Einstein Revisits His Theory of Light
By 1911, Einstein had already hypothesized that light consists of particles he called light quanta (later called photons). Moreover, he had shown that light has an inherent quality, whereby it exhibits both wave and particle properties. Although, he had seen further than anyone into the mysterious nature of light, it continued to perplex him:
“I do not ask anymore whether these [light] quanta really exist. Nor do I attempt any longer to construct them, since I now know my brain is incapable of advancing in that direction.”
However, Einstein would content himself with light’s strange behavior to focus on general relativity until November 1915, returning once again in July 1916 to letting light preoccupy his thoughts. The end result was a deeper understanding of the interaction between light and matter, which culminated with three papers, two in 1916 and the most prominent one in 1917.
Quantum, Light, and Atoms Before Einstein
As we have seen, Planck pioneered the quantum theory of light and matter. In his model, matter took on the intentionally ambiguous form of “resonators” – nothing more than an oscillating charge of small mass. A resonator’s interaction with light was “mostly” classical, in that its explicit interactions were between it and the classical electric field of light. The quantum portion of the theory was in the energy of the resonators, and its appearance in the theory was rather startling and without a mechanistic explanation. Bohr had brought quantum theory to the inside of the atom by quantizing the electronic orbits. He also provided a quantum theory of light and matter (at the atomic scale), where an electron jumping between orbits results in either the emission or absorption of light. Einstein’s quantum theory of light and matter would extend beyond both of their theories, while bringing together their best features.
Einstein was motivated by several factors. Undoubtedly, quanta and the “wave-particle duality” of light continued to weigh heavy on his mind. Bohr had provided a mechanism for the interaction of light and matter at the atomic scale, much richer in detail than Planck had with his resonator and light model, and Einstein wanted to explore the consequences further. Finally – and this point should be emphasized – it had been sixteen years since Planck derived his radiation law, and yet a completely quantum mechanical derivation of it was still nonexistence. In Einstein’s own words:
“[Planck’s] derivation was of unparalleled boldness, but found brilliant confirmation. … However, it remained unsatisfactory that the [classical mechanical] analysis, which led to [Planck’s Radiation Law], is incompatible with quantum theory, and it is not surprising that Planck himself and all theoreticians who work on this topic incessantly tried to modify the theory such as to base it on noncontradictory foundations.”
Indeed, despite its incredible success against ever-increasing experimental data, Planck’s theory was still tarnished by its “mostly classical” derivation. In fact, to be honest, the rigorously derived parts of the theory were all classical in nature; the quantum component (centered around the energy of the oscillators) wasn’t derived at all, but rather complete speculation.
Einstein’s Quantum Theory of Light
In 1905, Einstein arrived at light quanta by comparing an ideal gas to light. At that time, he had considered them as two separate systems, each at thermal equilibrium contained in their respective boxes. In 1916–7, he would consider them again, but this time as a “mixture” at thermal equilibrium in a single box. This time he wondered what frequency distribution of light (frequency spectrum or radiation law) was needed to maintain this system of matter and light in thermal equilibrium.
Bohr’s atom gave rise to the notion that an atom has electronic energy levels, which electrons can jump between as they absorb or emit a photon. Thus if we imagine the simplest atom, hydrogen, with its single electron, the quantum state of the entire atom is described by the electronic energy level occupied by the electron.
Einstein considered a collection of such atoms consisting of only two electronic energy levels. By now Einstein was a master at using statistical mechanics to probe physical problems and this time would be no exception. He asserted that the probability of finding the atoms of the system in either one of these two quantum states was given by the Boltzmann distribution. Moreover, he assumed that only three dynamical processes governed the transitions between the two energy states, each occurring with a certain transition probability. These three processes he named: spontaneous emission, stimulated emission and absorption.
Spontaneous emission occurs when an electron jumps from the higher to the lower energy state of the atom, emitting a photon in the process. The electron isn’t actually influenced to do this by light but rather does it naturally. As with other spontaneous processes, spontaneous emission is an irreversible process occurring naturally (without the slightest input of work) and resulting in an increase in entropy. On the other hand, stimulated emission occurs as a result of interacting with the light. Specifically, a photon “bumps” the electron on the way by, causing it to jump into the lower energy state once again emitting a photon (in addition to the one that passed by in the first place). Finally, absorption occurs when an electron absorbs a photon and jumps up to the higher energy state.
