#  PrimeQA: The Prime Repository for State-of-the-Art Multilingual Question
Answering Research and Development

Avirup Sil, Jaydeep Sen, Bhavani Iyer, Martin Franz, Kshitij Fadnis,
Mihaela Bornea, Sara Rosenthal, Scott McCarley, Rong Zhang, Vishwajeet Kumar,
Yulong Li, Md Arafat Sultan, Riyaz Bhat, Radu Florian, Salim Roukos
IBM Research AI  Corresponding author

###### Abstract

The field of Question Answering (QA) has made remarkable progress in recent
years, thanks to the advent of large pre-trained language models, newer
realistic benchmark datasets with leaderboards, and novel algorithms for key
components such as retrievers and readers. In this paper, we introduce
PrimeQA: a one-stop and open-source QA repository with an aim to democratize
QA research and facilitate easy replication of state-of-the-art (SOTA) QA
methods. PrimeQA supports core QA functionalities like retrieval and reading
comprehension as well as auxiliary capabilities such as question generation.
It has been designed as an end-to-end toolkit for various use cases: building
front-end applications, replicating SOTA methods on public benchmarks, and
expanding pre-existing methods. PrimeQA is available at:
https://github.com/primeqa.

##  1 Introduction

Question Answering (QA) is a major area of interest in Natural Language
Processing (NLP), consisting primarily of two subtasks: information retrieval
(IR) Manning (2008); Schütze et al. (2008) and machine reading comprehension
(MRC) Rajpurkar et al. (2016, 2018); Kwiatkowski et al. (2019a); Chakravarti
et al. (2020). IR and MRC systems, also referred to as retrievers and readers,
respectively, are commonly assembled in an end-to-end open-retrieval QA
pipeline (henceforth, OpenQA) Chen et al. (2017); Lee et al. (2019); Karpukhin
et al. (2020); Santhanam et al. (2022b) that accepts a query and a large
document collection as its input and provides an answer as output. The
specific role of the retriever is to identify documents or passages (i.e.,
contexts) that contain information relevant to the query, while the reader
component extracts a precise answer from such contexts.

While QA as a field has advanced rapidly, software to perform and replicate QA
experiments has mostly been written in silos. At the time of this writing, no
central repository exists that facilitates the training, analysis and
augmentation of state-of-the-art (SOTA) models for different QA tasks at
scale. In view of the above, and with an aim to democratize QA research by
providing easy replicability, here, we propose PrimeQA: an open-source
repository111 https://github.com/primeqa designed as an end-to-end toolkit,
with all the necessary tools to easily and quickly create a custom QA
application. We provide a main repository that contains easy-to-use scripts
for retrieval, machine reading comprehension, and question generation with the
ability to perform training, inference, and performance evaluation.
Additionally, several sibling repositories provide features for easily
connecting various retrievers and readers and creating a front-end user
interface (UI) for end users. PrimeQA has been designed as a platform for QA
development and research and encourages collaboration from everyone in the
field from beginners to experts. PrimeQA already has a growing developer base
with contributions from major academic institutions.

Our paper makes several major contributions:

  * •

We present PrimeQA, a first-of-its-kind repository for comprehensive QA
research. It is free to use, well documented, easy to contribute to, and
license friendly (Apache 2.0) for both academic and commercial usage.

  * •

PrimeQA provides the mechanism via accompanying repositories to create custom
OpenQA applications containing both retrievers and readers for industrial
deployment including a front-end UI.

  * •

We provide easy-to-use implementations of SOTA retrievers and readers that are
at the top of major QA leaderboards, with capabilities for performing
training, inference and performance evaluation of these models.

  * •

PrimeQA models are built on top of Transformers Wolf et al. (2020) and are
available on the Hugging Face model hub.222https://huggingface.co/PrimeQA

##  2 Related Work

One of the largest open source community efforts for NLP software is Papers
with Code Robert and Thomas (2022). Their mission is to create a free and open
resource for NLP papers, code, datasets, methods and evaluation tables. Their
focus is to cater to the wider NLP and Machine Learning community and not just
QA. Even though the QA section includes over 1800 papers with their code, the
underlying software components (written in various versions of both pytorch
and tensorflow, with no central control whatsoever) do not communicate with
each other. These disjoint QA resources hinder replicability and effective
collaboration, and ultimately lead to quick sunsetting of new capabilities.

