Regularization of $1/X^2$ potential in general case of deformed space with minimal length

TL;DR

This paper defines the inverse square position operator within a deformed Heisenberg algebra framework with minimal length and analyzes the energy spectrum of a particle in this potential, revealing slight dependence on deformation functions.

Contribution

It introduces a new definition of the inverse square position operator in deformed space and studies its spectral properties.

Findings

Energy spectrum slightly depends on deformation function

Analytical and numerical solutions obtained

Potentially relevant for quantum gravity models

Abstract

In general case of deformed Heisenberg algebra leading to the minimal length we present a definition of the square inverse position operator. Our proposal is based on the functional analysis of the square position operator. Using this definition a particle in the field of the square inverse position potential is studied. We have obtained analytical and numerical solutions for the energy spectrum of the considerable problem in different cases of deformation function. We find that the energy spectrum slightly depends on the choice of deformation function.

Equations

$$[\hat{X},\hat{P}]=i\hslash (1+\beta {\hat{P}}^2),$$

$$[\hat{X},\hat{P}]=i\hslash f(\hat{P}),$$

$$\hat{X}=\hat{x}=i\hslash \frac{d}{dp},$$

$$\hat{P}=g(p).$$

$$||\phi |{|}^2={\int }_{-b}^{b}dp|\phi (p){|}^2.$$

$$\frac{dg(p)}{dp}=f(P),$$

$$l_0=\frac{\pi \hslash }{2b}.$$

$$\phi (-b)=\phi (b)=0.$$

$${\hat{X}}^2{\varphi }_n(p)={\chi }_n{\varphi }_n(p).$$

$$D({\hat{X}}^2)=\left\{\phi (p),{\phi }^{\prime \prime }(p)\in L^2\left(-b,b\right),\phi \left(-b\right)=\phi (b)=0\right\}.$$

$$-{\hslash }^2\frac{d^2{\varphi }_n(p)}{d{p}^2}={\chi }_n{\varphi }_n(p),$$

$${\chi }_n=\frac{{\hslash }^2{\pi }^2}{4b^2}{n}^2,$$

$${\varphi }_n(p)=\frac{1}{\sqrt{b}}\sin \left(n\frac{\pi }{2b}(p+b)\right),$$

$$\frac{1}{{\hat{X}}^2}=\sum _{n=1}^{\infty }|{\varphi }_n⟩\frac{1}{{\chi }_n}⟨{\varphi }_n|.$$

$$\frac{1}{{\hat{X}}^2}\varphi (p)={\int }_{-b}^{b}d{p}^{\prime }\varphi (p^{\prime })K(p,p^{\prime }).$$

$$K(p,p^{\prime })=\sum _{n=1}^{\infty }{\varphi }_n(p)\frac{1}{{\chi }_n}{\varphi }_n^{\ast }(p^{\prime }).$$

$$K(p,p^{\prime })=\frac{2b}{h^2{\pi }^2}\sum _{n=1}^{\infty }\frac{1}{n^2}\left(\cos \left[\frac{\pi n}{2b}(p-p^{\prime })\right]-\cos \left[\frac{\pi n}{2b}(p+p^{\prime }+2b)\right]\right).$$

$$\sum _{n=1}^{+\infty }\frac{cos(n\theta )}{n^2}=\frac{\pi }{6}-\frac{\pi \theta }{2}+\frac{{\theta }^2}{4},\theta \in [0,2\pi ]$$

$$K(p,p^{\prime })=-\frac{1}{2{\hslash }^2}\left(|p-p^{\prime }|+\frac{p{p}^{\prime }}{b}-b\right).$$

$$\left(\frac{1}{2m}{g}^2(p)-E\right)\psi (p)-\alpha {\int }_{-b}^{b}K(p,p^{\prime })\psi (p^{\prime })d{p}^{\prime }=0,$$

$$F^{\prime \prime }(p)=-\frac{2m\alpha }{{\hslash }^2}\psi (p),$$

$$F(b)=F(-b)=0.$$

$$\left(\frac{g^2(by)}{b^2}+\epsilon \right)f^{\prime \prime }(y)=-{\alpha }_{0}f(y),$$

