- Sep 10, 2024
In general relativity, people usually choose one of the two major metric signatures. However, in certain cases, the imaginary time convention can be more convenient. Here is one of such cases: the derivation of the metric in Poincaré coordinates for the anti-de Sitter space.
- Jul 1, 2024
The smallest wave packet in the lowest Landau level exists, and is a Gaussian wave packet. This turns out to be related to the coherent state of the harmonic oscillator.
- 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.
- 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.
- May 1, 2023
For sake of rigor and generalizability, I feel it necessary to try to have a mathematical formulation for statistical ensembles. I chose measure spaces as the underlying mathematical structure of thermal systems and tried to justify the method of statistical ensembles by deducing them from some axioms.
- Mar 30, 2023
For sake of rigor and generalizability, I feel it necessary to try to have a mathematical formulation for statistical ensembles. I chose measure spaces as the underlying mathematical structure of thermal systems and tried to justify the method of statistical ensembles by deducing them from some axioms.
- Feb 2, 2023
We try solving the even function solutions to the time-independent Schrödinger equation for the potential V=−α∑j=−nnδ(x−ja) such that E<0 (bound states).
- Feb 1, 2023
The aliens intiated their attack to the earth! They shoot bullets with mass m and speed v from a far-awar planet. To defend, humans built a field U=α/r that can repel the bullets. What regions are safe? The answer turns out to be the interior of a circular paraboloid.
- Nov 20, 2022
We can derive the equation of motion for mechanical systems in a Galileo universe with ι time dimensions and χ space dimensions by generalizing the principle of relativity and Hamilton’s principle.