A “no math“ (but seven-part) guide to modern quantum mechanics

 A “no math“ (but seven-part) guide to modern quantum mechanics

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Quantum mechanics is complex, fold-your-brain stuff. But it <em>can</em> be explained.

Enlarge / Quantum mechanics is complex, fold-your-brain stuff. But it can be explained.
Aurich Lawson / Getty Images

Some technical revolutions enter with drama and a bang, others wriggle unnoticed into our everyday experience. And one of the quietest revolutions of our current century has been the entry of quantum mechanics into our everyday technology. It used to be that quantum effects were confined to physics laboratories and delicate experiments. But modern technology increasingly relies on quantum mechanics for its basic operation, and the importance of quantum effects will only grow in the decades to come.

As such, the time has come to explain quantum mechanics—or, at least, its basics.

My goal in this seven(!)-part series is to introduce the strangely beautiful effects of quantum mechanics and explain how they’ve come to influence our everyday world. Each edition will include a guided hike into the quantum mechanical woods where we’ll admire a new—and often surprising—effect. Once back at the visitor’s center, we’ll talk about how that effect is used in technology and where to look for it.

Embarking on a series of quantum mechanics articles can be intimidating. Few things trigger more fear than “a simple introduction to physics.” But to the intrepid and brave, I will make a few promises before we start:

  • No math. While the language of quantum mechanics is written using fairly advanced math, I don’t believe one has to read Japanese before you can appreciate Japanese art. Our journey will focus on the beauty of the quantum world.
  • No philosophy. There has been a fascination with the ‘meaning’ of quantum mechanics, but we’ll leave that discussion for pints down at the pub. Here we will focus on what we see.
  • Everything we encounter will be experimentally verified. While some of the results might be surprising, nothing we encounter will be speculative.

If you choose to follow me through this series of articles, we will see quantum phenomena on galactic scales, watch particles blend and mix, and see how these effects give rise to both our current technology and advances that are on the verge of making it out of the lab.

So put on your mental hiking boots, grab your binoculars, and follow me as we set out to explore the quantum world.

What is quantum mechanics?

My Mom once asked me, “What is quantum mechanics?” This question has had me stumped for a while now. My best answer so far is that quantum mechanics is the study of how small particles move and interact. But that’s an incomplete answer, since quantum effects can be important on galactic scales too. And it is doubly unsatisfactory because many effects like superconductivity are caused by the blending and mixing of multiple particles.

In many ways, the role of quantum mechanics can be understood in analogy with Newtonian gravity and Einstein’s general relativity. Both describe gravity, but general relativity is more correct—it describes how the Universe works in every situation we’ve managed to test. But 99.99 percent of the time, Newtonian gravity and general relativity give the same answer, and Newtonian gravity is much easier to use. So unless we’re near a black hole, or making precision measurements of time with an optical clock, Newtonian gravity is good enough.

Similarly classical mechanics and quantum mechanics both describe motions and interactions. Quantum mechanics is more right, but most of the time classical mechanics is good enough.

What I find fascinating is that “good enough” increasingly isn’t. Much of the technology developed in this century is starting to rely on quantum mechanics—classical mechanics is no longer accurate enough to understand how these inventions work.

So let’s start today’s hike with a deceptively simple question, “How do particles move?”

Kitchen quantum mechanics

Some of the experiments we will see require specialized equipment, but let’s start with an experiment you can do at home. Like a cooking show, I’ll explain how to do it, but you are encouraged to follow along and do the experiment for yourself. (Share your photos in the discussion below. Bonus points for setting the experiment up in your cubicle/place of work/other creative setting.)

To study how particles move, we need a good particle pea shooter to make lots of particles for us to play with. It turns out a laser pointer, in addition to entertaining the cat, is a great source of particles. It makes copious amounts of photons, all moving in nearly the same direction and with nearly the same energy (as indicated by their color).

If we look at the light from a laser pointer, it exits the end of the laser pointer and moves in a straight line until it hits an obstacle and scatters (or hits a mirror and bounces). At this point, it is tempting to guess that we know how particles move: they exit the end of the laser like little ball bearings and move in a straight line until they hit something. But as good observers, let’s make sure.

