15 August 2016 | by Piotr Migdał | 9 min read
A participant, Julia Amorós Binefa, and the no-cloning theorem - ICFO, 2012.
A few times I gave introductions to quantum mechanics for talented high-school students, in Poland (2011, 2016) and in Spain[^catalonia] (2011, 2012). I have been sharing the materials over the course of years. Now I have found some time[^finding_time] to clean it up a bit, hoping that you may find it useful or inspiring.
It certainly keeps inspiring me! First two workshops convinced me that quantum mechanics isn't that hard or bizarre and can be explained to motivated high-school students[^who_can]. Moreover, it resulted in the Quantum Game with Photons. It's still in development, but already playable at this stage (and certainly sufficient as a sandbox for interferometers - use
Before jumping into the actual content, let me explain my approach. Some is my teaching credo (I don't want to argue with others, as it is a matter of taste, not - fact), some are lessons learnt (often the hard way).
I use light polarization as the prototypical two-level quantum system, starting from vectors and 2x2 matrices (with straightforward calculations rather than abstraction), as it:
A classical[^classical] quantum mechanics introduction starts from describing position (and momentum) of a single particle. Sure, it has valuable pieces like calculating energies and orbitals, and there is the Heisenberg uncertainty principle[^heisenberg]. However, IMHO starting with classical mechanics (continuous variables) is the worst approach, as it:
Working with electrons as the two-state system (spin up, spin down) has some benefits (two-level system), however:
Moreover, when it comes to the mathematical tools, I try not treating them as a necessary evil or means to an end, as:
When it comes to delivery, I try to:
For experimental stuff, it's crucial to show first, explain later - if at all (otherwise you are killing the sense of suspension and awe):
The researchers' conclusion was that, in the context of strange toys of unknown function, prior explanation does, indeed, inhibit exploration and discovery.
An LCD screen + a plastic cup + a camera with a polarizer.
I did it four times:
Since there is a lot of redundancy, I will describe only 1. (3. was similar) and 2. (4. was similar), mentioning some modifications.
qualification problems as a core part
0. Linear algebra (in qualification problems)
Complex numbers (including radial representation), basics of linear algebra, quantum jargon (bras, kets and daggers).
1. Polarization of light
What is that, how can we act on it (wave plates, polarizers, rotation by a solution of sugar, ...); also: the 3 polarizers "paradox".
2. Quants and measurement
Single photons behave in the same as a classical wave, just they can't be measured partially, only 0/1. And then the quantum measurement is the most intuitive interpolation of the classical measurement.
3. Superposition vs mixture
How to tell the difference using interference. Detection of the light-sensitive bomb.
What is that? Why it is the most intuitive approach to more particles? Relation between interference, entanglement and looking at objects.
I was positively surprised that all points were easy and enjoyable for the participants. There was a slight slowdown with the tensor product, but it wasn't a blocker.
In 2016, I covered only 0., 1. and 2., all on-site. Because of the lack of preparation (0.) and more interesting parts (3. and 4.) it didn't go as well as the one of 2011 (one of my best educational experiences ever!).
*- an additional material that I used
**- an additional material that I didn't use (some of it was in 2012, though)
1. Classical cryptography
1.1. Naive letter/word relabeling
** show that natural languages can be easily detected (with analysis of ngrams - i.e. frequencies of short strings of letters)
1.2. Perfect XOR
but how to distribute the key?
2. Quantum cryptography
2.1. Measurement of the (classical) polarization of light
*introduction to complex numbers
decomposition of a vector in different bases
2.2. Introduction to quantum mechanics (of two polarizations)
superposition vs mixture
2.3. BB84 protocol
* 3 bases
** Quantum entanglement
** Information theory
The first historical drawing of "the light-twisting sugar solution as a mojito drink" - notes by Krzysztof Lis, Olsztyn, 2011.
Valerio Scarani, Six Quantum Pieces: A First Course in Quantum Physics
Leonard Susskind, Art Friedman, Quantum Mechanics: The Theoretical Minimum. What You Need to Know to Start Doing Physics.
Konrad Banaszek, Rafał Demkowicz-Dobrzański, Quantum information 1/2
Andrzej Dragan, Niezwykle Szczególna Teoria Względności (en: Extraordinarily Special Relativity)
∞mode, where you can freely simulate quantum mechanics of a single photon
Math and Physics Applets by Paul Falstad (unfortunately in Java)
Python (in Jupyter Notebook) for calculations and plots
Berthold-Georg Englert, Lectures on Quantum Mechanics - Volume 1: Basic Matters
David Griffiths, Introduction to Quantum Mechanics
I do not recommend Leonard I. Schiff, Quantum mechanics
Michael A. Nielsen, Isaac L. Chuang, Quantum Computation and Quantum Information
Carl D. Meyer, Matrix Analysis and Applied Linear Algebra
If you want to use any materials - feel free! And I would be even happier to hear that. If you have some other materials or references for an easy introduction you recommend - just send them.
I am considering writing a simple introduction to quantum mechanics, with interactive simulations (very likely with the Quantum Game engine). If you think it is a great idea, consider poking/teasing/tempting me. :)
I would like to thank Michał Kotowski for remarks on this blog post. And, of course, to everyone participating in my quantum mechanics workshops.
Before applying consult with your teacher or professor, as every misused didactic material may result in misunderstanding or discouragement.
[^catalonia]: Or rather: Catalonia! [^finding_time]: Read: stolen time from other projects or was doing white procrastination. [^who_can]: As a rule of thumb I can explain what quantum mechanics is in 3h, if the other person knows what matrices are. I did it to biologists and they are fine. [^classical]: Nomen omen! [^heisenberg]: Though, I have a strong preference in showing it as a Fourier transform property, for which QM is only incidental. Again, it's just a wave phenomenon (as interference or tunneling).