The birth of the old quantum theory

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Let’s talk about quantum mechanics.

We can only put off learning about quantum mechanics for so long and this is the place to start. Whatever we may want out of a theory of everything, it must be quantum mechanical in some limit. The universe is quantum mechanical, and as much as some would like it to be otherwise, you cannot argue with nature and nature is quantum mechanical. The question today is, how did we discover this remarkable fact?

There is an old saying, “success has many parents, but failure is an orphan.” There is nothing wrong with being an orphan, but the idea behind this expression is that when something works, many people will try and take credit for its success and when something fails, no one is lining up to say they were part of the effort.

Quantum mechanics, like success, has many proverbial parents. The theory was a long time coming and many, many people added pieces to our current understanding both big and small. If you want to learn as many of the details as possible by one author, I would recommend The Historical Development of Quantum Mechanics by Jagdish Mehra which was published in six volumes.

The curious case of Planck and the light bulb

In September 1882, the Friedrichstrasse in Berlin was first lit at night with artificial arc lights in a show of German engineering might. As with many projects, after the initial show was over, it was time to do the proper German thing and make it work better and more efficiently. This led the German Bureau of Standards to ask several physics and engineering professors around the country about how to make light bulbs more efficient. In a sentence, “how would you make a light bulb that gives the maximum amount of light while using the least amount of electrical power?”

We need to put ourselves in the shoes of scientists and engineers of the day and realize that very little was known quantitatively in electrical engineering, especially when it came to materials and best practices. Electricity was still firmly in the “tinkering around” stage of development without any real theory. For instance, the electron would not be discovered until 1897. Current in the electrical sense was then thought to be a positively charged (!) fluid that moved inside some materials but not others, an odd convention that we still use today.

Anyone can understand the light bulb problem and it is very practical. So how did this lead to quantum mechanics, a notoriously abstract subject? We can thank Max Planck, who at the time was starting to climb the German academic ladder and was regarded as an expert in thermodynamic questions.

Planck wanted to define the problem as best he could using mathematics and set out to predict how much light a hot filament should be giving off. This would eventually allow him to determine the absolute efficiency of whatever solution was provided. Planck was a brilliant physicist and he knew everything there was to know about light that could be learned from Maxwell’s equations, that is, that light was an electromagnetic wave and that different colors of light were carried by different frequencies and that the wavelength (color) and frequency of light were connected by a wave equation that involved the speed of light as a fixed quantity.

It was also known that there was more to light than just what we could see (to crack an old joke). Light was also infrared, ultraviolet, x-rays, etc. The entire spectrum was not yet known or characterized, but as any thermodynamic expert will tell you, everything with a temperature emits light. The goal became to calculate the optimal current that would produce the optimal temperature in the optimal filament that would produce the most amount of light in the visible part of the spectrum.

Seemed simple enough.

And that was the problem. Whenever Planck tried to calculate how much light, or in his calculation how much energy, should be produced by a light bulb he kept getting an infinite amount of energy at small wavelengths (large frequencies), in the region towards and above the ultraviolet. Whenever infinities come up in physics, something strange is happening and this was no exception.

In short, there was something wrong with the currently accepted theory, known as the Rayleigh-Jeans law. This led to what was known as the “ultraviolet catastrophe,” and it’s resolution directly involved the discovery of quantum mechanics. In what should be an excellent lesson for scientists of our own day, Planck decided he needed to leave the theory behind and see what the experimental data could tell him.

At the time, it was possible to measure the intensity of light that was produced by hot objects, even if they were not emitting light in the visible range. We knew the spectrum of light that was coming from the bulb (or any other object that was all at one temperature for that matter) and it wasn’t infinite in the ultraviolet. When compared to the expected curve from the Rayleigh-Jeans law, the law was approximately correct at large wavelengths (small frequencies) which translates to low energies, but something was going wrong at short wavelengths.

Planck had a very practical, often used idea for dealing with situations like this in thermodynamics. He “artificially” introduced the idea that light came in tiny little packets, packets that would eventually be called quanta. The idea was to introduce quanta as a natural cutoff in energy, complete the calculation successfully, and at the end take the limit as the size of these packets becomes infinitesimally small. This is not so different from how integral calculus is defined.

And it worked! Well… almost. Planck was able to recreate and now completely predict the measured curve of light intensity with only one input, the temperature of the object and he could solve the problem at hand about the light bulb. However, no matter how hard he tried, he could not get rid of the quanta.

This is an often misunderstood concept. Does this mean that we can only have certain energies for little bits of light, what we call photons? No, energy is still continuous with frequency. This is why in the history of physics, this discovery is known as “old quantum theory” because the discrete energy levels that are now typically associated with a quantum mechanical description of atoms would come much later. Free photons can have any energy.

What Planck discovered was that every time you asked for a photon of a certain frequency (energy), you got one such photon. Not 1.7 or 3.2 such photons. So, if energy was to be carried away by photons from say, a light bulb, they could only be carried by whole numbers of photons. Thus, if there was a certain amount of energy emitted from a light bulb, only photons less energetic than the total energy content can be emitted at all, because they must be emitted in whole numbers. As photons escaped, less energy was left over, and only lower and lower energy photons could be emitted at all. No more ultraviolet catastrophe.

There is a common analogy often given having to do with money. For the analogy to work, we have to imagine that you are in a swap market where no one can give you any change (not too much of a stretch) and that you are the kind of person who will strictly not overpay for a purchase. If you don’t have the exact amount of cash on hand, you will not make a purchase and no one will cut you a deal and drop their price.

Although currency is different around the world, on small scales, it is not infinitely divisible. In the US, the smallest coin in circulation is a penny (one cent piece). It would not make sense to price a single item less than a penny. Although unit costs could be a fraction of a penny, you can only spend in units of pennies. On the other hand, if you wanted to buy something for twenty dollars, a twenty dollar bill would work. Or two tens. Or 2000 pennies. Or a nickel and 1995 pennies. You get the idea. But, if you only had a fifty, you could only buy something for sale for fifty dollars.

If the light bulb has a certain “energy budget”, it cannot send out photons in any old combination of frequencies because it is making an exact purchase. An individual photon can have any particular energy, but when an object is emitting light, it does so such that the total energy is an integer count of the available photons. This cuts off the number of high energy photons that can be emitted (or they would blow the budget) and solved the ultraviolet catastrophe.

So, what was the answer? A light bulb needs to heat it’s filament to about 3200K for optimal emission of visible light. The specifics of the material can relate how much resistance it has, therefore how much energy it will absorb from the current passing through it, which can be related to its temperature. As you can see, this particular answer is only part of an engineering solution. There is still a great deal of materials research needed to create a cheap, long lasting bulb and it needs to be connected to a uniform and highly available electric grid. Of course, modern LED light bulbs work very differently, but still depend on quantum mechanics in an entirely different way.

The discovery that photons come issued in quanta would eventually lead Einstein to solve another great mystery of the day, the photoelectric effect. The door was open for quantum mechanics.

Dr. Bryan J Field is a theoretical high-energy physicist and Associate Professor. “A Theory of Everything” (MIT Press, Spring 2023). https://www.bryanjfield.com