Recently, among the most used repositories for NLP users have been the
Transformers repository Wolf et al. (2020). However, while being widely
adopted by the community, they lack a distinct focus on QA. Unlike PrimeQA,
they only support one general script for extractive QA and several stand-alone
python scripts for retrievers. Similarly FairSeq, Ott et al. (2019) and
AllenNLP Gardner et al. (2018) also focus on a wide array of generic NLP tasks
and hence do not solely present a QA repository allowing anyone plug and play
components for their custom search application. There exists several toolkits
catered to build customer-specific search applications NVDIA (2022); Deepset
(2021) or search-based virtual assistants IBM (2020). However, while they have
a good foundation for software deployment, unlike PrimeQA, they lack the focus
on replicating (and extending) the latest SOTA for QA on academic benchmarks
which is an essential component needed in order for us to make rapid progress
in this field.

##  3  PrimeQA

Figure 1: The PrimeQA Repository: the core components and features. Core Models | Extensions
---|---
Retriever
BM25 Robertson and Zaragoza (2009) | Dr.DECR * Li et al. (2022)
DPR Karpukhin et al. (2020) |
ColBERT Santhanam et al. (2022b) |
Reader
General MRC Alberti et al. (2019b) | ReasonBERT
FiD Izacard and Grave (2020) | OmniTab Jiang et al. (2022a)
Boolean* McCarley et al. (2022) |
Lists |
Tapas Herzig et al. (2020a) |
Tapex Liu et al. (2021) |
Question Generation
Table QG Chemmengath et al. (2021) |
Passage QG |
Hybrid QG |
Table 1:  A non-exhaustive list of core PrimeQA models for the three main
supported tasks (left) and their various extensions (right) available on our
Hugging Face model hub: https://huggingface.co/PrimeQA. * SOTA leaderboard
systems.

PrimeQA is a comprehensive open-source resource for cutting-edge QA research
and development, governed by the following design principles:

$\bullet$ Reproducible: Users can easily reproduce SOTA published results and
combine or “plug and play” the corresponding methods with companion modules,
e.g., a reader with a retriever. The various supported applications and the
associated PrimeQA components are listed in Table 1.

$\bullet$ Customizable: We allow users to customize and extend SOTA models for
their own applications. This often entails fine-tuning on users’ custom data,
which they can provide through one of several supported data formats, or
process on their own by writing a custom data processor.

$\bullet$ Reusable: We aim to make it straightforward for developers to
quickly deploy pre-trained off-the-shelf PrimeQA models for their QA
applications, requiring minimal code change.

$\bullet$ Accessible: We provide easy integration with Hugging Face Datasets
and the Model Hub, allowing users to quickly plug in a range of datasets and
models as shown in Table 1.

PrimeQA in its entirety is a collection of four different repositories: a
primary research and replicability333 https://github.com/primeqa/primeqa
repository and three accompanying repositories444
https://github.com/primeqa/create-primeqa-app ,555
https://github.com/primeqa/primeqa-orchestrator ,666
https://github.com/primeqa/primeqa-ui for industrial deployment. Figure 1
shows a diagram of the PrimeQA repository. It provides several entry points,
supporting the needs of different users, as shown at the top of the figure.
The repository is centered around three core components: a retriever, a
reader, and a question generator for data augmentation. These components can
be used as individual modules or assembled into an end-to-end QA pipeline. All
components are implemented on top of existing AI libraries.

###  3.1  The Core Components

Each of the three core PrimeQA components supports different flavors of the
task it has been built for, as we detail in this section.

####  3.1.1 Retriever: run_ir.py

Retrievers predict documents (or passages) from a document collection that are
relevant to an input question. PrimeQA has both sparse and SOTA dense
retrievers along with their extensions, as shown in Table 1. We provide a
single Python script run_ir.py that can be passed arguments to switch between
different retriever algorithms.
Sparse: BM25 Robertson and Zaragoza (2009) is one of the most popular sparse
retrieval methods, thanks to its simplicity, efficiency and robustness. Our
Python-based implementation of BM25 is powered by the open-source library
PySerini.

