Teaching Notes on A Curious Observer's Guide to Quantum Mechanics

Because this series covers a wide range of quantum phenomena and modern experimental examples, these articles can be used as a starting point for teaching and discussing Quantum Mechanics to a wide range of students. Despite the disguise of non-technical language, these articles grew out of a graduate physics class I taught for a number of years. I’ve also used these articles as the backbone of a ‘physics for poets’ class, and I think they’d be fun as part of an undergrad physics major reading club.

In these teaching notes I provide sketches for how the articles can be expanded for students at different levels and links to some of the underlying papers. I’ll start by discussing a few of the themes that run through all of the articles, then provide notes for each article in turn.

Themes

For those of you who have studied and taught Quantum Mechanics, there are a number of non-traditional choices made in this article series. For better or worse, these decisions are very purposeful. The three principle themes I had are:

1. Quantum mechanics applies to all particles, all the time. Students are often led to believe that QM only applies to fundamental particles at atomic scales. Light (photons) is often implicitly assumed to be classical and composite particles being quantum mechanical is sidestepped. I try and attack this with examples very early on. Even in the first article we see a neutron interferometer, and throughout the series I try and incorporate neat examples of QM behavior of composite particles and large scales. I also never use the phrase “classical limit.” This phrase is poorly defined intellectually, and misused more of often than not. I want to stress that QM is fundamental to how the world works, and if we look closely we see its consequences all around us.

2. I don’t use the phrase “wave-particle duality” and instead use my phrase “particles move like waves and hit like particles.” These two phrases say the same thing, but with slightly different emphasis. The traditional wave-particle duality highlights the mathematical step of interpreting the wave function as a probability and can be useful for teaching young physicists first confronting wave functions. However, concentrating on the duality can block the formation of a workable mental model and in popular discussion too often leads to tiresome philosophical discussions of what QM “means.” I have found, even with graduate students, that the phrase “moves like a wave and hits like a particle” helps students get to a workable mental model. It emphasizes treating everything as a wave function until an interaction occurs, with a visualizable (and accurate) model. As I wanted to quickly get to more advanced ideas like particle bunching (HBT) and energy indeterminism (wave packets); I have chosen a phrase that allows students to move forward more easily.

3. With modern technology quantum mechanics is important and accessible. This is my motivation for writing the article series. I’ve gotten bored of rehashing examples from more than a hundred years ago (Bohr atom, photoelectric effect, Planck spectrum), when there are such beautiful modern examples to use. I’ve tried to work as many of these examples into the articles as I can, and I’ll provide links to some of the underlying experimental papers below.

1: Curious Observer's Guide to Quantum Mechanics

An introduction to quantum mechanics, and in many ways this first article is very traditional. We start with Young’s double slit experiment, and I’ve found it very helpful to have students actually make the two slits themselves—it gives them a sense of power and involvement.

I then introduce the Mach-Zender interferometer, and in classes I’ve actually built one of these. The advantage over the double-slit is that the particle paths are macroscopically distinct (part of my theme of QM applying at all scales), and it is much less intellectually subtle than a Michelson interferometer (though harder to align). This is also setting up experimental variants I’ll use in articles 2 & 3, so reduces future cognitive load.

I try and move as quickly as I can to observing individual photons and wave-particle duality (without using those words).

Lastly, we then talk about color and I try and tightly link wavelength (via stripe spacing) and the energy of the particles. This is the photoelectric effect, but with wave nature, particle nature, wavelength, and energy as closely coupled as I can. By having these all in one experiment (with implicit MKIDs detectors), it helps drive home that these attributes are interlinked. And by using this set of observables I’ve pushed the uncertainty principle into its own article (#3).

I then finish with the neutron interferometer to hammer home that these effects are common to all particles. This allows me to introduce wave motion of composite particles very early, and the slowing of the particles due to gravity (gravitational potential energy) to reinforce that even massive particles have wave-like motion.

Additional resources:

A history of neutron interferometry https://doi.org/10.1016/j.physb.2006.05.200, and a couple of my favorite papers https://doi.org/10.1103/PhysRevLett.34.1472 and https://doi.org/10.1103/PhysRevLett.42.1103. Technical, of course, but can be good for background or discussions with advanced students.

