Einstein’s Miracles Part 1: Quanta

By Caleb Todd

“There is nothing new to be discovered in physics now. All that remains is more and more precise measurement.”

—Lord Kelvin, 1900

“Hold my beer.”

—Albert Einstein, at some point in 1905 probably

As the 19th century came to a close, there was a strong sense that physics was complete. Isaac Newton had long since formulated his three laws of motion, which described the behaviour of all physical objects. James Clerk Maxwell’s electromagnetic theory explained how light and electrically charged matter interacted. Even systems too complex for analysis from first-principles were being conquered using the tools of statistical mechanics. Sure, there were a few wrinkles to iron out, but physicists had formalisms to deal with any conceivable problem. These formalisms — collectively called the tools of classical physics — appeared to work. To many, it seemed that there was nothing really new to be discovered.

However, within the first five years of the new century, all illusions of the completeness of physics were utterly dispelled. The wrinkles in classical mechanics instead turned out to be loose threads which, when pulled on, would unravel the entire tapestry. A series of death blows were dealt to the very foundations of how we understood the universe, and the chief executioner was a little-known patent clerk called Albert Einstein. In 1905, fresh out of his PhD, the 26-year-old Einstein published four papers that revolutionised physics. Producing any one of them was itself an extraordinary achievement; to publish all four in a single year was nothing short of miraculous. 1905 is now called Einstein’s annus mirabilis — his “miracle year” — and is arguably the most inspired single year in the history of science.

There are three main topics that a university course on so-called ‘modern physics’ will cover: quantum mechanics, atomic physics, and special relativity. Quantum mechanics was established by the first of Einstein’s 1905 papers. The existence of atoms was proven by his second. His third paper founded the field of special relativity. Einstein’s fourth paper presented what is now the most famous equation on Earth: E = mc2. Einstein’s work in 1905 was seminal in taking physics beyond the classical domain. A series of articles this year, of which this is the first, will discuss what each of these papers really mean. I hope to convey how extraordinary they are as intellectual achievements, and how significant they have been to physics as a whole.

(1)

“On a Heuristic Viewpoint Concerning the Production and Transformation of Light”

Annalen der Physik, June 9th, 1905

To understand why this paper was so important in the early days of quantum mechanics, we must first discuss how classical physicists understood light.

Maxwell’s equations were developed to describe how electric and magnetic fields interact with each other and with charged particles. The moment Maxwell discovered that his equations predicted the existence of electromagnetic waves that travelled at the speed of light was one of the greatest moments in physics. The nature of light had finally been revealed. Young’s double-slit experiment had confirmed that light was fundamentally a wave long before [1], but now physicists knew what was ‘waving’ — the electric and magnetic fields.

Imagine you and a friend are holding one end of a rope each. Now suppose you begin rapidly moving your end up and down so as to induce wave motion in the rope. How much energy you put into the rope wave can be varied continuously. You can increase or decrease how vigorously you move the rope by any amount you choose (within the limits of your strength). Such is the case with any wave in classical physics, and such was assumed about electromagnetic waves. However, this assumption caused some issues; in particular, it was responsible for the ‘ultraviolet catastrophe’. If you assume that electromagnetic waves can transfer energy to matter in arbitrary amounts, a neat piece of mathematics demonstrates that the intensity of light being radiated at each wavelength increases vastly as you decrease the wavelength [2]. Not only does this prediction disagree with experiments (as shown in Fig. 1), it also implies the matter is radiating infinite energy, which is an impossibility. The assumption must be incorrect.

Figure 1: The intensity of radiated light from an ideal blackbody. The black curve shows the prediction obtained when assuming light and matter can exchange arbitrary quantities of energy at a temperature of 5000 K. The blue, green, and red curves show the experimental measurements at 5000 K, 4000 K, and 3000 K respectively. There is a clear disagreement between the classical prediction and the true values, particularly at short wavelengths — this is referred to as the ultraviolet catastrophe.

Max Planck resolved the ultraviolet catastrophe in 1901 by proposing an alternative theoretical starting point. He assumed that energy would be exchanged between light and matter only in discrete chunks — so-called “quanta” of energy [3]. In our rope analogy, this would correspond to you being able to increase the amplitude of your oscillation only in increments of, say, 10 cm. You could have 40 cm tall waves or 50 cm tall waves, but not 45 cm tall waves. An absurd assumption at face value, but it leads to correct mathematical predictions regarding the intensity of radiation from matter [3].

