Why everyone can — and should — learn quantum mechanics

It’s not rocket science — it’s a fundamental part of how our world works, and not as complex as you might fear

Perhaps no other area of physics has enjoyed as much attention from scientists and non-scientists as quantum mechanics. The fame of quantum mechanics theories stands in juxtaposition to the physical “weirdness” they manifest – even some of the scientists who discovered these theories were set aback by the startling consequences. It’s no wonder Einstein remarked, “The more success the quantum theory has, the sillier it looks.” But as “silly” as it may seem, the physical implications of quantum mechanics are real, and not nearly as complicated nor inaccessible as they might seem.

Energy Comes in Chunks

We are all familiar with the way the burner of an electric stove goes from being faint red to flaming bright red as the temperature rises. If we could increase the temperature even higher, we would eventually see the burner shifting from its reddish glow to more of a bluish hue. In essence, what we are observing is a very specific relationship between the temperature of a hot object (e.g., stove burner) and the light (thermal radiation) it gives off: as the temperature increases, the light emitted from the burner shifts to a higher frequency. And although our eyes only see a particular color, it’s actually a range of colors, or a frequency spectrum, that’s emitted. This seemingly mundane physical phenomenon left twentieth-century physicists paralyzed for answers, and it would ultimately provide the very first peak into the bizarre world of quantum mechanics.

In 1900, some six years of work had led Max Planck to the correct mathematical form of the frequency spectrum known as Planck’s Radiation Law. Indeed, it was an amazing accomplishment worthy of a Nobel Prize in and of itself. However, the law provided nothing in the way of actual physical insight. So the questioned remained: What’s it about the interaction of matter and radiation that results in the frequency spectrum? Planck needed to know, and so he pushed forward. What he found would change physics and our understanding of nature forever: matter can only emit or absorb energy in specific “chunks”! In other words, the energy values allowed are discrete rather than a continuous distribution. So, if an atom’s energy goes up or down during its interaction with light, it must do so in specific increments, no more, no less. Let me give you an analogy.

Imagine a big, empty box. Outside the box, there are balls of varying sizes. Now, let the box represent matter, and the balls represent energy. According to classical mechanics, matter can absorb energy in any amount, so we’re free to place balls of any size into the box until it’s completely full. That is, it doesn’t matter what balls I use to fill the box – I just need to fill it up. However, according to quantum mechanics, energy can only be absorbed in specific increments. Therefore, I’m restricted to a specific-sized ball – say a tennis ball – and the box can only be filled with this “energy quantum.”

Reluctant with the physical implications of his theory that involved the enigmatic energy quanta, Planck – and pretty much everyone else – instead focused on its remarkable accuracy. It would be almost eight years after Planck first presented his quantum theory of discrete energy before he would begin to come around to the idea that it represented the true nature of energy. Be that as it may, Planck probably never fully accepted that idea, longing rather for the days of the “old familiar physics” (classical physics).

In his 1905 paper “On a Heuristic Point of View Concerning the Production and Transformation of Light,” Einstein introduced the idea that light at low intensity (or low density) behaves as a particle (photon), rather than as an electromagnetic wave, a theory proposed by James Clerk Maxwell in 1864 and verified by Heinrich Hertz in 1887. The physics community turned its back on Einstein’s photon concept for almost twenty years before ultimately embracing it.

Undeterred, Einstein continued to investigate the true nature of light. In 1909, using a fluctuation theory approach he developed, he found that light simultaneously behaves as both a wave and a particle as it fluctuates in energy and momentum. This work led him to the bold conclusion:

It is therefore my opinion that the next stage in the development of theoretical physics will bring us a theory of light that can be understood as a kind of fusion of the wave and [particle] theories of light.

Once again, Einstein stood alone in his opinion.

Then, in 1923, Louis-Victor-Pierre-Raymond de Broglie came up with the amazing concept that Einstein’s wave–particle duality actually applied to all quantum particles, in particular electrons. He published three papers and wrote his PhD thesis around the theory. In de Broglie’s theory, every quantum particle has a wavelength associated with it. De Broglie imagined the wave as accompanying the particle, guiding or “piloting” its movements. His mathematics was simple, but the physics was far-reaching. The implications of this proposal was so radical, in fact, that had it not been for Einstein’s keen response (Einstein had been sent de Broglie’s thesis for review), de Broglie might not have been awarded his PhD.

In 1925, Einstein was working on his theory of the quantum ideal gas. Once again, as he had done in 1909 for light, he appealed to his fluctuation theory approach and found that with respect to particle fluctuations, a quantum ideal gas behaves as both wave and particle. The wave–particle duality had now come full cycle for Einstein, and it was de Broglie’s work that he cited as providing the physical insight. In 1927, Clinton Davisson and Lester Germer confirmed de Broglie’s theory and the wave–particle duality, when they showed that a beam of electrons directed at a nickel crystal caused the electrons to diffract (like waves). For this work, de Broglie received the Nobel Prize in 1929.

