Photon energy comes in packets

Question

From the HyperPhysics page on the Photoelectric Effect:

According to the Planck hypothesis, all electromagnetic radiation is quantized and occurs in finite "bundles" of energy which we call photons. The quantum of energy for a photon is not Planck's constant $h$ itself, but the product of $h$ and the frequency. The quantization implies that a photon of blue light of given frequency or wavelength will always have the same size quantum of energy.

I still don't understand how does a photon look like or how does light/EM waves are quantized.

If in some unbounded system (not in a box, but vacuum with no boundaries), I have "light" and nothing else, and the energy of the system is $E_0$, How do I know how many photons exist in that system?

One way to look at it is: $$ E_0=ℏω_0 $$ it would mean there is one photon with energy of $ℏω_0$. However, one could also see it as

$E_0=2ℏω_1$ where $ω_1=ω_0/2$.

And it would mean that there are two photons with frequency $ω_1$. In other words, how can energy be quantized/discrete when frequency itself is continuous.

I know there are similar post out there but I couldn't really understand it. In some of the post, they mentioned the photoelectric effects, but from what I understood, it just meant that light contains energy, and energy is transferable. There are also answers on Planck hypothesis, for me that is just the mathematical formulation to fit the experimental data. But the intuition behind "light is quantized" is still a mystery to me.

Answer 1

Energy as such is not quantized, we can see that from the spectrum of a black body: It is a continuous spectrum, so all values of frequency and thus energy are possible.

However, the energy of photons which are emitted by atoms when an electron in an atom's hull switches from a higher to a lower level, is only permitted to attain some particular values. The origin of this is the fact that an atom is a bounded system so the electron in its hull is not free. A free electron's energy could again attain any value.

Now, to describe the discrete energy levels in an atom it was necessary to have a model where electromagnetic waves can carry some characteristic energy, the photon model. With a classical wave model, such discrete energy levels cannot be described. That doesn't mean, however, that any system shows discrete energy levels, only bound states like atoms do so.

Answer 2

Your question relates to Planck's original work where he defined the Planck constant.

First, you must build on thermodynamics which tells us how systems in equilibrium distribute energy among themselves. So, a system with fixed energy could have one high energy photon, or it could have two photons with half as much energy - but neither system is in thermal equilibrium. Given abundant mechanisms for redistributing the energy into other frequencies the distribution of the photons will be the so-called black-body radiation spectrum. If light was not quantized then thermal effects would favor all energy making its way into the higher frequencies. If, however, the amount of energy in a given frequency is quantized then the energy density will peak at a finite frequency.

Answer 3

Radiation appears quantized only in confinement. The same is for electrons- they appear to move like waves in confinement. In confinement there are many participants and very small space to move. It could also be a small number, but moving very fast in a small space, giving the same effect.

As a result of confinement, the Coulomb inverse square changes to Hook's law- the space spring law. Hook's law is seen in every vibratory system.(The proof is simple, take three equally spaced points on a line. Give the middle a nudge keeping the ends fixed and you see the force;f=k/r^2 changing to f=Kr in the limit of small displacement). The result of vibration is that the whole force field becomes harmonic causing an electron to behave as a wave. And the energy levels to become discrete(characteristic of vibrating systems), ending in light behaving as quanta or particles as Einstein found in his photon work. But when in the open, electrons behave as particles with a well defined path- as one can see in numerous experiments, ranging from the vacuum tube to the cloud chamber. And photons/radiation revert back to the wave behavior, as Huygens found.

The easiest way to imagine a photon is think of it as the probability of finding that photon energy at a point. This way one can avoid questions about size and the rest.