$$f(1)=f(-1)=0.$$

$$[X,P]=i\hslash ,P\in [-b,b].$$

$$\left(y^2+\epsilon \right)f^{\prime \prime }(y)=-{\alpha }_{0}f(y),$$

$$f_{even}(y)=C_1(y^2+{\epsilon }^2){}_2{F}_1\left(a,b;c;-\frac{y^2}{\epsilon }\right),$$

$$f_{odd}(y)=C_{2}y(y^2+{\epsilon }^2){}_2{F}_1\left(a+1/2,b+1/2;c+1;-\frac{y^2}{\epsilon }\right),$$

$$a=\frac{3}{4}+\frac{\kappa }{2},$$

$$b=\frac{3}{4}-\frac{\kappa }{2},$$

$$c=\frac{1}{2},$$

$$\kappa =\sqrt{1/4-{\alpha }_0}.$$

$${}_2{F}_1\left(a,b;c;-\frac{1}{\epsilon }\right)=0,$$

Full text

Keywords: deformed Heisenberg algebra, minimal length, singular potential.

PACS numbers: 03.65.Ge, 02.40.Gh

1 Introduction

Idea of the minimal length have been attracted a lot of attention recently. This interest was inspired by the studies in string theory and quantum gravity, which suggest the existence of the minimal length as a finite lower bound to the possible resolution of length [1, 2, 3]. Minimal length can be achieved by modifying usual canonical commutation relations [4, 5, 6, 7]. The most popular deformed algebra is the one proposed by Kempf [4]

[TABLE]

leading to minimal length $\hbar\sqrt{\beta}$.

The one of the important problem concerning to minimal length hypothesis is to find quantum-mechanical (or classical) system with high sensitivity to deformation with minimal length. The purpose is not only in the estimation of the upper bound of minimal length but is in proving or refuting of the hypothesis of minimal length on the basis of comparison with experimental data.

In deformed space with minimum length coordinate representation does not exist. Due to this fact, it is especially interesting to study the effect of the minimum length on systems with singular potentials, since such systems are expected to have the nontrivial sensitivity to minimal length.

The implications of non-zero minimal length were considered in the context of the following problems with singularity in potential energy: hydrogen atom [8, 9, 10, 11, 12, 13, 14], gravitational quantum well [15, 16, 17], a particle in delta potential and double delta potential[19, 18], one-dimensional Coulomb-like problem [19, 21, 20], particle in the singular inverse square potential [22, 23, 24].

Particle in the field inverse square potential $1/R^{2}$ considered in paper [22, 23, 24] seems to be espesially interesting. The main feature of this system is the fact that such a system has no bound states in the undeformed case but do have the ones in case of deformed algebra with minimal length. In present paper we consider particle in one dimensional inverse square potential $1/X^{2}$ in general case of deformed algebra. The main aim this work is to study the dependence of the spectrum of the system on the choice of deformed algebra.

The paper is organized as follows. In section II we brief about general deformed algebra. In section III we present the definition of the $1/X^{2}$ operator based on the functional analysis of square position operator. In section IV Schrödinger equation in momentum representation for the particle in $1/X^{2}$ potential is reduced to the solving of the differential equation with boundary conditions. Section V contains analysys of energy spectrum for considered system in some particular cases of deformation. Finally, in section V we conclude obtained results.

2 Deformed algebras and minimal length

Let us consider a modified one-dimensional Heisenberg algebra which is generated by position $\hat{X}$ and momentum $\hat{P}$ hermitian operators satisfying the following relation

[TABLE]

where $f$ are called functions of deformation. We assume that it is strictly positive ($f>0$), even function.