Nature Is Fundamentally Probabilistic

In 1916–1917, Einstein made great strides in understanding the way light interacts with matter. His deep insight would lead him to another conclusion, one he found quite unsettling. Einstein found that when an atom spontaneously gives off a photon — a phenomenon called “spontaneous emission” — the direction in which the photon is emitted and its momentum are determined purely by “chance.” In other words, there is no way to know this information with complete certainty. At the time, Einstein saw this as a flaw in his theory. Later, it would be clear that he had come across the uncertainty inherent in (what would later be called) quantum mechanics.

After Erwin Schrödinger inaugurated quantum mechanics with his famous wave equation in 1926, the biggest challenge plaguing it was the role of the “wavefunction” — the function that is related to the probability of finding a particle at a given position in space (and other physical properties). While mathematically the wavefunction solves the wave equation, it wasn’t clear what it meant physically. Along with Schrödinger, several others – such as Paul Dirac, Eugene Wigner, and Max Born – were also contemplating the physical meaning of the wavefunction. It was Max Born’s work that clearly defined the role of the wavefunction and the concept of quantum probability. He summed it up nicely: “The motion of particles follows probability laws….”

In short, this means that the motion of a quantum particle (electrons, photons, etc.) is not governed by deterministic equations as in the case of a classical particle (or object). Consequently, a quantum particle doesn’t have a well-defined path that it moves along, with well-defined values for key physical properties (such as position, momentum, energy, and the like) at every instant in time. Instead, these physical quantities and many others are solely determined by a quantum probability, which, in turn, is directly related to the wavefunction..

To be sure, the use of probabilities as a tool to make a physical problem more tractable to solve was nothing new. However, quantum probability is a wholly different beast, as it isn’t merely a mathematical approach to gain insight into physical reality. In the quantum world, this probabilistic nature is the physical reality. And it means that the only thing you can know about a quantum particle is the probability of finding it in a certain quantum (micro) state.

For Einstein, quantum probability would put an end to his relationship with quantum mechanics. He had lead the way for almost twenty years, and he had even introduced his own probabilities into it, but now he would be completely unforgiving. In response to a letter from Born, he said:

Quantum mechanics is very impressive. But an inner voice tells me that it is not yet the real thing. The theory produces a good deal but hardly brings us closer to the secret of the Old One. I am at all events convinced that He does not play dice.

Now, according to Born, a quantum particle doesn’t follow a deterministic path; rather, its quantum state is completely controlled by quantum probability. Werner Heisenberg wondered: What would we see if we tried to measure the position and momentum of an electron at a given instance in time?

Heisenberg had developed his version of quantum mechanics, separate from Schrödinger’s wave mechanics, known as matrix mechanics. Using that approach and an ingenious thought experiment, Heisenberg showed that certain pairs of properties (e.g., position and momentum in the same direction) can’t be determined with precision. Specifically, he found that nature had set a lower bound: the product of their uncertainties can’t be smaller than Planck’s constant. (Today, we know that it’s actually Planck’s constant divided by 4p.) In practice, we either know one property almost exactly and in turn know nothing of the other, or we strike a compromise, where we know a little bit about both properties.

Understand this isn’t due to our lack of ability to measure these properties. Rather, it means that for a quantum particle these pairs of properties at a given instance in time exist only in a fuzzy (not-well-defined) manner. Indeed, Born’s quantum probability and Heisenberg’s Uncertainty Principle are two independent blows against causality in quantum mechanics.

Quantum Particles Are Indistinguishable

Ever since its introduction in 1900, Planck’s Radiation Law was known to be less than a full-fledged quantum theory. In essence, it was largely derived using classical mechanics and then ending with Planck’s energy quanta hypothesis; a true quantum theory would be completely free of classical mechanics. With his work in 1916–1917, Einstein had come closer than anyone to a full quantum derivation of Planck’s Radiation Law, but in the end, he had to make assumptions that caused him to fall short too. It would be an unknown physicist who would provide the solution, probing even deeper into the quantum world.

In 1924, Satyendra Nath Bose derived Planck’s Radiation Law in a manner completely free of classical mechanics artifacts. At the heart of Bose’s theory was Einstein’s photon concept and that photons of the same frequency are identical; that is, they’re indistinguishable. The indistinguishability nature means that if we could actually see two photons up close, they would look exactly the same. To put it another way, nature provides no way to tell two photons of the same frequency apart. This may seem a bit obvious, but an illustration could prove otherwise.

Consider two coins (e.g., quarters): “coin 1” and “coin 2.” For the most part, two quarters look pretty much the same. However, a closer look will reveal distinguishing features between them. Perhaps there’s a slight difference in color, or a mark on one but not the other. In the end, there will always be some way to tell them apart. This isn’t true for photons, and this indistinguishability has real physical consequences.

