Happy 100th Birthday, Quantum Mechanics!

It is hard for me to believe that quantum mechanics is being celebrated as a century old this year! We physicists still tend to call it a branch of “modern physics,” but I wonder how long that name will survive? Last June, the United Nations acknowledged this benchmark and officially declared 2025 as the International Year of Quantum Science and Technology. The Physics Teacher joined the festivities by issuing a call for papers on “Quantum Science and Technology in the Introductory Physics Classroom” with an eye toward publishing a special collection on the topic. We received a good number of manuscripts in response, and are looking forward to sharing them with you, starting with about 10 articles in this issue of TPT. I am grateful to our TPT Guest Editors for the Quantum Collection for helping to shepherd this wonderful group of papers forward:

• Kevin Valson Jacob, Wheaton College, Wheaton, IL,

• Farai Mazhandu, The Oklahoma School of Science and Mathematics, Oklahoma City, OK,

• Zac Patterson, The Ohio State University, Columbus, OH,

• Ann-Marie Mårtensson-Pendrill, University of Gothenburg, Gothenburg, Sweden, and

• Michael Zheng, St. John’s University, Queens, NY.

I appreciate having this extra level of expertise to call on, especially on a subject so rich with complexities. Please express thanks to these individuals should you cross paths!

Quantum ideas are really the reason why one of my favorite courses to teach is the algebra-based introductory physics course, often filled with prospective and potential medical and health science students. This may come as a surprise to some, especially since many colleagues complain about this course—their laments are often related to the (perhaps unfair) stereotype that the students in the class often care more about getting an “A” than about learning the material. But I generally like that the students come to the class with a lot of motivation to do well. The more important reason, though, that I like to teach the “pre-meds” is that I feel much more compelled to spend a good fraction of the course on topics of modern physics, since it is often the last bit of physics that these blossoming medical science initiates will see. In contrast, the physics students in our calculus-based courses often have one or more additional physics courses after the one-year introductory sequence, so the need to include more 20th century physics in that course seems less urgent. (I recognize that this logic has flaws, but it is where my head is at oftentimes.)

So, I take this to mean that I have fuller license to teach the “pre-meds” more about the Bohr model and nuclear physics and quantum mechanics more generally, and less about, say, rotational motion and refrigerator cycles and magnetism. The Bohr model especially appeals to me for lots of reasons, not the least of which is that it is so remarkably successful as a model for describing the atom, getting the beautiful and famous red, aqua, and purple spectral lines of hydrogen so astonishingly correct.¹ In addition, Bohr’s paper wherein he describes the model is such an interesting piece of physics, showcasing as it does what I like to call “the three usual ways of checking your answer”: units, numerical values, and limiting cases.

Bohr starts his 1913 paper² by noting the apparent necessity of adopting the structure suggested in Rutherford’s famous alpha-particles-on-gold experiment, namely that an atom consists of a heavy, compact, positively charged nucleus at the center, with light negatively charged particles bound to the nucleus by Coulombic attraction, at a relative distance much greater than the size of the nucleus.

Despite the necessary adoption of the Rutherford model, Bohr also notes the “inadequacy of classical electrodynamics in describing the behaviour of systems of atomic size” in this model, first indicating that a length scale comparable to the suspected size of atoms cannot be developed from the defining constants of classical electrodynamics. Bohr further comments that this picture changes if one introduces into the model a quantity foreign to electrodynamics (namely Planck’s constant, h ≈ 6.6 × 10^–34 J s, which, for example, does not appear in Maxwell’s classical equations), because this constant is “of such dimensions and magnitude, that it, together with the mass and charge of the particles” can readily be arranged into a combination with units of length and numerical value on the order of the size of atoms [namely the Bohr radius, h²/(4π²mke²) ≈ 0.5 × 10⁻¹⁰ m]. It is especially striking and satisfying to me that Bohr begins the discussion of his celebrated model with an appeal to dimensional analysis, and by showing that the numerical values involved match atomic distance scales (two of the three usual ways!). Then, he turns to the third of the “three usual ways,” a limiting case argument to flesh out the reasoning for his model.

Note that in Bohr’s picture the electron is not said to be “orbiting” the nucleus as that would require that it have a centrifugal acceleration and thus, according to Maxwell’s equations, would necessarily mean that it would emit electromagnetic radiation as it orbited—rather, Bohr introduces the phrase “stationary states” to describe a new view of an atom’s constituents and their configuration. He posits that electromagnetic radiation is released only when an electron transitions from a higher-energy to a lower-energy stationary state (rather than due to centrifugal acceleration), essentially describing a new mechanism for emitted radiation from this system. The energy of light emitted by this newly proposed mechanism is parametrized by a constant c (not the speed of light). Bohr fixes the value of c by insisting that his new mechanism of light production results in the same spectrum as that predicted by Maxwell’s equations in the “slow vibration” limit, that is, in the limit when the electron is very far removed from the nucleus (but still bound to the nucleus). The value of c that results (c = ½ is required), and the entire mechanism for producing light, is then assumed to hold as one moves back into the inner reaches of the atom. Because this model reproduces the well-known Balmer spectral lines for mono-atomic hydrogen (red, aqua, violet …) with spectacular precision, it was immediately clear that Bohr was on to something. So, Bohr uses limiting case analysis to fix the value of a parameter in a speculative theory in an attempt to better understand nature,³ to learn something new; this is especially satisfying to someone who has become a staunch advocate of limiting case analysis as a routine feature⁴ needed in physics education. It’s a great story for the physics classroom…. And much of the rest of quantum mechanics seems to operate this way, with a certain magical omniscience, whether it’s Dirac’s prediction of positrons, or Feynman diagrams for computing cross-sections, or Gell-Mann’s eightfold way for describing baryons—which, in part, explains why I like teaching this material so much!

What parts of these quantum realities can be brought into the introductory classroom effectively? Well, as I have said, I like for my students to see the details of the Bohr model, but I know others object in principle.⁵ It’s hard to know definitively what bit of quantum might be appropriate for the first-year course, and in fact, that’s one of the reasons why we decided to put forth a call for papers on quantum in the introductory physics classroom. In response, we did receive a nice collection of manuscripts with a wide variety of approaches to bringing more quantum into the first course, starting with Gianluca Li Causi’s “Explaining Quanta with Optical Illusions” on page 7. Other papers tackle the interpretation of the photoelectric effect and how particles can be found in a classically forbidden region, while others address quantum computing pedagogy and improving resources available to secondary educators. I hope that you find something useful here in your pursuit to better educate your students.

Sincerely, Gary White