We consider a representation leaving position operator undeformed

[TABLE]

The position and momentum operators in representation (3) act on square integrable function $\varphi(p)\in\it{L}^{2}(-b,b),(b\leq\infty)$. The norm of $\varphi$ is given by

[TABLE]

From the fact that operators $\hat{X}$ and $\hat{P}$, written in representation (3), have to satisfy the commutation relation (2), we obtain the following differential equation for $g(p)$

[TABLE]

with $P=g(p).$ Function $g({p})$ is an odd function defined on $[-b,b]$, with $b=g^{-1}(a)$. Here $a$ represents the limits of momentum $P\in[-a,a]$. Minimal length for the deformed algebra is [25]

[TABLE]

Thus, if $b<\infty$ nonzero minimal length exists and if $b=\infty$ the minimal length is zero. Here it is important to note that the result obtained in [25] concerning the minimal length depends on the choice of boundary condition for wavefunction. Formula (6) was derived with assumption of zero boundary conditions

[TABLE]

Alternative choice of the boundary conditions may lead to elimination of the minimal length even for finite $b$ [26].

3 Definition of $1/X^{2}$ operator

To define square inverse distance operator let us consider the eigenproblem for $\hat{X}^{2}$

[TABLE]

In representation (3) the square distance operator has the form $\hat{X}^{2}=-\hbar^{2}\frac{d^{2}}{dp^{2}}$ and acts on the domain

[TABLE]

Now the eigenproblem (8) writes

[TABLE]

with $\phi_{n}(\pm b)=0$.

Note, that this problem is rather similar to the problem of particle in a box in undeformed space. The solution of the problem is

[TABLE]

with $n=1,2,\dots\ .$

We define the square inverse distance operator as

[TABLE]

From this definition we see that the action of the square inverse distance operator ${1}/{\hat{X}^{2}}$ on any function belonging to its domain can be presented by

[TABLE]

with

[TABLE]

being the kernel of square inverse distance operator . Substituting the expression for $\chi_{n}$ and $\phi_{n}(p)$ from (11) and (12) into (15) and using product-to-sum identity for sine functions we write

[TABLE]

Using the following formula[Abramovic]

[TABLE]

we can finally write the kernel of square inverse distance operator in the following form

[TABLE]

Thus, (14) together with (18) presents the definition of the square inverse distance operator in representation (3). Note, that $K(\pm b,p^{\prime})=0$, which means that the action of the square inverse distance operator returns the wavefunction belonging to the domain of square distance operator (9).

4 $1/\hat{X}^{2}$ quantum well and minimal length

The Schrödinger equation for the particle in potential $-\alpha/\hat{X}^{2}$ with minimal length can be written as

[TABLE]

where $K(p,p^{\prime})$ is given by (18). By changing $p$ for $-p$ in (19) we can show that the Hamiltonian commutes with the parity operator, thus its eigenfunctions can be chosen as even or odd functions, i.e. as parity eigenstates.

Differentiating twice the Shrödinger equation we obtain

[TABLE]

with $F(p)=\left(g^{2}(p)-2mE\right)\psi(p).$

Expressing $\psi(p)$ from (20) and substituting it into (19) we obtain that both even and odd eigenfunctions have to satisfy the following conditions

[TABLE]

From these conditions the energy spectrum can be found.

In the dimensionless form equation (20) can be written as

[TABLE]

with $y=\frac{p}{b},\ y\in[-1,1]$, $\varepsilon=-\frac{2mE}{b^{2}}$, $\alpha_{0}=\frac{2m\alpha}{\hbar^{2}}$ and $f(y)=F(by)$. Function $f(y)$ have to satisfy the following conditions with

[TABLE]

Thus, the problem is reduced to the solving of equation (22) with boundary conditions (23).

5 Energy spectrum for special cases of deformation function

In this section we present energy spectrum of considerable problem for some special examples of deformation function.

5.1 Example 1

Let us consider the simplest deformed commutation relation leading to minimal length

[TABLE]

In this case $g(p)=p$ and equation (22) writes

[TABLE]

Equation (25) has two linearly independent solutions finite at $y=0$, even and odd with respect to $y$:

[TABLE]

with

[TABLE]

Here$\ {}_{2}F_{1}$ denotes hypergeometric function. The energy spectrum can be found from the conditions

[TABLE]

In the case of weakly attractive potential $0<\alpha_{0}\leq 1/4$, which correspondes to the case of real $\kappa$, equation (32) has one solution $\epsilon_{0}$, while (33) has no solutions (see Fig.1).