Let’s challenge the particles with an obstacle course by cutting thin slits in aluminum foil with razor blades. In the aluminum foil I’ve made a couple of different cuts. The first is a single slit, a few millimeters long. For the second I’ve stacked two razor blades together and used them to cut two parallel slits a few tenths of a millimeter apart.

Horizontal slits in aluminum foil made with razor blades. The upper slit is from a single blade, while the lower is from two blades taped together.

Enlarge / Horizontal slits in aluminum foil made with razor blades. The upper slit is from a single blade, while the lower is from two blades taped together.
Miguel Morales

In a darkened room, I setup my laser pointer to shoot across the room and hit a blank wall. As expected I see a spot (provided the cat’s not around). Next, I put the single slit in the aluminum foil in the laser’s path and look at the pattern on the wall. When we send the light through the single slit, we see that the beam dramatically expands in the direction perpendicular to the slit—not along the slit.

Laser light passing through the single horizontal slit is spread vertically

Enlarge / Laser light passing through the single horizontal slit is spread vertically
Miguel Morales

Interesting. But let’s press on.

Now let’s put the closely spaced slits into the laser beam. The light is again spread out, but now there is a stripey pattern.

Laser light passing through the two horizontal slits produces the distinctive stripes of quantum mechanics.

Enlarge / Laser light passing through the two horizontal slits produces the distinctive stripes of quantum mechanics.
Miguel Morales

Congratulations! You’ve just spotted a quantum mechanical effect! (whoo hoo animated emoji) This is the classic double-slit experiment. The stripey pattern is called interference, and is a telltale signature of quantum mechanics. We will see a lot of stripes like these.

Now you have probably seen interference like this before, since water and sound waves show exactly this kind of striping.

Water waves from two sources (one visible in green, the other hidden behind the presenter). The circular waves overlap into regions of extra strength (bright stripes) and regions where the waves cancel each other out (dark bands). The formation of stripes is a signature of wave motion.

Enlarge / Water waves from two sources (one visible in green, the other hidden behind the presenter). The circular waves overlap into regions of extra strength (bright stripes) and regions where the waves cancel each other out (dark bands). The formation of stripes is a signature of wave motion.

In the photo above, each ball creates waves that move out in a circle. But a wave has both a peak and a trough. In some places the peak of the wave from one of the balls always coincides with the trough from the other (and vice versa). In these areas the waves always cancel out and the water is calm. In other locations the peaks of the waves from both balls always arrive together and add up to make a wave that is extra tall. In these locations the troughs also add up to be extra deep.

So does the fact that we are seeing stripes when our laser pointer goes through two slits mean that particles are waves? To answer that question, we’re going to have to look more closely.

The professional kitchen

The double slits served two purposes: each slit spread the beam out (as we saw when we went through just one slit), while the two closely spaced slits provided the particles with a choice as to which slit to pass through. We can use a different experimental setup to more clearly separate the act of spreading the beam and forcing a choice, and this bigger setup will allow us to study in detail what is happening. (This is the moment in the cooking show where they pull out some obscure cooking tool—like a crème brûlée torch—that you’re unlikely to have at home.)

The professional version of the two slits in aluminum foil. Here the light beam is split by a half silvered mirror and sent into left and right paths. These paths are then recombined to show stripes.

Enlarge / The professional version of the two slits in aluminum foil. Here the light beam is split by a half silvered mirror and sent into left and right paths. These paths are then recombined to show stripes.
Miguel Morales

Keeping the laser, let’s use a lens to spread the beam out, and then a half-silvered mirror to give the particles a one-way-or-the-other choice. We then use a couple of good mirrors and another half-silvered mirror to recombine the beams before they reach a wall or screen, as shown above. If one of the mirrors is mis-aligned ever so slightly, we will again see stripes (places where waves add together and cancel out). In the 38 second video below I show the working version.

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What I find fun about this setup is that it is big. I don’t need a microscope to mess with it. For convenience, I made this table top sized, but we could have made it huge. In the LIGO experiment used to detect gravitational waves, the light is sent down arms 4 km long, and with radio light we’ve used the Cassini spacecraft as it approached Saturn as a mirror. Quantum mechanics is not limited to the microscopic world.

