Light Quanta
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Light was the only field in nature that was shown to be a field and measured as such before we recognized its associated quanta. In all other cases, the reverse was true in history.
In our courses so far, we derived light waves using fields at the end of our studies of electricity and magnetism. We spent time in optics showing how light waves interfere and diffract. With all that proof of light being waves, and the associated mathematics in place, there were those two experimentally troublesome results that demanded an explanation right near the turn of the 20th century - the spectrum of a blackbody, and the photoelectric effect. As we saw in the last two sections, neither of these experiments agreed with the theory of light waves from electromagnetism. Something was missing. Some crucial piece of understanding. That missing piece was quantization.
Max Planck's Solution to Blackbody Radiation
In 1900, Max Planck showed that the spectrum of a blackbody could be resolved by a novel assumption. Until his time it was assumed that the light was given off by atomic oscillators of any value of amplitude. After all, they were oscillating charges, and it was known for decades at that point (due to the work of Maxwell) that oscillating charges give rise to light. In fact, we used that idea to derive a light wave in our EM studies, and in application, we use that principle to broadcast radio waves even today throughout our world. So how can it be wrong?
While the part about atomic or molecular oscillators remained true, Planck found that if he only allowed the oscillators to oscillate at discrete energy states and corresponding amplitudes, he could reproduce the spectrum of a blackbody! He suggested that the allowable molecular oscillation energies were in multiples of a value dependent on frequency only, or where the constant h is now called Planck's constant. This began the quantum revolution! Planck was suggesting that the continuous states of nature - at least molecular oscillators - were illusions, and that such energy states are discrete, or that molecular oscillators have quantized energy states, the energies of which were given by his equation. This caught everyone's attention.
By making such a novel assumption, suddenly theory matched experiment exactly... yet the theory certainly had some strange implications.
Einstein's Nobel Prize
One person who was paying attention was Einstein. In addition to all the contributions Einstein made to relativity, he was the one to figure out what was going on with the photoelectric effect. It spite of the work in relativity for which Einstein is famous, it was for this work on the photoelectric effect that he was awarded his Nobel Prize. As an interesting historical side-note, it is worth mentioning that upon receipt of the Nobel prize - during which it is customary to give a lecture on the topic for which the award was won - Einstein gave a relativity lecture.
Einstein's solution to the photoelectric effect was ultimately the same solution that Max Planck used to understand blackbody radiation, but seen from another angle. He reasoned that if the molecular oscillators of a solid could only be in quantized states, then maybe the light they give off is given off in discrete quanta as well! Maybe all light in nature is actually a stream of uncountably many discrete quanta.
Einstein therefore suggested that the light incident on a metallic surface in the photoelectric effect was actually a stream of quanta, each with energy , rather than a continuous EM wave. He proposed that the photoelectric effect (which had that same term hf in it) is really not dependent on a light wave, but on the individual quanta of the wave, which we today call photons. The truth is, he didn't see it this clearly. But decades of experimentation and hindsight brings clarity.
The interaction between one photon and one electron during the photoelectric effect determines whether the electron is ejected from the metallic surface. When the frequency of light is high enough (and photon energy high enough) to overcome the binding energy that we refer to as the work function for a bulk metal, the electron is set free. Below that frequency, nothing happens because the individual light quanta don't have sufficient energy to liberate electrons. Above a certain threshold frequency, when electrons are ejected, any excess of energy is manifested as left-over kinetic energy for the electron.
To reiterate in mathematical terms, the equation derived from experiments is where the first term on the right is the incoming photon energy and the second term is the energy required to eject an electron. Think of this as energy conservation. Light delivers energy hf. From that energy is deducted the amount necessary to remove the electron from the metal which is . What's left over is the kinetic energy of the electron.
The kinetic energy is experimentally measured in terms of a stopping potential, or the voltage necessary to stop the electron from bridging a gap in vacuum between the metallic surface from which it started, to another surface. A voltage difference across the plates acts the same as gravity acting against a ball shot upward into the air. As the ball rises, it loses kinetic energy in exchange for gravitational potential energy. In the case of the electron, it loses kinetic energy in exchange for electric potential energy. The difference is that the voltage can be adjusted, which is akin to controlling how strong gravity is. At just the right value, the electron does not bridge the gap. That is when , where is the stopping potential or voltage, and e is the charge of the electron.