The new Quantum Mechanics (a.k.a. QM) idea we introduced last time is that energy comes in lumps – quanta.  Another name for a quantum is a “wavicle”, a made-up word I like to use because it conveys both important parts of its behaviour: particle-like (“corpuscular”) and wave-like. Which behaviour you see depends on how you probe the quantum and at what energy. In the case of photons, energy E is proportional to frequency f; the constant of proportionality is known as Planck’s constant and denoted h (which is very small in SI units).

Two hallmark experiments helped convince physicists that all subatomic particles (including photons, electrons, etc.) display wavelike properties. These were Young’s double-slit experiment, which showed interference, and the Davisson-Germer experiment, which showed diffraction. These two behaviours (example pictures are in the notes) are characteristic of physical systems of waves, as anyone who has played with water in the bathtub/swimming pool will know. Diffraction is bending of waves around a small obstacle or spreading out of waves from a small opening. It can occur because waves are extended objects, not residing just at a one point in space. Interference patterns arise when (at least) two wave sources are present, and happen when waves from the different sources can either pile up together, adding to make a bigger wave amplitude, or cancel each other out. The pretty pattern is what you see in the region where waves from the two (or more) sources overlap.

In Young’s double-slit experiment, he set up a first screen with a teeny hole (“slit”) in it, to let sunlight in. His second screen, behind the first, had two teeny holes in it, equidistant from the slit in the first screen. Finally, a detector was placed behind the second screen. Young was looking to see what pattern eventuated. If photons (the particles used in this experiment) were corpuscular – i.e. like classical bullets – then they would have piled up on the screen in two peaks at the exact locations of the two slits. What Young saw was instead an interference pattern with alternating light and dark fringes! These arise at places where waves either add or cancel out, depending on the difference in distance between the two slits and the detector position, giving a light or dark fringe accordingly.

The Davisson-Germer experiment had a very different nature. Electrons from a heated filament were accelerated via an electric voltage, and allowed to strike a target made of Nickel metal.  If the impinging electrons were classical bullets, they would ricochet off the wall in the style of a tennis ball reflecting off a concrete wall. The angle of reflection would be the same as the incident angle, measured from the perpendicular to the target.  But this is not what Davisson and Germer saw at Bell Labs in the 1920s.  They actually saw a series of peaks, at a variety of angles, not just the classical reflection angle! This is diffraction and could only arise from wavelike behaviour of the electrons.

In 1924, Louis de Broglie presented a very bold hypothesis for the wavelength of any quantum to his PhD committee. They wanted to fail him, because he had almost no evidence to back up his contention. It was later realized de Broglie was amazingly prescient – his formula turned out to be correct! It says that the quantum wavelength λ is given by λ=h/p, where p is the momentum and h is Planck’s constant. So for a wavicle with higher momentum, its wavelength is shorter. Or a long wavelength results from low momentum. Quantum wavelengths tend to be noticeable only for subatomic wavicles because h is so small in SI units.

Physicists built up firm evidence for the wavelike behaviour of electrons and other important subatomic animals in the 1920s, 30s, and beyond. One special thing to notice is the requirement of having an integer number of wiggles of an electron wave upon making a full revolution around the nucleus. This is a necessary consistency condition to ensure that the wave matches back up onto itself properly. (Otherwise, it would end up cancelling itself out!) This need to fit an integer number of wiggles around a finite sized atom is the fundamental reason why you see quantum numbers. These are discussed in chemistry textbooks as the principal quantum number n and the two angular momentum quantum numbers m and l. There are 3 because there are 3 dimensions of space. (The shapes of various possible wavefunctions for hydrogen are illustrated in a figure.)

Physicists have found it very useful to encode the wavelike behaviour of quanta in something called the wavefunction Ψ. This little chap, who is different for each particle with particular mass and spin, obeys a differential equation involving both changes in time and variations over space. In the case of massive quanta in the non-relativistic regime, the equation is called the Schrodinger equation, which essentially expresses the story of the energy budget for the system.

If relativity is added to QM, the result is known as QFT or Quantum Field Theory. In QFT, the object Ψ becomes a quantum field, a thingie capable of representing the quantum physics of (say) multiple electrons and their antiparticles at the same time. The equation governing the electron-positron field is known as the Dirac equation. QFT is a very powerful apparatus, and learning it is a graduate-level endeavour. It is a quintessential tool for any researcher studying subatomic physics, experimentalists and theorists alike.

One thing physicists discovered while studying wavefunctions is that the Schrodinger (or Dirac, etc.) equation computes how Ψ evolves into the future and across space. BUT what Ψ gives physicists is only the probability for detecting the particle in question! That probability is proportional to |Ψ|² and has to add up to 100%, counting all possible outcomes. This might not look like much conceptually, until you realize that this quantum mechanics is inherently probabilistic – what you actually end up detecting is purely up to random chance! Quantum mechanics breaks classical determinism – the older Newtonian-era idea of the universe as a giant clockwork machine goes out the window! All you can predict precisely as a physicist is the probability; the actuality happens in a purely random way. Albert Einstein found this concept extremely disturbing, and he never accepted it – he famously said “God does not play dice with the universe!”. Stephen Hawking, decades later, came up with a very pithy rejoinder – “It was Einstein who was confused, not the quantum theory!”. This amusing anecdote reminds us that physics is beholden to experiment: what we need to explain theoretically is what is, not what we would like necessarily.

The most famous statement of the physical uncertainty inherent in QM is the Heisenberg Uncertainty Principle. It originates in the wavelike nature of quanta, which “fuzzes out” the physics compared to our classical expectations. Unfortunately the HUP is often misquoted, especially by social scientists who want to co-opt physics terms to make themselves sound more authoritative. (Instead, they just make themselves look foolish, through their dire lack of science literacy!). What the HUP actually says is this: specific pairs of variables are affected by a minimum uncertainty – if you get too precise measuring one variable in the pair (e.g. momentum) you lose precision in measuring the other (e.g. position) and vice versa. Mathematically this is expressed as ΔpΔx ≥ h . Energy and time are another pair of physical observables affected by Heisenberg uncertainty. This is not fundamentally a measurement disruption problem, it’s  more that a quantum doesn’t even have a precise position and momentum at any given time. Physics is fuzzier in the microscopic realm than we are used to in our more macroscopic human world.

Here is the narrated slideshow as a mp4 video, as an aac audio file, and as a PDF file of slides.