But let’s play with our human sized experiment, and see some more quantum mechanical effects.

Slow mo

The first thing I’d like to do is to look very carefully at the stripey pattern. Instead of just looking at the screen, let’s use a sensitive camera so we can watch it as it develops. When the laser is bright, the camera will immediately capture stripes. But if we make the room very dark and turn down the strength of the laser, we will notice that light is a not smoothly distributed, but appears at individual ‘points’—in slow motion we can see the arrival of individual photons. They look like small red paintballs hitting the camera sensor.

Since this is a video camera, we have a record of where the photons hit. If we play the video, it looks like we are watching someone make a pointillism painting. Individual spots are appear on the screen without an obvious pattern. If we keep where they landed lit up by summing the video frames, the spots start to fill in the stripey pattern we saw when we had the laser turned all the way up.

The stripy pattern is there, even if we only generate it one photon at a time.

Which way did he go?

In both the double slit experiment and the bigger lab setup, the light can take two different paths: the left slit/path or the right slit/path. But what we see is the result of individual particles hitting the screen. So which path do the individual particles take?

If I block either of the paths, the stripes go away. The same thing happens with the aluminum foil if you block one of the slits with an index card (though this requires a very steady hand). We only see stripes if the light can travel both paths and interact before hitting the screen.

This gets even stranger if we turn down the intensity of the light. We can turn the laser down so low that only one particle of light is traveling through the experiment at a time. No matter how low we turn the light, the stripes only appear when each individual photon can take both paths. (Even if we limit it to one particle at a time as in the video above, the particles will slowly add up to the pattern seen above.)

Even though a single photon hits the screen like a little paintball, where it hits the screen (in the bright stripes) shows that that every photon took both paths.

If your head isn’t hurting yet, you’re not paying attention. But, it is important to register all of the markings on a strange bird before looking in the guidebook, so let’s look a little more closely before we explain what is happening.


What if we use a different color laser pointer? Replacing the red laser pointer with a green one, we see the same set of stripes but now the stripes are closer together. The stripes are even closer together for a blue laser pointer. As the wavelength of the light gets shorter, the spacing of the stripes gets narrower.

The stripes from sending red, orange, and green lasers through the same pair of slits. Notice how the green dots are closer together.

Enlarge / The stripes from sending red, orange, and green lasers through the same pair of slits. Notice how the green dots are closer together.

We can repeat these experiments, looking at the arrival of the photons in slow mo and blocking different paths, and everything works the same way with the exception of the stripes being closer together.

But looking at the slow mo video of the particles hitting the camera begs the question of how hard are the photons hitting? Are there hard hits and soft hits? The standard camera sensor we’ve been using can’t measure how hard the photons hit. But there are (expensive) detectors that can measure how much each pixel heats up when it is hit by a photon. The heat deposited by the photon directly tells us how hard the photon hit—how much energy it had.

If we put one of these fancy detectors in our experiment, we will see that all of the red photons have the same energy. All green photons have more energy than the red photons, but the same energy as all the other green photons. Blue photons are even more energetic. The color directly tells us how hard a photon will hit the detector.

So we have a pair of observations associated with color. As the color becomes more blue, the spacing of the stripes becomes closer and the energy of each impact with the screen increases. Energy is closely associated with the spacing of the stripey patterns.

So what is happening?

The stripes we are seeing are a hallmark of waves. Sound waves and water waves both exhibit this kind of striping. If you have ever heard loud and quiet places at a concert or while listening to the stereo and walking around the room, you’ve heard the stripes of wave interference. You can recreate this experience by playing a single note from a synthesizer through a pair of stereo speakers. The two speakers act like the two slits in the aluminum foil; by moving your head left to right you can hear the loud (bright) and soft (dim) parts of the stripes (try a note a couple of octaves above middle C). We saw the same wave effect in water waves earlier.

As the pitch of the sound gets higher—or the water waves become shorter in length—the distance between the bright stripes will get closer, just like when we changed the laser color. And if we block the waves in one of the paths (by pulling the speaker cord on one of the speakers or stopping one of the sources of waves in water) the stripes will go away. All the experiments we did that tested how the particles moved from the laser pointer to the screen show that the particles move like waves.