Dense: Modern neural retrievers have utilized dense question and passage
representations to achieve SOTA performance on various benchmarks, while
needing GPUs for efficiency. We currently support ColBERT Santhanam et al.
(2022b) and DPR Karpukhin et al. (2020): both fine-tune pre-trained language
models to train question and passage encoders Devlin et al. (2019); Conneau et
al. (2020). They utilize FAISS Johnson et al. (2017) for K-nearest neighbor
clustering and compressed index representations, respectively, and support
multilingual retrieval with the question and the documents being in the same
Lee et al. (2019); Longpre et al. (2021) or different languages (cross-
lingual) Asai et al. (2021).

####  3.1.2 Reader: run_mrc.py

Given a question and a retrieved passage—also called the context—a reader
predicts an answer that is either extracted directly from the context or is
generated based on it. PrimeQA supports training and inference of both
extractive and generative readers through a single Python script: run_mrc.py.
It works out-of-the-box with different QA models extended from the
Transformers library Wolf et al. (2020).

Extractive: PrimeQA’s general extractive reader is a pointer network that
predicts the start and end of the answer span from the input context Devlin et
al. (2019); Alberti et al. (2019b). It can be initialized with most large pre-
trained language models Devlin et al. (2019); Liu et al. (2019); Conneau et
al. (2020). In addition, our reader is extremely versatile as it can provide
responses to questions with list answers Khashabi et al. (2021), yes/no
responses to Boolean questions Clark et al. (2019, 2020a); Kwiatkowski et al.
(2019b), answer spans found in tables Herzig et al. (2020b) and in multimodal
(text+image) documents Mathew et al. (2021). Examples of several extractive
readers along with their extensions are provided in Table 1.

Generative: PrimeQA provides generative readers based on the popular Fusion-
in-Decoder (FiD) Izacard and Grave (2020) algorithm. Currently, it supports
easy initialization with large pre-trained sequence-to-sequence (henceforth,
seq2seq) models Lewis et al. (2019); Raffel et al. (2022). With FiD, the
question and the retrieved passages are used to generate relatively long and
complex multi-sentence answers providing support for long form question
answering tasks, e.g., ELI5 Petroni et al. (2021); Fan et al. (2019).

####  3.1.3 Question Generation: run_qg.py

Data augmentation through synthetic question generation (QG) can be a powerful
tool for improving QA model generalization Alberti et al. (2019a); Sultan et
al. (2020); Gangi Reddy et al. (2022a), including in domain adaptation Shakeri
et al. (2021); Gangi Reddy et al. (2021, 2022b), domain generalization Sultan
et al. (2022) and few-shot learning Yue et al. (2022) settings. QG can also
circumvent the problem of not having labeled data in the target domain of
application. Question generators take a span of text (e.g., a sentence) from a
document as input, hypothesize an answer (e.g., a named entity) and generate a
question as output. PrimeQA’s QG component Chemmengath et al. (2021) is based
on SOTA sequence-to-sequence generation architectures Raffel et al. (2022),
and supports both unstructured and structured input text through a single
Python script run_qg.py. A list of available question generators and their
extensions are provided in Table 1.
Unstructured Input: Our first variant of QG is a multilingual text-to-text
model capable of generating questions in the language of the input passage. It
fine-tunes a pre-trained T5 language model Raffel et al. (2022) on publicly
available multilingual QA data Clark et al. (2020b).

Structured Input: We also provide QG capability over tables, for which the
generator is trained on examples of SQL and natural language question pairs
extracted from the popular Table QA dataset Zhong et al. (2017a). As in
Chemmengath et al. (2021), during inference, PrimeQA uses a controllable SQL
sampler to select SQL queries for a given table and answer text, and then
applies the trained QG model to generate natural language questions.

###  3.2 Entry Points

We cater to different user groups in the QA community by providing different
entry points to PrimeQA, as shown in Figure 1.

$\bullet$ Top-level Scripts: Researchers can use the top level scripts,
run_ir/mrc/qg.py, to reproduce published results as well as train, fine-tune
and evaluate associated models on their own custom data.