Notes:

I use ‘stripes’ instead of ‘fringes’ throughout, as fringe is actually rather specialized jargon (think of a fringed 70’s jacket or the fringe of a crowd). Stripes brings a clearer mental image for students, with no loss of accuracy.

2: The Particle Melting Pot

I’m leaning on two separate ideas in this article. One is that it does not matter if a particle is interfering with itself or with another particle (if that question even makes sense when not in a number state). The other is the idea of temporal interference—the first step to a wave packet.

I have found that students often struggle with mental models that implicitly differentiate between one particle taking both paths and one particle interfering with another particle, when both occur and work fine. So I wanted to discuss this head on and early.

The examples using multiple lasers lead to interference of different energies and temporal interference patterns. This then paves the way for talking about wave packets in the next article.

Discussion ideas:

The optical comb is a beautifully subtle example, and lots of discussion questions and assignments can be generated from it.

First, I use the fact that it does not matter how a quantum state is made. While the white pulses could have been made with a bunch of mode-locked lasers, the fact that we made the pulses with a pulsed laser is immaterial. The behavior after the prism will be the same as long as the input state is the same.

Second, the fact that photons after the prism are random in time just messes with peoples’ minds. A discussion variant is to imagine attenuating the pulsed beam into the single photon limit, then splitting it into two paths with a half-silvered mirror (or using a switchable mirror). The first path is sent to a diode to time when photons appear, the second path is sent through a prism with diode(s) on individual color beams. The first path diode sees the photons in time with the pulses, but their color is indeterminate. The second path diodes see photons randomly in time, but their color is known.

One can go farther and actually do the QM, either analytically or numerically. Start with a single pulse with a central color. To make this take a color (sine wave) and envelope it with a very short function (Gaussian works well). Take the Fourier Transform of this pulse and show that you then have a wide contiguous spread of colors (energies) centered around the original color—the pulse is white. Now create a series of pulses by convolving your original temporal pulse with a comb of delta-functions in time. This makes a time series of identical pulses. Take the Fourier Transform of this to show that the output is now a series of delta-function colors enveloped by the initial pulse color envelope. Changing the various parameters shows how the optical comb is sensitive to various effects, and is great review of Fourier multiplication and convolution. This allows all sorts of discussion. Also note that a single pulse is white (contiguous colors), but a train of pulses has discrete colors. Imagine enveloping the train of pulses (say let 10 pulses through a shutter), what does the color look like? 100 pulses? What does this mean for how narrow the colors in an optical comb are, relative to when the laser was turned on (in practice this is limited by the atomic clock precision, but in principle…)?

We often talk of a wave packet as a small contiguous envelope in time and the corresponding lump of contiguous colors, and this leads directly to the uncertainty principle. But there is no need for the envelope or color distributions to be contiguous, and the optical comb drives this home because it is disjoint in both color and time! The more colors in frequency (holding spacing constant) the narrower the individual pulses in time become, and vice versa. So the energy-time uncertainty principle still holds.

3: Rose Colored Glasses

In this article I directly dive into wave packets, but I motivate it from an experimental perspective. Coherence length is something that I’ve found even graduate students struggle with—it is so tempting to fall back on an internal variable model where each photon has a pre-determined color, we just don’t know what it is. But we know that internal variable models are wrong, and by linking it to particle bunching I try and highlight the necessity of energy indeterminacy.

While I have not had the courage to dive into Bell inequalities for this audience, we can get close with particle bunching. Particle bunching (or anti-bunching) really doesn’t make sense unless we have wave packets, and wave packets inherently have energy indeterminacy. Internal variable models don’t cleanly explain particle bunching.

I’ve found that discussing this with students is incredibly fruitful. At a university it is fairly straightforward to set up a white light interferometer where you can show the fringes disappear, and show that different filters and/or light sources have very different coherence lengths. And explicitly setting up a photon bunching experiment should be within the reach of most advanced labs. The key is to stress that internal variable explanations are not tenable.

For physics majors, the Heisenberg uncertainty principle and its link to commutation operators is deep and beautiful. But it can also be fun to look at this as just a natural experimental result of particles moving as waves—an enveloped wave naturally has a range of energies (colors) due to the math of Fourier Transforms.