Analitical expression for $\varepsilon_{0}$ can be found from (32) using the following property for hypergeometric function

[TABLE]

with

[TABLE]

With the assumption of small $\varepsilon_{0}$ condition (32) reads

[TABLE]

From (36) we obtain the energy

[TABLE]

which is in very good coincidence with the numerical results (see Fig.2). It is interesting that in case of infinitesimal coupling constant $\alpha_{0}$ there is bound state with infinitesimal energy $\varepsilon\approx\pi^{2}\alpha_{0}^{2}/4$, while in case of $\alpha_{0}=0$ obviously there is no bound state with zero energy. This fact correspondes to the vanishing of the eigenfunction (26) in the limit of $\alpha_{0}$ to zero.

In case of strongly attractive potential $\alpha_{0}>1/4$, parameter $\kappa$ become complex

[TABLE]

Using (5.1) in the limit of small $\varepsilon$ conditions on energy spectrum (32) and (33) take the form

[TABLE]

with

[TABLE]

Both equations (39) has many solutions accumulating at $\varepsilon=0$. The energy spectrum consists of two branches corresponding to even and odd eigenfunctions

[TABLE]

with $n,m=0,1,2,\dots$. However this formulas incorrectly describes the energy spectrum of the system, when the condition of smallness of the values of energy $\varepsilon\ll 1$ is not fulfiled (see Fig.3).

Analitical behavior for large energies can be obtained from the Bohr-Sommerfeld quantization rule in deformed space with minimal length (see [27])

[TABLE]

with $n\in N$ and $\delta\in[0,1)$. For considerable system $H(X,P)=\frac{P^{2}}{2m}-\frac{\alpha}{X^{2}}$ from equation $H(X,P)=E$ we find

[TABLE]

Thus, we write quantization rule as

[TABLE]

In dimensionless form (46) reads

[TABLE]

From (47) we obtain the energy spectrum

[TABLE]

In the limit of very strong attraction $\alpha_{0}\rightarrow\infty$ energy spectrum linearly depends on coupling constant $\alpha_{0}$

[TABLE]

Comparison of the numerical results and the one’s obtained from Bohr-Sommerfeld quantization rule with $\delta=0$ for ground and first excited state is presented on Fig. 4.

From Fig. 4 we see that asymptotic behavior at $\alpha_{0}\rightarrow\infty$ of the energy spectrum obtained from WKB aproximation method given in (49) coinsides with the numerical results.

Thus, one dimensional inverse square potential is regularized in deformed space with minimal length. Regularization of Coulomb potential in frame of electrodynamics in space with minimal length is presented in [28].

5.2 Example 2

Let us consider the following deformation function

[TABLE]

In this case equation (22) takes the form

[TABLE]

Both even and odd with respect to $y$ solutions of later equation can be presented by confluent Heun function

[TABLE]

Energy spectrum can be found from the conditions

[TABLE]

In the limit of $\varepsilon\rightarrow 0$ using the following relation between hypergeom and confluent Heun function

[TABLE]

it can be shown that (55) and (56) coinsides with (32) and (33) respectively.

Thus, the energy spectrum for this type of deformation function is close to the one obtained by a cutoff regularization. Such a conclusion can also be made on the basis of the numerical calculation (see Fig. 5)

5.3 Example 3

We also provide the numerical calculations of the energy spectum for the following deformation functions

[TABLE]

and provide the comparison with the corresponding results for the first example of function of deformation.

As we can see from Fig. 6 the values of the energy for the ground state and the first exited state slightly depends on the choice of deformation function in case of coupling constant $\alpha_{0}<1$. This conclusion is valid also for the case for the function of deformation considered in Example 2. Ground state energy is almost the same for considered types of deformations (including Example 2) in wide range of coupling constant. The maximal difference of the ground state energies is $\frac{\varepsilon_{max}(\alpha_{0})-\varepsilon_{min}(\alpha_{0})}{\varepsilon_{min}(\alpha_{0})}<6\%$ for any choice of $\alpha_{0}$.

Let us consider two Hamiltonians of the particle in the inverse square potential for two different derormed space.