$\bullet$ Jupyter Notebooks: These demonstrate how to use built-in classes to
run the different PrimeQA components and perform the corresponding tasks.
These are useful for developers and researchers who want to reuse and extend
PrimeQA functionalities.

$\bullet$ Inference APIs: The Inference APIs are primarily meant for
developers, allowing them to use PrimeQA components on their own data with
only a few lines of code. These APIs can be initialized with the pre-trained
PrimeQA models provided in the HuggingFace hub, or with a custom model that
has been trained for a specific use case.

$\bullet$ Service Layer: The service layer helps developers set up an end-to-
end QA system quickly by providing a wrapper around the core components that
exposes an endpoint and an API.

$\bullet$ UI: The UI is for end-users, including the non-technical layman who
wants to use PrimeQA services interactively to ask questions and get answers.

###  3.3 Pipelines for OpenQA

PrimeQA core components and entry points make it intuitive for users to build
an OpenQA pipeline and configure it to use any of the PrimeQA retrievers and
readers. This is facilitated through a lightweight wrapper built around each
core component, which implements a training and an inference API. The
retrieval component of the pipeline predicts relevant passages/contexts for an
input question, and the reader predicts an answer from the retrieved contexts.
PrimeQA pipelines are easy to construct using the pre-trained models in the
model hub and our inference APIs.

An example of such a pipeline can be connecting a ColBERT retriever to an FiD
reader to construct a long form QA (LFQA) system. This pipeline uses the
retriever to obtain supporting passages that are subsequently used by the
reader to generate complex multi-sentence answers. A different pipeline can
also be instantiated to use an extractive reader instead that is available
through our model hub.

##  4 Services and Deployment

Figure 2: PrimeQA’s end-to-end application. Each container contains a
development (blue), build (yellow) and deployment (green) stack.

Industrial deployment often necessitates running complex models and processes
at scale. We use Docker to package these components into micro-services that
interact with each other and can be ported to servers with different hardware
capabilities (e.g. GPUs, CPUs, memory). The use of docker makes the addition,
replacement or deletion of services easy and scalable. All components in the
PrimeQA repository are available via REST and/or gRPC micro-services. Our
docker containers are available on the public DockerHub and can be deployed
using technologies such as OpenShift and Kubernetes.

In addition to the main PrimeQA repository, we provide three sibling
repositories for application deployment:

primeqa-ui is the front-end UI. Users can personalize the front-end UI by
adding custom organization logos or changing the display fonts.

primeqa-orchestrator is a REST server and is the central hub for the
integration of PrimeQA services and external components and the execution of a
pipeline. For instance, the orchestrator can be configured to search a
document collection with either a retriever from PrimeQA such as ColBERT, or
an external search engine such as Watson Discovery.777
https://www.ibm.com/cloud/watson-discovery

create-primeqa-app provides the scripts to launch the demo application by
starting the orchestrator and UI services.

Figure 3: A custom OpenQA search application built with PrimeQA. Additional
screenshots are in Appendix A.

Figure 2 illustrates how to deploy a QA application at scale using the core
PrimeQA services (e.g. Reader and Retriever) and our three sibling
repositories. We provide this end-to-end deployment for our demo, however
users can also utilize PrimeQA as an application with their own orchestrator
or UI.

Figure 3 shows an OpenQA demo application built with PrimeQA components. Our
demo application provides a mechanism to collect user feedback. The thumbs up
/ down icons next to each result enables a user to record feedback which is
then stored in a database. The user feedback data can then be retrieved and
used as additional training data to further improve a retriever and reader
model.

##  5 Community Contributions

While being relatively new, PrimeQA has already garnered positive attention
from the QA community and is receiving constant successful contributions from
both international academia and industry via Github pull requests. We describe
some instances here and encourage further contributions from all in the
community. We provide support for those interested in contributing through a
dedicated slack channel 888https://ibm.biz/pqa-slack, Github issues and PR
reviews.