However, we also saw the light hitting the screen like little paintballs. They hit at a spot and deposit a set amount of energy depending on their color. All of the experiments about how the particles hit the camera show that they behave like little ball bearings.

This is the fundamental mystery of quantum mechanics: particles move like waves and hit like particles.

The immediate question is whether this is something special associated with light particles, or whether it is true of all particles. The short answer is it is always true. It is true for any kind of particle, under any circumstance, all the time. Photons, electrons—even molecules. Particles always move like waves and hit like particles.

Why physicists are rarely invited to parties

Clearly this statement—that all particles move like waves and hit like particles—requires intensive testing. My favorite example involves neutrons; we can repeat the experiment we did with laser light to see if neutrons behave the same way. To do this, we need two things: a good neutron pea shooter, and some neutron mirrors.

We are all very lucky that neutron laser pointers don’t exist. But there is a very good source of neutrons: your friendly neighborhood nuclear power plant. Nuclear reactors create copious amounts of neutrons, and the plant operators already pipe some of the neutrons out of the reactor area because they need to measure the neutrons to keep track of the reactor. If you bat your eyelashes and get security clearance, they will let you pipe off some of the neutrons too.

You then need to throw away both the low energy and high energy neutrons to make a stream of neutrons of the same energy (like photons of the same color in a laser). Once you’ve done that, you have a great neutron pea shooter that produces a lot of neutrons of nearly the same color (it is probably for the best that it is shackled to an immovable nuclear reactor).

Now we need some neutron mirrors. One can use crystals to reflect neutrons, but because the wavelength of the neutrons is very small, alignment of the mirrors becomes a critical issue. This is where the semi-conductor industry comes to our rescue. Computer chips are built on silicon wafers that are cut from large, perfect crystals.

Instead of slicing these crystals into wafers, we can cut away half of the crystal to leave three ‘fins’ that act like the mirrors of an interferometer and are held in perfect alignment by the remaining crystal.

When we send our single-energy neutrons through the crystal mirrors, we see exactly the same pattern of interference that we saw with light. We can repeat all of the experiments we did with light and we get exactly the same results. Even when we send one neutron at a time through the crystal, we see the wave-like stripes. With the right hardware, we can extend this beyond photons and neutrons to test many different kinds of particles. And when we do, we find that all particles move like waves and hit like particles.

Neutrons are a particularly interesting particle for several reasons. First, they’re heavy and slow. This makes them more sensitive to gravity. We can rotate the crystal so instead of left and right paths we have top and bottom paths as shown below. As a neutron climbs uphill, it slows down (becomes more red), and as it descends it speeds up (becomes more yellow). So the neutron wave on the upper path will travel more slowly than the neutron wave on the bottom path.

We still see the stripes when the waves recombine, but because the neutron wave on the upper arm was slow and arrived a little late the stripes are shifted side-to-side. The amount of this shift is one way to measure the strength of gravity.

(While very precise, there are currently better ways of measuring gravity than with neutrons. However, many of them use exactly the same kind of experimental setup, just with the neutrons replaced with Rubidium atoms and mirrors made out of light.)

Neutrons are also interesting because they are composite particles—a neutron is made up of three quarks. Even though it is made up of sub-particles, it still moves like a wave. Modern experiments have taken this much farther and regularly send Cesium atoms (more than 180 protons+neutrons+electrons), and even large molecules like Bucky Balls (60 atoms) and phthalocyanine with thousands constituent particles through similar interferometer setups. Even these huge composite particles move like waves and produce the telltale stripes in an interferometer.

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While getting a narrow energy range becomes very hard as the mass increases, these large composite objects behave exactly the same way as the photons from our laser pointer. A single Bucky Ball will move like a wave and take all of the paths through the experiment, then hit like a particle. The telltale stripes of wave motion, and the energy-dependent stripe spacing, show that even larger objects move like a waves until they reach the detector, where they hit like particles.

