe more predicable and deterministic; betting odds improve.

[Bored chemist]

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.

[hamdani yusuf (OP)]

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

In a stable orbital motion, the radius depends on centripetal force. In electromagnetic interaction, it depends on electric charges of the particles involved. Since the charges can only change in multiplication of integer, so does the radius.

[evan_au]

Since 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 fixed

So how come you can have several different permitted radii, if the product of the charges is fixed?
- That's not it!

In 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 orbits

In 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

[Bored chemist]

In a stable orbital motion,

In classical physics, any orbital radius is allowed- as long as the velocity matches it.

[hamdani yusuf (OP)]

F centripetal = F coulomb
m.v²/r = k.q1.q2/r²
m.ω².r = k.q1.q2/r²
ω².r³ = k.q1.q2/m
The value in the right side is quantized, but the left side is not. But the equation above only take electrostatic force into account. The additional force due to electrodynamic (magnetic) hasn't been included. I guess that the formula for electrodynamic force can be set to make the radiation energy per frequency quantized.

[Bored chemist]

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

[hamdani yusuf (OP)]

Since 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 fixed

So how come you can have several different permitted radii, if the product of the charges is fixed?
- That's not it!

In 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 orbits

In 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

I want to know how far we can follow classical physics until it inevitably fails and no reasonable assumptions can be put into it to make it work and agree with observations.

[hamdani yusuf (OP)]

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

If we accept that mass and charges are quantized, while k is constant, then it is necessary that value of (k. q. q/m) is quantized.
Planck's law only suggests for quantized radiation energy, while the frequency itself is not quantized so it can have any value.

[evan_au]

ω².r³ = k.q₁.q_2/m

I agree that everything on the right side is quantized.

So that leaves an infinite number of combinations of ω & r on the left hand side, ie neither ω nor r are quantized by this equation.

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".
I have seen suggestions that all atoms would collapse into neutrons within femtoseconds!
The Earth would collapse into a neutron planet within 15 minutes.

See: https://en.wikipedia.org/wiki/Ultraviolet_catastrophe

[hamdani yusuf (OP)]

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

There are many versions of classical physics, other than Newtonian and Maxwellian theories. Ultraviolet catastrophe is not the only failure of Maxwellian physics.
I want to know what's the cause of that quantization in a more fundamental level,  rather than taking it as a postulate.

[hamdani yusuf (OP)]

A Simple Method For Measuring Plancks Constant

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.

[Bored chemist]

A Simple Method For Measuring Plancks Constant

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.

Interesting, but more or less wrong.
The brightness vs voltage curve for an LED is a smooth function, quite close to an exponential.
So there's no voltage where the light suddenly "turns on".
Here are some examples. (The light output is proportional to the current).
https://www.researchgate.net/figure/Current-versus-voltage-I-V-curves-for-pixels-of-differing-diameter-from-8-8-micro-LED_fig1_260327577

[hamdani yusuf (OP)]

Here's the diagram from the research mentioned in BC's post.

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.

It seems like the experiment doesn't explicitly support the quantization of energy.

[hamdani yusuf (OP)]

Imagine an electron oscillating up and down in 1 Hz frequency. What's the amplitude of the oscillation which is corresponding to minimum energy transfer, which is h? Why can't we get lower than that while still non zero?

[hamdani yusuf (OP)]

Dimensional analysis tells us that energy has mass and length in it. In the equation above, they reside in Planck's constant. So the quantification of energy must come from some things that determine the value of Planck's constant.

Let's analyze Planck's formula by plotting it in a graphic. To avoid unnecessary complications, the units are chosen so that every constant has value of 1.
https://www.wolframalpha.com/input?i=plot++1+f%5E3%2F%28e%5E%281+f%29+-1%29+from+0+to+20

What will happen if the Planck's constant were smaller or bigger than that?

https://www.wolframalpha.com/input?i=plot++0.5+f%5E3%2F%28e%5E%280.5+f%29+-1%29+from+0+to+20

https://www.wolframalpha.com/input?i=plot++2+f%5E3%2F%28e%5E%282+f%29+-1%29+from+0+to+10

It seems that the curve maintains it's shape, and only the scales in x and y axes change. It means that quantization of radiation energy can't be derived solely from Planck's formula. It must come from somewhere else.

[hamdani yusuf (OP)]

Imagine an electron oscillating up and down in 1 Hz frequency. What's the amplitude of the oscillation which is corresponding to minimum energy transfer, which is h? Why can't we get lower than that while still non zero?

Let's approach the problem from different direction, while keeping it simple.
Imagine an electron oscillating up and down in 1 Hz frequency, and the amplitude of the oscillation is 1 meter. Is there any way to determine its radiation power?

[hamdani yusuf (OP)]

SI unit analysis tells us that h is stated in Joule second per cycle. The cycle is often omitted because it has no dimension. But IMO, this can lead to confusions. Why can't we say that the unit is Joule second per cycle square?

According to current quantum theories, the minimum non-zero energy can be radiated by an antenna transmitting 1 Hz electromagnetic wave is h Joule. But there's no minimum power, hence the power output can be made arbitrarily close to zero, given an extremely long period to radiate it.

[hamdani yusuf (OP)]

This video tries to find analogy for quantification of electromagnetic radiation frequencies in atomic spectrum in wave physical framework. But it doesn't explicitly state why the energy for a certain frequency is quantized.

Trying to find analogies between the wave energy confined in a string and matter interacting with light.

0:00 Intro
6:38 Experiments with waves in a string
15:40 Analogies with electron waves
22:50 Changing the standing wave mode in a string using phase
26:49 A hypothetical model for demonstrating quantized wave behavior
32:26 Elastically Inertial Poetry

When I learned about Balmer formula, I also imagined about beat frequency. This video shows how to visualize it.