Neural Retrievers: ColBERT, one of our core neural retrievers, was contributed
by Stanford NLP. Since PrimeQA provides very easy entry points into its core
library, they were able to integrate their software into the retriever script
run_ir.py independently. Their contribution to PrimeQA provides the QA
community with the ability to obtain SOTA performance on OpenQA benchmark
datasets by performing ‘late interaction’ search on a variety of datasets.
They also contributed ColBERTv2 Santhanam et al. (2022b) and its PLAID
Santhanam et al. (2022a) variant. The former reduces the index size by 10x
over its predecessor while the latter makes search faster by almost 7x on GPUs
and 45x on CPUs.

Few shot learning: The SunLab from Ohio State University provided the ability
to easily perform few-shot learning in PrimeQA . Their first contribution,
ReasonBERT Deng et al. (2021), provides a pretrained methodology that augments
language models with the ability to reason over long-range relations. Under
the few-shot setting, ReasonBERT in PrimeQA substantially outperforms a
RoBERTa Liu et al. (2019) baseline on the extractive QA task. PrimeQA gives
any researcher or developer the capability to easily integrate this component
in their custom search application e.g. a DPR retriever and a ReasonBERT
reader.

Table Readers: Beihang University and Microsoft Research Asia contributed
Tapex Liu et al. (2021) as the first generative Table reader within PrimeQA.
Tapex proposes a novel table pre-training strategy based on neural SQL
executor and achieves the SOTA on Wiki-SQL Zhong et al. (2017a) and Wiki-
TableQuestions Pasupat and Liang (2015a). Their contribution reused the
seq2seq trainer from Transformers Wolf et al. (2020) for a seamless
integration into PrimeQA. Another contribution comes from LTI CMU’s NeuLab
which integrated OmniTab Jiang et al. (2022b). It proposes an efficient pre-
training strategy combining natural and synthetic pre-training data. This
integration happened organically as OmniTab builds on top of Tapex in PrimeQA.
Currently, their model produces the best few-shot performance on Wiki-
TableQuestions, making it suitable for domain adaptation experiments for
anyone using PrimeQA.

Custom search app for Earth Science: Joint work between NASA and University of
Alabama in Huntsville, involved creating a custom search application over
scientific abstracts and papers related to Earth Science. First, using the top
level scripts in PrimeQA, they easily trained an OpenQA system on over 100k
abstracts by training a ColBERT retriever and an extractive reader. Then, they
were able to quickly deploy the search application using the create-primeqa-
app and make it available publicly999http://primeqa.nasa-impact.net/qa.

##  6 Conclusion

PrimeQA is an open-source library designed by QA researchers and developers to
easily facilitate reproduciblity and reusability of existing and future works.
This is an important and valuable contribution to the community as it enables
these models to be easily accessible to researchers and end-users in the
rapidly progressing field of QA. Our library also provides a ‘service layer’
that allows developers to take pre-trained PrimeQA models and deploy them for
their custom search application. PrimeQA is built on top of the largest NLP
open-source libraries and tools and provides simple python scripts as entry
points to easily reuse its core components across different applications. Our
easy access and reusability has already garnered significant positive traction
and enables PrimeQA to grow organically as an important resource for the rapid
state-of-the-art progress within the QA community.

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##  Appendix A Appendix

(a)  (b)  (c)

Figure 4: PrimeQA Applications

###  A.1 PrimeQA Applications

Figure 4 shows a screen-shot of three PrimeQA applications.

Datasets
---
OpenNQ
XOR-TyDi Asai et al. (2021)
ELI5 SQuAD Rajpurkar et al. (2016)
TyDiQA Clark et al. (2020b)
NQ Kwiatkowski et al. (2019c)
ELI5
SQA Iyyer et al. (2017)
WTQ Pasupat and Liang (2015b)
DocVQA Mathew et al. (2021)
WikiSQL Zhong et al. (2017b)
Table 2:  A list of some of the supported datasets in PrimeQA retriever | simple python script
---|---
reader | inference APIs
unstructured qg | inference APIs
pipeline |  inference APIs.
Table 3:  Links to PrimeQA


# Two quantum particles trapped in three dimensions harmonic oscillator and
interacting via finite range soft-core interaction

