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Maths and physics

What I studied at university, and still a real interest. I do less of it now, but I work it into my everyday work wherever I can.

University College London · 2020

BSc Mathematics and Physics

I studied mathematics and physics together at UCL and graduated in 2020. The parts that stayed with me are quantum mechanics and pure mathematics, and since then the maths hiding inside music and inside cryptography. I taught it for years too, tutoring A-level and GCSE maths from 2015 to 2022.

Quantum mechanicsPure mathematicsMaths of musicCryptography

Three proofs and an experiment

Step through a proof by contradiction, by construction and by induction. Then fire some electrons.

Pure maths

Three ways to prove something

Proof by contradiction $\sqrt{2}$ cannot be written as a fraction.

1
Assume the opposite: that it can, with whole numbers $a$ and $b$ cancelled down until they share no factor.
$\sqrt{2} = \frac{a}{b}$

Where the number comes from: the diagonal of a square of side 1. Swing it down onto a ruler and it lands on a point no fraction names.

the wave where they landed the prediction0 fired

Slits apart (d)Speed (p)

Quantum mechanics

One electron, two slits

Electrons are fired one at a time. Each lands as a single dot, like a particle. But the dots pile up in stripes, which only waves make. Each electron seems to go through both slits and interfere with itself. Check which slit it takes, and the stripes go.

$λ = \frac{h}{p}$
Every particle has a wavelength, set by its momentum. Faster means shorter.
$P = {| ψ_{1} + ψ_{2} |}^{2} = {| ψ_{1} |}^{2} + {| ψ_{2} |}^{2} + 2 Re (ψ_{1}^{*} ψ_{2})$
Unwatched: add the two waves first, then square. The extra term on the end is the stripes.
$d \sin (θ) = n λ Δ y = \frac{λ L}{d}$
Bright stripes sit where the two paths differ by a whole number of wavelengths. Move the slits apart, or speed the electrons up, and the stripes crowd together.