I again try and stress that this applies to all kinds of particles, and I really do love the Jeltes et al. result. I think it should be in every intro quantum textbook.

Bandwidth is normally explained just using classical waves. Which is fine, but we know there are particles and they become important as the signal strength becomes small. To me the traditional explanation is limited to the high-intensity regime, and I like to emphasize the wave-packet roots. Further, the bandwidth limit applies equally well to a beam of electrons where there is no classical wave analog.

Additional resources:

The Jeltes et al. paper can be found here https://doi.org/10.1038/nature05513 , with more links from this group at http://www.nat.vu.nl/~wim/Cold_Atoms/HBT.html

4: Looking at the Stars

This is the most conceptually challenging of the articles. There are really two separate take home messages: a) wave functions are big, b) there is a spatial pattern to the particle bunching (HBT).

As an interferometrist I really wanted to discuss how large wave functions get. I have found that even physicists implicitly assume that a wave function can’t span an interferometer that is tens to thousands of km across. Not only can it, but it must for modern telescopes and interferometers to work.

There is a subtle issue of when you are in the single/few photon regime. At radio frequencies the Cosmic Microwave Background and other sources mean that you are always in the high intensity limit (if ~10^-26 Watts/m² is ‘high intensity’). But as our alignment and timing improves we can increasingly move to higher frequency where there is less background emission and start to work in the single photon regime. The transition to the single photon regime has important technical consequences on whether the noise is lower if you sample the electric field and interfere digitally, or interfere the electric fields and then detect the power. While quantum mechanics smoothly converges to the classical result for strong radio signals, it really is a quantum system at heart regardless of frequency.

And the fact that HBT works for pions is just fantastic. Not only is it such a cool example, your nuclear physics colleagues will love you if you introduce HBT with pions.

Discussion questions:

There are some very subtle discussion topics you can dive into with more advanced students:

How big is the wave function of a photon emitted from a single atom/interaction on a star? Well, radiation is dipolar so you can get the correct answer by looking at the radiative term leaving the atom, and the answer is that it covers most of the sphere. What we see from a star is the overlap of huge numbers of dipole radiation wave functions. The ‘single photon limit’ just means that the amplitude of all the wave functions added together is very small when it reaches us.

This quickly leads to particle indistinguishability. The question of where a photon emitted by a particular atom on the star landed is not a sensible question. When the wave functions from many emitted photons overlap the particle identities become irrevocably scrambled. This is of course true of all many particle systems, and is the basis of much of statistical mechanics. While I find students are okay with indistinguishable particles in a box, they are more troubled by indistinguishable photons emanating from a distant star.

Even if the individual photons leaving the stellar atoms had narrow energies (and thus a long wave packet), the scrambling of photon identities means that the coherence length of the wave function when it reaches us is very short—the photons are white when they arrive.

5: Catching a Wave

Because atomic spectra played such a foundational role in the early development of quantum mechanics it is often taught first. However, while historically important the conceptual connection between discrete spectral lines and the difference in electron wave function energies is quite subtle—even before discussing selection rules.

I have found that it works better to discuss atomic spectra after students have a firm grounding in the the wave-like motion of particles. Particles always move like waves, even when the wave is trapped, and it is the shape of the trap that leads to the discrete energy levels.

When I approach it this way students don’t find the discreteness of electron energies in an atom surprising—it is a natural consequence of the wave-like motion demonstrated by all particles. Further, they find quantum dots straightforward. The first time I used this approach to teach electron harmonics to my physics for poets class, I had this huge lead up to quantum dots expecting students to say “oh wow!” Instead they saw the discrete spectra of quantum dots as totally natural—electrons move like waves, of course the emission is discrete when trapped, why was I making such a big deal about it? With a strong wave conceptual foundation they didn’t have trouble with this idea.

Pictorially, you will notice that all of my wave functions show the full motion of the trapped wave harmonic (transparent) with an extremum drawn as a heavy line. I find this is important, as it signals to students that this is a snapshot of a rapidly wiggling wave—it implicitly reminds them of the dynamics of the wave function.

The linked article on quantum dots in televisions is a very nice way of talking about quantum mechanics in our technological lives.