M. A. Shahzad  Department of Physics, Hazara University, Pakistan.
<EMAIL_ADDRESS>
###### Abstract

We study the exactly solvable quantum system of two particles confined in a
three-dimensional harmonic trap and interacting via finite-range soft-core
interaction by means of the separation of variables and ansatz method.
Supposing the solution in the form of ansatz $\Psi(r)=r^{a}e^{-\lambda
r^{2}}\psi(r)$ we transform the time independent Schrödinger equation into
Kummer’s differential equation whose solution are given in the form of
confluent hypergeometric function. We also discuss that in the absence of
central force potential, the quantum system map into the problem of two
quantum particle trapped in one-dimension harmonic oscillator and interacting
through finite distance soft-core potential. In such special case the eigen
value equation become the Weber’s differential equation and its solution are
also given in the form of confluent hypergeometric function.

##  I Introduction

-

Exactly solvable potentials play a very important role in various fields of
Physics. These potentials served as useful tools in modeling realistic
physical problems, and offered an interesting field of investigation in
different fields of applications in Physics.
The quantum system of particles with short range interactions in a harmonic
oscillator has been well studied. Identical particles in such a trap are
referred to as harmonium J , which has been studied as an exactly solvable
model for the artificial helium atom, where the interaction of the electrons
with the nucleus has been replaced by a harmonic confinement. Cold atomic
gases offer a promising route towards ultimate brightness conditions. In
particular Rydberg crystals are very close to the quantum plasma regime, and
extraction of the electrons could be done with a minimum amount of heating,
such that the electrons could be trapped in Paul traps and Ponderomotive
traps.
Exactly solvable quantum model bloch1  of two ultra-cold bosons confined in a
harmonic trap and interacting via contact forces play important role to
understand strongly correlated many body systems. Particularly, exact
solutions of the two particle model were essential for understanding of the
Tonks-Girardeau limit of infinite repulsions between particles bloch2 .
Recently, P. Kościk and T. Sowiński studies the exactly solvable model of two
indistinguishable quantum particles (bosons or fermions) confined in a one-
dimensional harmonic trap and interacting via finite-range soft-core
interaction bloch3 . They shown that independently on the potential range, in
the strong interaction limit bosonic and fermionic solutions become
degenerate.
Here, we present a generalization of the one-dimensional P. Kościk and T.
Sowiński model of two quantum particles (bosons as well as fermions)
interacting via the force of a finite range to the case of two particles in
three-dimensions harmonic potential. We present analytically the solutions of
the Schrödinger equation for the two quantum particle interacting via soft-
core interaction and trapped in three-dimension harmonic oscillator. We map
the three-dimensions eigenvalue equation into the Kummer’s differential
equation and presents its solution in term of confluent hypergeometric
function.

##  II The Eigenvalue problem

Consider a quantum system consists of two identical particles of mass $m$
interacting through soft-core potential and trapped in harmonic potential of
frequency $\omega$. The Schrödinger equation for the system is

$\displaystyle-{\hbar^{2}\over 2m}\Big{[}{\partial^{2}\Psi\over\partial
x_{1}^{2}}+{\partial^{2}\Psi\over\partial
x_{2}^{2}}\Big{]}+\Big{[}{m\omega^{2}\over
2}\big{(}x_{1}^{2}+x_{2}^{2}\big{)}+V(x_{1}-X_{2})\Big{]}\Psi$
$\displaystyle\qquad=E\Psi$ (1)

where the interaction potential $V(x)$ is

$V(x)=\begin{cases}V&\text{if }|x|<a\\\ 0&\text{if }|x|\geq a\end{cases}$

Using center of mass coordinate,

$R={x_{1}+x_{2}\over 2},\qquad r=x_{2}-x_{1}$ (2)

With the introduction of these new variables, the Schrödinger equation can
written as a sum of two independent single particle equation with Hamiltonian
$H=H_{R}+H_{r}$,

$\displaystyle-{\hbar^{2}\over 4m}{d^{2}\Psi\over
dR^{2}}+m\omega^{2}R^{2}\Psi=E\Psi$ (3) $\displaystyle-{\hbar^{2}\over
m}{d^{2}\Psi\over dr^{2}}+V(r)\Psi=E\Psi$ (4)