Using these three processes as the basis for the interaction of light and matter in thermal equilibrium, Einstein was able to arrive at the desired frequency distribution, and found it to be given by none other than Planck’s Radiation Law; Planck’s Radiation Law is the equilibrium spectrum for light in thermal equilibrium with matter. Out of the three processes, stimulated emission hadn’t been described before until Einstein’s 1916 work; the other two were already contained in Bohr’s model.
It turns out that stimulated emission is crucial in obtaining the correct form of the radiation law, without it one obtains Wien’s Radiation Law; evidently, stimulated emission is important in correctly obtaining the low-frequency contributions to the radiation law. Another gem of Einstein’s theory was that Bohr’s Frequency Rule was obtained as a natural consequence. However, Einstein wasn’t finished yet.
Light Has Momentum
As discussed before, a system of ideal gas atoms at thermal equilibrium will have a Maxwell distribution for the velocities. In 1917, Einstein asserted the same was true for thermal equilibrium of his mixture of atoms and light, and set out to find the frequency distribution that made this possible. He used the same approach as in 1909, when studying momentum fluctuations of light.
Recall that there he considered a “small” mirror, moving in only one direction, and light in thermal equilibrium. For his current study, an atom from the mixture plays the role of the small mirror, and the resulting fluctuation equation is the same. Using this as his starting point he was able to show that Planck’s Radiation Law is the correct form of the frequency distribution needed to maintain the Maxwell distribution for his mixture at thermal equilibrium. The result is particularly interesting because Einstein arrived at it by only considering the interactions between the atoms and light.
In other words, the collisions between the atoms themselves played no role in his calculation. Therefore, while a collection of ideal gas atoms alone at thermal equilibrium attain a Maxwell distribution for the velocities, so do the gas atoms of Einstein’s mixture interacting with light (according to Einstein’s three processes) at thermal equilibrium.
However, one of the most important consequences of Einstein’s theory was centered on the momentum of a photon. In Einstein’s 1905 work about light quanta, the focus was on the energy of a photon of light, while its momentum was of no consequence. In 1909 this changed, and Einstein showed that the momentum fluctuations due to light involved both particle and wave terms. This was an amazing result that positioned Einstein to be able to write down both the energy and the momentum of a photon. He didn’t do this at that time, and as to why he didn’t, we’ll never know.
It wasn’t until his works of 1916–7 that he completed the picture of the photon by endowing it with both energy and momentum. The momentum played a key role in Einstein’s aforementioned result involving the Maxwell distribution. Einstein notes that:
“The most important result [of the present work] seems to me, however, to be the one about the momentum transferred to the molecule in spontaneous or induced radiation processes.”
According to Einstein, regardless of if the atom absorbs or emits a photon, the amount of momentum transferred is:
the photon energy divided by the speed of light. For absorption to occur, one imagines an incoming photon traveling in a specific direction and “bumping” into the atom. It’s in this direction that the momentum is transferred to the atom.
For stimulated emission, the atom is once again “bumped” by an incoming photon, but this time it causes the atom to emit a photon of its own. As before the direction of the momentum transfer is determined by the incoming photon, but this time it will be in the opposite direction. However, in spontaneous emission there isn’t an incoming photon involved, the atom simply emits a photon at will.
Quantum Probability: The Beginning
So in what direction is the momentum transfer? According to Einstein, the direction is determined only by “chance”. There’s simply no way of knowing for sure as there was for the other two processes. Indeed, Einstein had hit upon the uncertainty that was inherent in what would later be quantum mechanics. This work would serve as a turning point in two regards. First, it would establish the physical reality of light quanta for Einstein, once and for all. Writing to a friend shortly after this work Einstein says:
“I do not doubt anymore the reality of radiation [light] quanta, although I still stand quite alone in this conviction.”
Einstein remained alone until 1923 when the experimental work of Arthur Compton (1892–1962) showed:
“… very convincingly that a radiation [light] quantum carries with it directed momentum as well as energy.”
Finally, his 1917 work would be the beginning of Einstein’s departure from what would later be (the post-quantum theory of) quantum mechanics. Einstein had commented that the chance, or probabilistic nature of spontaneous emission was a shortcoming of the theory, although he maintained confidence in the approach he had used. Once again writing to his friend, Einstein says:
“I feel that the real joke that the eternal inventor of enigmas has presented us with has absolutely not been understood as yet.”
If in 1917 Einstein truly saw the probabilistic nature as a shortcoming, later he would be less forgiving. His last contribution to quantum theory (and considered by many to be his last significant scientific contribution as well) would occur in 1925, and thereafter he would turn his back forever on quantum mechanics, arguing that its probabilistic nature was a fundamental flaw.
