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

TL;DR

This paper presents an exact solution for a two-particle quantum system in three dimensions with harmonic confinement and finite-range soft-core interaction, using separation of variables and special functions.

Contribution

It introduces an analytical approach to solve the Schrödinger equation for two interacting particles in 3D harmonic traps with finite-range interactions, extending previous models.

Findings

Solutions expressed via confluent hypergeometric functions.

Mapping to 1D harmonic oscillator in absence of central potential.

Eigenvalue equations derived from Weber's differential equation.

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 Schr\"{o}dinger 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.

Equations

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

$$=E\Psi$$

$$V(x)=\{\begin{array}{ll}V& \text{if }|x|<a\\ 0& \text{if }|x|\ge a\end{array}$$

$$R=\frac{x_1+x_2}{2},r=x_2-x_1$$

$$-\frac{{\hslash }^2}{4m}\frac{d^2\Psi }{d{R}^2}+m{\omega }^2{R}^2\Psi =E\Psi$$

$$-\frac{{\hslash }^2}{m}\frac{d^2\Psi }{d{r}^2}+V(r)\Psi =E\Psi$$

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

$$\displaystyle\Big{[}-{d^{2}\Psi\over dx^{2}}-{d^{2}\Psi\over dy^{2}}-{d^{2}\Psi\over dz^{2}}\Big{]}$$

$$\displaystyle+\Big{\{}{1\over 4}(x^{2}+y^{2}+z^{2})+V(x,y,z)\Big{\}}\Psi(x,y,z)=E\Psi(x,y,z)$$

$$\displaystyle-\Big{[}{1\over r^{2}}{\partial\over\partial r}\Big{(}r^{2}{\partial\Psi\over\partial r}\Big{)}+{1\over r^{2}\sin\theta}{\partial\over\partial\theta}\Big{(}\sin\theta{\partial\Psi\over\partial\theta}\Big{)}\Big{.}$$

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

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

$$\displaystyle{1\over Y}\Big{\{}{1\over\sin\theta}{\partial\over\partial\theta}\Big{(}\sin\theta{\partial Y\over\partial\theta}\Big{)}+{1\over\sin^{2}\theta}\Big{(}{\partial^{2}Y\over\partial\phi^{2}}\Big{)}\Big{\}}=-l(l+1)$$

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

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

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

$$u_n(r)=N_n{e}^{-r^2/4}{H}_n(r/\sqrt{2}).$$

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

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

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

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

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

$$P(r)=\sum _{i=0}^p{a}_i{r}^i,\underset{r\to \infty }{\lim }u(r)=0.$$

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

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

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

$$M(a,b,x)$$

$$+\frac{a(a+1)(a+2)}{b(b+1)(b+2)}\frac{x^3}{3!}+....{=}_1{F}_1(a;b;x)$$

$$\chi =4n_r+2l+3=n$$

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

Full text

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

[TABLE]

where the interaction potential $V(x)$ is

[TABLE]

Using center of mass coordinate,

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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;

[TABLE]

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

[TABLE]

where the effective potential

[TABLE]

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

[TABLE]

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

[TABLE]

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 ${}_{1}F_{1}$,

[TABLE]

[TABLE]

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

[TABLE]

The corresponding Schroödinger equation can be written as

[TABLE]

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

[TABLE]

where

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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,

[TABLE]

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.

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.