Equation (3) have the form of schrödinger equation with harmonic potential,
whose solution can be easily find. The second equation has an additional term
due to soft-core interaction $V(x)$ and can be rewritten using natural units
of the an external harmonic oscillator, that is

$\Big{[}-{d^{2}\over dr^{2}}+{1\over 4}r^{2}+V(r)\Big{]}\Psi=E\Psi$ (5)

In three dimensions, the above time independent Schrödinger equation takes the
form

$\displaystyle\Big{[}-{d^{2}\Psi\over dx^{2}}-{d^{2}\Psi\over
dy^{2}}-{d^{2}\Psi\over dz^{2}}\Big{]}$ $\displaystyle+\Big{\\{}{1\over
4}(x^{2}+y^{2}+z^{2})+V(x,y,z)\Big{\\}}\Psi(x,y,z)=E\Psi(x,y,z)$ (6)

In spherical coordinates, the time independent Schrödinger equation is given
by

$\displaystyle-\Big{[}{1\over r^{2}}{\partial\over\partial
r}\Big{(}r^{2}{\partial\Psi\over\partial r}\Big{)}+{1\over
r^{2}\sin\theta}{\partial\over\partial\theta}\Big{(}\sin\theta{\partial\Psi\over\partial\theta}\Big{)}\Big{.}$
$\displaystyle+{1\over
r^{2}\sin^{2}\theta}\Big{(}{\partial^{2}\over\partial\phi^{2}}\Big{]}+{1\over
4}r^{2}\Psi+V(r)\Psi=E\Psi$ (7)

Using method of separations of variables, we seek a solution of the form
$\Psi(r,\theta,\phi)=R(r)Y(\theta,\Phi)$. Substituting in the above equation
and separating yields the radial and angular equations;

$\displaystyle{d\over dr}\Big{(}r^{2}{dR\over
dr}\Big{)}-r^{2}\Big{[}{r^{2}\over 4}+V(r)-E\Big{]}R=l(l+1)R$
$\displaystyle{1\over
Y}\Big{\\{}{1\over\sin\theta}{\partial\over\partial\theta}\Big{(}\sin\theta{\partial
Y\over\partial\theta}\Big{)}+{1\over\sin^{2}\theta}\Big{(}{\partial^{2}Y\over\partial\phi^{2}}\Big{)}\Big{\\}}=-l(l+1)$

The solution of angular equation is called spherical harmonics. Using
$u(r)=rR(r)$, equation (8) can be written as

$-{d^{2}u\over dr^{2}}+\Big{[}V(r)+{1\over 4}r^{2}+{l(l+1)\over
r^{2}}\Big{]}u=Eu$ (8)

where the effective potential

$V_{eff}=V(r)+{1\over 4}r^{2}+{l(l+1)\over r^{2}}$ (9)

As a special care with $l=0$, equation (8) has the form of Weber differtail
equation

$\Big{[}-{d^{2}\over dr^{2}}+{1\over 4}r^{2}\Big{]}u=-\Big{[}\eta+{1\over
2}\Big{]}u$ (10)

with $\eta=-E-V-1/2$ for $|r|<a$ and $\eta=-E-1/2$ for $|r|\geq a$. For non-
interacting partiales (V=0), we get a solution for harmonic oscillator, that
is

$u_{n}(r)=N_{n}e^{-r^{2}/4}H_{n}(r/\sqrt{2}).$ (11)

where $H_{n}(x)$ is the Hermit polynomial.
For $V\neq 0$, the solution of Weber differtial equation can be written in
term of the confluent hypergeometric function ${}_{1}F_{1}$,

$u^{(+)}(r)=e^{-r^{2}/4}{{}_{1}F_{1}}\Big{[}{{\eta+1}\over 2};{1\over
2};{r^{2}\over 2}\Big{]}$ (12)

$u^{(-)}(r)=re^{-r^{2}/4}{{}_{1}F_{1}}\Big{[}{{\eta+2}\over 2};{3\over
2};{r^{2}\over 2}\Big{]}$ (13)

These function are divergent in the infinity, $r\rightarrow\pm\infty$, and
hence the appropriate solutions exists only in the region $|r|<a$.