Planck’s proposal is now regarded as the birthplace of quantum mechanics, but Planck himself thought very little of his technique. Einstein, however, dared to consider what would happen if you took the idea of quanta seriously. Rather than supposing only that energy left and entered the electromagnetic field in discrete chunks, he proposed that light itself was separated into discrete chunks. Rather than a continuous electromagnetic wave, completely distributed throughout space, light is instead composed of localised, particle-like packets that we call photons. The transfer of energy quanta as per Planck’s work was really the absorption or emission of these photons. This is the “heuristic viewpoint concerning the production and transformation of light” of which the paper’s title speaks.

On what basis, though, could Einstein make this claim? Planck’s theory required only that the exchange of energy happens discretely — for light itself to be discretised is a far stronger statement. Einstein’s approach was to show how his new theory could explain phenomena beyond Planck’s radiation intensities. There were two principle phenomena he dealt with: he showed that the entropy of a light field behaved as a gas of particles, and he used his quantum theory to explain the photoelectric effect. While the photoelectric effect is the most famous result from this paper, Einstein’s explanation of it does not necessarily require light itself to be quantised — it depends only on the exchanges of energy being quantised, as in Planck’s paper. So, let’s talk about entropy.

The entropy of a system, in essence, describes how many ways that system can be configured without changing its macroscopic state. For example, you could switch the position of two atoms, but the temperature or volume (indeed, any important large-scale quantity) will be unchanged. The more configurations exist for a given state, the more likely it is that the system will be found in that state, so entropy tells you how probable different states are for a given system. For example, given a gas of helium molecules confined in a box, you can determine the probability that all of the molecules are found in the top half of the box at any given time using entropy.

Einstein demonstrated a correspondence between the entropy of light and that of a gas of particles; in particular, their volume-dependence. If an ideal, low-density gas of N particles is confined in a box of volume V0, then the probability, P, that all N particles will be simultaneously found in a sub-volume, V < V0, is P = (V/V0)N. Einstein showed that the entropy of a low-density light field in a box had exactly the same form as that of the ideal gas. Moreover, by comparing the two expressions he deduced that the probability of all the light being found in a sub-volume V < V0, is P = (V/V0)E/(hf), where E is the total energy in the light field, h is Planck’s constant, and f is the frequency of the light (i.e. its colour, determined by its wavelength). The product hf is precisely the size of one of Planck’s quanta of energy (hence why the constant h is given his name), which, if Einstein is right about the light field itself being quantised, would make E/(hf) exactly the number of photons in the box. In other words, light in a box behaves exactly as a collection of discrete particles. Light is quantised.

This result had exceptional significance in the development of quantum physics. As you may have guessed, quantum physics gets its name from the discretisations — the quantisations — which occur as a motif in the theory. One of the core features of quantum mechanics is that quantities (like energy or the amount of light) must often take on discrete values, and photons were the first quanta to be (knowingly) discovered. From this beginn ing, supplied by Einstein and Planck, quantum theory would rewrite virtually everything we thought we knew about physics. Indeed, quantum theory and general relativity together now form the basis for all of physics as we understand it. Quantum theory was accepted only slowly at first, but it would eventually become the most influential idea of the 20th century.

This paper, Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt in its original language, won Einstein the Nobel Prize in 1921. It would prove to be his only Nobel prize, but a strong case can be made that at least one of his other papers in 1905 deserved to win him a second. In the next edition of the UoA Scientific, we will delve into Einstein’s work on the statistical mechanics of atoms and how he finally put to rest the question of their existence — a result we now take for granted.

1. This quote is almost certainly apocryphal, but who wants to let facts get in the way of a good story?

2. You can’t prove I’m wrong.

3. This fact will be extremely significant when we come to discuss Einstein’s third paper.

4. Imagine you have a friend if needed

References

[1] T. Young, “I. The Bakerian Lecture. Experiments and Calculations Relative to Physical Optics,” Philosophical transactions of the Royal Society of London, no. 94, pp. 1–16, 1804

[2] M. Vazquez and A. Hanslmeier, Ultraviolet radiation in the solar system, vol. 331. Springer Science & Business Media, 2005.

[3] M. Planck, “On the Law of Distribution of Energy in the Normal Spectrum,” Annalen der Physik, vol. 4, no. 553, p. 1, 1901.