It is very easy to extend this article into the full atomic wave functions (3D), and really go into depth with atomic spectra and the shape of the periodic table. These ideas were just a little too advanced for a popular article series, but is a rich vein for college students.

For physics students this is an interesting place to discuss the complex nature of wave functions. Looking at the oscillatory nature of the electron wave functions it is natural to assume that when the (real) wave function moves through zero everywhere that the particle does not exist (or sometimes students will think of it sloshing). But because it is a complex wave and we are squaring the wave function, while there is oscillation the probability distribution of where an electron (in a single eigenstate) will be found is constant in time. One way of visualizing this is to think of the wave as like a jump rope viewed from the side with the real axis vertically and the imaginary axis into the page. A jump rope oscillates and looks like a guitar string when viewed from the side, but the amplitude of the jump rope wave is constant in time. This complex nature of quantum wave function is important, and thinking of wave functions as rotating into and out of the page can keep students from making conceptual errors.

This mental model is a little tricky when looking back at the free particle wave packets of the previous article where there is a wave within a smooth envelope (the envelope does not follow the oscillations of the wave). A mental picture of the wave as a spiral—like a wire wrapped around a glass bottle and viewed from the side—may be helpful for students. For massive particles you still get oscillations within the wave packet (the bottle rotates) but for photons the wave and envelope propagate together so the wave is ‘frozen’ within the envelope.

The complex nature of wave functions is necessary to simultaneously describe the oscillations and the envelope and the associated dynamics.

6: Quantum Spooks

This was the most difficult of the articles to get right, and I tried a number of different approaches. In the end I decided to concentrate on three intertwined ideas:

• Deterministic measurements. For a single particle this just means that repeated measurement will give the same result (vertically polarized photons will pass a vertical polarizer 100% of the time). But I found this phrasing a useful extension for (fully) entangled particles where there is a deterministic relationship between particles.

• Mutually random measurements. This is really talking about basis sets for describing a quantum state. If deterministic measurements are about choosing within one basis set (above); mutually random measurements are related to the existence of multiple orthogonal basis sets. Without using mathematical language, this is saying you can sort your measurements into sets that are internally deterministic (repeatable) and mutually random (orthogonal). That this is a general feature of quantum mechanics is something I find useful to point out even for undergraduate physics majors.

• Entanglement. Just the fact that multiple particles can be in a shared quantum state.

While entanglement is very strange, I find it important to stress that not only is it experimentally verified, it is required. Entanglement is a by-product of no internal variables; and thus is closely related to particle bunching, the HBT effect, and particle indistinguishability. These ideas all come as a set, and without them our world would be very different.

For advanced students this is a jumping off point for talking about basis sets and entanglement and Bell’s inequality. Lots and lots of fun.

For non-physicists, I find careful demos with polarizers works well. I’ve actually made demonstration-size versions of the blue and green framed glasses with polarizing sheets. I never talk about partial rotations, as that is conceptually quite subtle. Instead I keep my sheets in the two orthogonal basis sets (blue +; green x) and run through the examples in this article.

7: The Quantum Century

Quantum technology is coming. This last article does not really present any new quantum phenomena, but instead concentrates on what it will mean to be technologically literate when our world is infused with quantum technology.

I think there are three natural directions one could head in discussions.

• Living in a quantum world. It is a little hard to predict what our world will be like in 10 or 20 years, but it can be fun to think about. What are the kinds of technologies that will be commonplace? What kind of education will students need to live in such a world? How long will it be until departments of quantum engineering are commonplace, and kids perform quantum experiments in middle school?

• What did I miss? What are some of the best experiments and/or technologies that should have been discussed in this article series and were not? Part of what determined my list was trying to come up with clear and accurate qualitative descriptions. How does one accurately describe some of these examples? I personally find many of the condensed matter examples difficult to translate to a general audience. If people mail me examples that should be included I’d be happy to keep a list and post them.

• A launchpad for more advanced topics. Particularly for physics students, this can be a jumping off point to talk about all sorts of advanced topics. Superconductivity; types of quantum computers; comparison of Schrödinger, Dirac, and Feynman integral approaches to QM; physical chemistry; quantum mechanics in biological systems (chlorophyl) and medicine (MEG); benefits and challenges to society (superconductivity and green energy, cryptography and security); to name just a few.