With Coulomb interaction of the form $V=1/r$, the effective potential is

$V_{eff}={1\over r}+{1\over 4}r^{2}+{l(l+1)\over r^{2}}$ (14)

The corresponding Schroödinger equation can be written as

$-{d^{2}u\over dr^{2}}+\Big{[}{1\over r}+{1\over 4}r^{2}+{l(l+1)\over
r^{2}}\Big{]}u=Eu$ (15)

The properties of Eq.(15) with effective potential given in Eq.(14) were
discussed in detail in the literature C1 ; C2 ; C3 ; C4  and the corresponding
quantum system is referred to as harmonium. The Schroödinger equation Eq.(15)
is not analytically solvable, but rather the problem is quasi-exactly
solvable. In such quasi-exactly solvable limit the wave functions can be
written as

$u(r)=r^{l+1}e^{-r^{2}/4}P(r)$ (16)

where

$P(r)=\sum_{i=0}^{p}a_{i}r^{i},\qquad\lim_{r\rightarrow\infty}u(r)=0.$ (17)

is a polynomial of finite degree. With this Ansatz, Eq. (15) gives a
recurrence relation for the coefficients $a_{i}$, which can be solved when $i$
is given.

Now, Eq.(8) can be rewritten as

${d^{2}u\over dr^{2}}+\Big{[}\chi-{1\over 4}r^{2}-{l(l+1)\over
r^{2}}\Big{]}u=0$ (18)

where $\chi=E-V$ for $|r|<a$ and $\chi=E$ for $|r|\geq a$. Substituting

$u(r)=r^{l+1}e^{-r^{2}/2}F(r)$ (19)

in equation (18), we find that $F(x)$ satisfies Kummer’s differential equation

$x{{d^{2}F\over{dx^{2}}}}+(b-x){{dF\over{dx}}}-aF=0$ (20)

where $x=r^{2}$, $a=2l+3-\chi$, and $b=l+3/2$. The solution of Kummer’s
differential equation are confluent hypergeometric function F1 ; F2

$\displaystyle M(a,b,x)$ $\displaystyle=1+{a\over b}x+{a(a+1)\over
b(b+1)}{x^{2}\over 2!}$ $\displaystyle+{a(a+1)(a+2)\over
b(b+1)(b+2)}{x^{3}\over 3!}+....=_{1}F_{1}(a;b;x)$ (21)

with a polynomial structure of when $a=-n_{r}$, with $n_{r}=0,1,2,....$. The
quantum number $n_{r}$ is called the radial quantum number. From
$a=2l+3-\chi$, we have

$\chi=4n_{r}+2l+3=n$ (22)

The positive integer $n$ is called the principle quantum number. The solution
of the above equation for angular momentum $l$ can be written in term of
confluent hypergeometric function,

$u=r^{l+1}e^{-r^{2}/2}M\Big{(}{{2l+3-n}\over 4},l+{3\over 2},r^{2}\Big{)}$
(23)

Figure (1) shows the plot of the radial wave function $u(r)$ as a function of
$r$ for differential value of $n$ and $l$. It should be noted that the radial
wave functions with lowest value of $l$ for a given value of $n$, has no nodes
for $r>0$. The outermost maximum of the each wave function is seen to occur at
decreasing distances from the origin as $n$ increases.

Figure 1:  Radial wave function $u(r)$ (Eq.(23)) as a function of $r$ for
$n=1,3$ and $l=0,1$.

##  III Conclusion

We studied the exactly solvable quantum system of two particles trapped in a
three-dimensional harmonic oscillator and interacting via finite-range soft-
core interaction using method of separation of variables. The time independent
Schrödinger equation are mapped into the Kummer’s differential equation and
its solution are presented in the form of confluent hypergeometric function.
With in the limit of special case $l=0$, we obtained a model of two quantum
particle trapped in one-dimension harmonic oscillator and interacting through
finite distance soft-core potential. The eigen value equation become the
Weber’s differential equation which was basically used to solve Laplace
equation expressed in parabolic coordinates, and its solution are given form
of confluent hypergeometric function. The results will enabled us to
investigate different properties of the system in a whole range of parameters
between limiting cases of Busch et al. and hard-core models. The model can be
used to understand ground states of many-body system in three-dimension.

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