Dec 22, 2023
The conformal map w(z) transforms the trajectory with energy −B in potential U(z):=A∣dw/dz∣2 into the trajectory with energy −A in potential V(w):=B∣dz/dw∣2. I will prove this beautiful result and show some implications of it.
Nov 9, 2023
For a plane lattice, there is only a finite number of different rotational symmetries that are compatible with the discrete translational symmetry. For example, the 5-fold rotational symmetry is not one of them. Why is that? It turns out that whether an m-fold symmetry is compatible with translational symmetry is the same as whether φ(m)≤2.
Oct 18, 2023
When someone asks you why it is −i here instead of i or the other way around, you can say that this is just a convention. My professor of quantum mechanics once asked the class similar a question, and I replied with this letter.
Jun 13, 2020
Joukowsky transformation of a circle centered at (1,1) of radius 1 is a curve resembling a heart.
Jan 6, 2020
In this article, I try exploring an idea: using complex numbers to combine pairs of canonical variables into complex variables: c:=αq+iβp. It turns out that we can write canonical equations dtdc=−2iαβ∂c∗∂H, Poisson brackets
{f,g}=−2iαβ(∂c∂f⋅∂c∗∂g−∂c∗∂f⋅∂c∂g), and canonical transformations
∂c′∗∂c∗=∂c∂c′,∂c′∗∂c=−∂c∗∂c′ in these complex numbers. Finally, I show two examples of using them in real problems: a free particle, and a harmonic oscillator.