# Analytic results of the excited electronic states at $\upsilon=1/3$ and the Laughlin-Jain microscopic wave function approaches

## TL;DR

This paper analytically investigates the energies of excited electronic states in a two-dimensional electron gas under strong magnetic fields, focusing on the fractional quantum Hall regime at filling factor 1/3, using complex coordinate methods.

## Contribution

It provides exact analytical results for energies of excited states at specific filling factors, extending previous numerical work and exploring the connection between IQHE and FQHE.

## Key findings

- Analytical energies for systems up to 10 electrons at $
u=1$ and $
u=1/3$
- Good agreement with previous numerical and integral representations
- Insights into the relation between fractional and integer quantum Hall effects

## Abstract

In this work we studied the properties of a two-dimensional electronic gas subjected to a strong magnetic field and cooled at a low temperature. We reported exact analytical results of energies at the ground state. The results are for systems up to $N_{e}=10$ electrons calculated in the integer quantum Hall effect (IQHE) regime at the filling factor $\upsilon=1$. To accomplish the calculation we used the complex polar coordinates method. Note that the system of electrons in the quantum Hall regime relied heavily on the disk geometry for finite systems of electrons with arbitrary values of $N_{e}=2$ to $10$ particles. The results that we obtained by analytical calculations are in good agreement with those reported by Ciftja [Ciftja O., J. Math. Phys., 2011, 52, 122105], where the representation for certain integrals of products of Bessel functions is obtained. In the end, we have studied the composite fermions energies for the excited states for several systems at $\upsilon =1/3$ and the correspondence between the fractional quantum Hall effect (FQHE) and the IQHE.