Back at the Visitor’s Center

Congratulations on surviving your first guided walk through the spooky woods of quantum mechanics! So what did we see on our first excursion? We observed that when we send one particle at a time through an experiment, it will take both paths just like a wave, but when we detect it, it will hit like a particle. How hard it hits, its energy, is related to the wavelength of the wave. As the energy goes up (shorter wavelength), the stripes become more closely spaced. This behavior is consistent for all kinds of particles, from photons of light to electrons to composite particles like neutrons, atoms, or molecules. It is a fundamental feature of the way nature works.

Now that we’re back at the visitor’s center, let’s talk about how these observations apply to our technological world. The wave nature of particles appears everywhere. The iridescence of hummingbird feathers and soap bubbles, the anti-reflection coatings on camera lenses, and the optics of electron microscopes all rely on the wave-like quantum motion of particles.

But a good technological example is the optical gyroscope. Gyroscopic guidance systems are used in airplanes, satellites, and rockets. They originally used physical spinning hardware to keep track of orientation, but they have nearly all been replaced by optical gyroscopes because they are cheaper, more sensitive, and more reliable.

In the neutron interferometer, we noticed that the stripes shifted when the wave that went through the upper arm arrived at the final mirror a little late—in that case because gravity slowed down the neutrons in the upper arm. We can get a similar effect using light if the interferometer rotates while the photon is traversing the experiment.

Because the final mirror has moved during the experiment, the clockwise arm is slightly longer than the counter-clockwise arm, so the clockwise wave arrives a little late and the interference pattern shifts. The faster the experiment rotates, the farther the stripes shift. Measuring how far the stripes shift directly tells us the rotation speed. Changes in orientation in whatever the device is attached to—be it a car, a plane, or a satellite—will rotate the interferometer, and add an additional shift to the stripes.

To make the gyroscope more robust, we can make all the lasers, mirrors, and paths out of fiber optics. And to make it more sensitive, we can have the light travel in many clockwise and counter-clockwise loops before recombining. Fiber gyroscopes work because a photon of light moves like a wave and will take both the clockwise and counter-clockwise paths and produce stripes when recombined. Fiber gyroscopes rely on quantum mechanics to work.

The coming weeks

One of my goals in these articles is to show some of the wonders hiding in the quantum mechanical woods. In this article we stayed pretty close to the nature path, but in the coming weeks I’d like to go deeper into the woods and show you things that are usually reserved for physics graduate students. I think we can do this without the math that is usually used, and really see some of the beauty that arises when you look closely at the natural world.

In next week’s article, we will expand on the idea of particle waves and look at particle mixing. This will lead us to understanding how continuous wave LIDAR works, and we’ll see one of the great inventions that’s only recently made its way out of the lab—the optical comb. In the subsequent weeks, we’ll look at particle introverts and extroverts, interference on extra-galactic scales, artificial atoms, quantum cryptography, and more. So come back next week for another hike into the quantum mechanical woods.


But which path did the particle really take? The experiments show that the particles really take both paths. Despite much confusion (even among some physicists), this is the answer. But the question is based on a faulty mental image. The question assumes that a particle is really a little ball bearing, and thus must have chosen one path or the other. But this mental image is wrong. Particles really behave like waves when in motion. Asking which path a tsunami wave took when traveling between Hawaii and California really makes no sense—it is spread out. Similarly, asking which path the particle really takes makes no sense; it moves like a wave so it naturally takes all of the available paths.

But isn’t the stripy pattern we see with light a classical effect? Yes and no. In quantum mechanics, there is nothing special about a photon of light vs. any other kind of particle—they all move like waves and hit like particles and they will all make interference stripes. What is different is the history of our understanding. Before the invention of quantum mechanics there was a wave-theory of light—Maxwell’s electrodynamics.

But at the time electrons and protons were understood as little ball bearings, and no wave-like classical theory for electrons and protons was ever developed (all other particles were discovered after the invention of quantum mechanics). Our lack of a wave-like classical theory for electrons and protons is largely a fluke of history. So while quantum mechanics sees all particles as behaving in the same way, due to history, there is a simplified classical theory for photons but no simplified theory for the other particles. To me, the stripes clearly show the wavelike motion of particles, and this is one of the hallmarks of quantum mechanics, so it is a quantum mechanical effect.

Miguel F. Morales is a Professor of Physics at the University of Washington in Seattle.