Upon flipping our quarters, we can describe a given outcome of them landing on their heads (H) or tails (T), according to four possible physical states: (H1,H2), (H1,T2), (T1,H2), and (T1,T2). If our quarters were indistinguishable, like photons, there would only be three possible physical states – (H,H), (T,T), and (H,T) – since (H1,T2) and (T1,H2) are now the same. In other words, indistinguishability has changed the statistical outcomes. Bose’s amazing insight was to realize that these statistical outcomes had real physical consequences for photons. With this, Bose provided the first fully quantum derivation of Planck’s Radiation Law, solved the overall mystery of light (that had eluded even Einstein), and became the father of quantum statistics. Surprisingly, he was never awarded the Nobel Prize.

Einstein immediately recognized the implication of Bose’s quantum statistics. Whereas Bose had used his approach for photons, Einstein was ready to extend it to molecules:

If Bose’s derivation of Planck’s radiation formula is taken seriously, then one will not be allowed to ignore [my] theory of the ideal gas; since if it is justified to regard the radiation [light] as a quantum gas, then the analogy between the quantum gas [light] and the molecule gas has to be a complete one.

Einstein wrote three papers on the quantum theory of the monoatomic ideal gas. The first paper, published in 1924, was pivotal for beginning to establish the equivalence between light and atoms. The second paper, which was published in 1925, is the most significant of the three Einstein published on the subject. Here, he confronts the indistinguishability matter head on by detailing the difference between his quantum theory and classical mechanics, acknowledging the very real physical consequences of indistinguishability and providing a compact formula for determining the number of statistical outcomes for indistinguishable particles.

With the work of Bose and Einstein, we became aware of the indistinguishable nature of quantum particles and the resulting physical consequences.

A Quantum System Exists in a Superposition of States

If everything we discussed so far isn’t mind-blowing enough, then let me leave you with one more to ponder. In physics, the physical meaning of the mathematical solutions to key equations is often elusive. The solution to the Schrödinger wave equation is no exception. Taken verbatim, it allows one to conclude that a quantum system exists in more than one physical state – at the same time! As an example of this superposition of states, let’s consider the double-slit experiment.

In the double-slit experiment, we have an “electron gun” that fires electrons at a wall that has two holes (or slits), which are the same size and just big enough to let the electrons pass through. Moreover, the holes are at the same distance from the gun and the same distance from the center of the wall. In other words, with respect to the gun and the holes, everything is symmetrical. Finally, we’re not actually aiming at either hole, but rather we’re firing at random.

As the electrons head towards the holes, some will pass though, and some won’t. Those that do go through will hit another wall located further down, which acts as a backstop. Here, their final position will be recorded by a detector, and the overall information is then processed by a computer. We want to get good statistics, so we fire a lot of electrons at those holes. From the aggregate information of all the electron positions, the computer will reveal a pattern, or distribution, from which we will learn the probability of finding an electron at a given position on the back wall when fired randomly at the two holes. So, what does the distribution look like?

Well, long story short, it doesn’t look like what one expects for particles simply passing through holes. In fact, the distribution formed by all the electron positions shows an interference pattern. This is surprising because it’s waves, not particles, that show interference patterns. Oh, but wait, quantum particles do exhibit wave–particle duality as we discussed earlier. So, what’s the wave that’s causing this interference? The overall motion of the electrons and their final position at the back wall will be governed by the quantum probability.

With that mystery solved, there’s one last question to ask: Would we still get the interference pattern if we fired one single electron at a time at the two holes? Yes, it doesn’t matter if we fire several or just one at a time, we still get the interference pattern. Evidently, a single electron encountering the two holes interferes – in some way – with itself. In other words, the single electron seems to be in a superposition of states, causing it to pass through both holes at the same time.

This is so strange that we decide to place a detector next to each hole so that we can watch which one an electron passes through. Good news: we do observe an electron passing through either one or the other hole; bad news: the interference pattern has now disappeared. In other words, when we’re not looking at the holes, we have the interference pattern, but when we’re looking, the interference pattern (superposition) disappears, and we’re left with the distribution one expects for “true” particles.

If the physical consequences of quantum mechanics are wreaking havoc with your senses, don’t worry, you’re not alone.

# # #

Dr. Scott Bembenek has a PhD in theoretical chemical physics, was a National Research Council Fellow, and works as a computational chemist doing drug discovery research. He is also the author of “The Cosmic Machine: The Science That Runs Our Universe and the Story Behind It,” which tells the inspiring story of how scientific discoveries (and the key players of those discoveries) shaped the world as we know it today. To learn more about Dr. Bembenek and his work, visit http://scottbembenek.com and connect with him on Twitter.