[TABLE]

with $p\in[-b,b]$ for both the Hamiltonians. Howewer the definition of the inverse square potential does not depends on the choice of deformation function $g(p)$ we conclude that $\langle\phi|H_{b}|\phi\rangle>\langle\phi|H_{a}|\phi\rangle$ if $g^{2}_{b}(p)>g^{2}_{a}(p)$ for all the values of $p$. If we consider ground state eigenfunction $\phi_{b}$ of the Hamiltonian $H_{b}$ we can write for the energy of ground state of the Hamiltonian $H_{b}$ the following relation $E_{b}=\langle\phi_{b}|H_{b}|\phi_{b}\rangle>\langle\phi_{b}|H_{a}|\phi_{b}\rangle$. But we know that $\langle\phi_{b}|H_{a}|\phi_{b}\rangle>\langle\phi_{a}|H_{a}|\phi_{a}\rangle=E_{a}$, with $\phi_{a}$ being the ground state eigenfunction of the Hamiltonian $H_{a}$. Finally we conclude that if $g^{2}_{b}(p)>g^{2}_{a}(p)$ for all the values of $p$ the following condition is hold $E_{b}>E_{a}$. This result can also be applicable for excited states. On Fig. (7) we present plots of kinetic energies $g^{2}(p)$ for the consedered examples of deformed space. We see that among considered deformed spaces the one given in Ex. 3b and Ex. 3a have the biggest and the smallest ground state energy correspondingly. This results is also confirmed by numerical calculations.

6 Conclusion

In this paper we have studied the general case of deformed Heisenberg algebra leading to the minimal length. We have proposed the definition of square inverse position operator (14). This definition has been obtained from the functional analysis of the square position operator in deformed space with minimal length. Using definition of the square inverse position operator we have studied quantum particle in the field of inverse square potential in the general case of deformed Heisenberg algebra leading to the minimal length. The problem has been reduced to solving of the equation (22) with boundary condition (23).

Particular examples of the deformation function have been studied. In the simplest case of deformation function (24), which corresponds to cutoff procedure in momentum space, we have obtained the eigenfunctions in terms of hypergeometric function and transcendental equation for energy spectrum. In the case of weakly attractive potential $0<\alpha_{0}\leq 1/4$ energy spectrum consists of one energy level (32) corresponding to even eigenfunction with respect to momentum. We have derived the aproximate analytical formula (37) for this energy level which is in very good coincidence with the numerical calculations. It is interesting that in case of infinitesimal coupling constant $\alpha_{0}$ there is bound state with infinitesimal energy, while in case of $\alpha_{0}=0$ there is no bound state with zero energy. In case of strongly attractive potential $\alpha_{0}>1/4$ the energy spectrum consists of two branches of energies accumulating at $\varepsilon=0$ corresponding to even and odd eigenfunctions. We have obtained aproximate analytical formulas for this two branches in the assumption of small energies. From the numerical solutions for energy spectrum we have concluded that this approximation is valid in case of $\alpha_{0}<1$. From numerical calculation it have been shown the linear dependence of the energies on coupling constant $\alpha_{0}$ in the limit of big $\alpha_{0}$. The same dependence have been obtained analytically from the Bohr-Sommerfeld quantization rule for deformed space with minimal length.

In the second example of deformation function (50) we have obtained the eigenfunctions in terms of confluent Heun function and transcendental equation for energy spectrum. In the approximation of small energies we have shown that energy spectrum for deformation function (50) coinsides with the one obtained in the case of deformation function (24). This conclusion have been confirmed by numerical calculation.

Finally numerical solution of the equation (22) with boundary condition (23) have been obtained for two more examples of deformation functions (58) and (59). The values of the energy for the ground state and the first exited state slightly depends on the choice of deformation function in case of coupling constant $\alpha_{0}<1$. This conclusion is valid for all considered examples of the functions of deformation. Ground state energy is almost the same for considered types of deformations in wide range of coupling constant. The maximal difference of the energies does not exceed $6\%$.

7 Acknowlegments

This work was partly supported by the grant of the President of Ukraine for support of scientific researches of young scientists (F-75).