- Jun 30, 2024
The partition function of a hydrogen atom diverges (only considering bound states). However, we can regularize it to get finite answers. Different regularizations give the same result. They largely agree with the physical arguments for the case of the hydrogen atom at room or cold temperature, but this should be considered a mere coincidence. The results from regularized partition functions cannot generally be trusted.
- 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.