Piotr Migdał

Quantum mechanics for high-school students

15 August 2016 | by Piotr Migdał | 9 min read

No-cloning theorem with Júlia Amorós Binefa in 2012
No-cloning theorem with Júlia Amorós Binefa in 2012

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 mode).

My approach

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.

from When should you teach children, and when should you let them explore? - The Economist.

LCD screen + plastic cup + polarizer
LCD screen + plastic cup + polarizer

An LCD screen + a plastic cup + a camera with a polarizer.

Course outlines

I did it four times:

  1. Jun-Jul 2011, Castelldefels, Spain, ICFO for Jóvenes y Ciencia
  2. Aug 2011, Olsztyn, Poland, 7th edition of Wakacyjne Warsztaty Wielodyscyplinarne
  1. Jun-Jul 2012, Castelldefels, Spain, ICFO for Jóvenes y Ciencia
  1. May 2016, Serock, Poland, Wielodyscyplinarny Obóz Naukowy of Krajowy Fundusz na rzecz Dzieci

Since there is a lot of redundancy, I will describe only 1. (3. was similar) and 2. (4. was similar), mentioning some modifications.

Quantum optics, 7th Scientific Summer School (2011)


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.

4. Entanglement

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!).

Quantum cryptography and quantum information, ICFO (2011)


1. Classical cryptography

1.1. Naive letter/word relabeling

2. Quantum cryptography

2.1. Measurement of the (classical) polarization of light

Quantum Zeno effect with polarization rotation - notes by Krzysztof Lis
Quantum Zeno effect with polarization rotation - notes by Krzysztof Lis

The first historical drawing of "the light-twisting sugar solution as a mojito drink" - notes by Krzysztof Lis, Olsztyn, 2011.


Simple books

Interactive stuff




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).