From Number Theory to Physics.

Saturday, 27 April 2019

From Number Theory to Physics.

I have asked a question here:

I want to see an example which is related to (integral) quadratic forms or theta series.

@Kiryl Pesotski answered me in some comments as following:

For example, you may want to compute the partition function for the pure point spectrum of the Hamiltonian where the eigenenergies are quadratic in the quantum numbers. This is e.g. particle in a box or the leading order correction due to the $x^3$ and $x^4$ powers to the spectrum of the harmonic oscillator. These series can be written in terms of the Jacobi theta functions.

I mean you have the energies being something like $E_{n}=\alpha+\beta{n}+\gamma{n^{2}}$ , where $n \in \mathbb{Z}$ , or $\mathbb{N}$ . The partition function is given by the seres $Z=\sum_{\forall{n}}e^{-\beta{E_{n}}}$ , such sreis are used to represent Jacobi theta functions.

The other example is the heat equation. The Jacobi theta function solves it. E.g. $\partial_{t}u=\frac{1}{4\pi}\partial^{2}_{x}u$ is solved by $u(x, t)=\theta(x, it)$ , where $\theta(x, \tau)$ is the jacobi theta function.

I have not any physical knowledge;

Can any one explain his answer for me in more details?

Also I have list other related questions from the highest vote to the lowest:
[None of them answers my question; except the $6^{\text{th}}$ question; which I feel a connection.]