# Few-body quantum method in a $d$-dimensional space

## TL;DR

This paper compares two methods for analyzing quantum systems under continuous confinement from three to two dimensions, highlighting their equivalence and potential for broader applications in multi-particle and varied confinement scenarios.

## Contribution

It introduces and relates two approaches for continuous quantum confinement, simplifying numerical calculations and enabling applications beyond two-body systems.

## Key findings

- The two methods are mathematically equivalent.
- The second method simplifies numerical implementation.
- Potential for extending to multi-particle systems and different potentials.

## Abstract

In this work we investigate the continuous confinement of quantum systems from three to two dimensions. Two different methods will be used and related. In the first one the confinement is achieved by putting the system under the effect of an external field. This method is conceptually simple, although, due to the presence of the external field, its numerical implementation can become rather cumbersome, especially when the system is highly confined. In the second method the external field is not used, and it simply considers the spatial dimension $d$ as a parameter that changes continuously between the ordinary integer values. In this way the numerical effort is absorbed in a modified strength of the centrifugal barrier. Then the technique required to obtain the wave function of the confined system is precisely the same as needed in ordinary three dimensional calculations without any confinement potential. The case of a two-body system squeezed from three to two dimensions is considered, and used to provide a translation between all the quantities in the two methods. Finally we point out perspectives for applications on more particles, different spatial dimensions, and other confinement potentials.