1 Introduction

Despite the exploit of its experimental predictions and its successes, the field theory still suffers from certain divergences, which cannot be eliminated by mathematical regularization and physical renormalization methods. This is why the theory of noncommutative has been considered as a possible alternative for understanding many physical phenomena such as ultraviolet and infrared divergence. As a result, the study of quantum mechanics on a noncommutative space has recently been very well received. Historically, the first article on quantum mechanics on a noncommutative spacetime was published in 1947 by Snyder [1, 2]. In his model, Snyder suggested a noncommutative spacetime model using the projective geometry approach of de Sitter momentum space with two universal constants. Snyder defined the spacetime coordinates operators \(X_{\mu }\) as 4-generators of “translation” of dS algebra and identified the quantity of energy \(P_{\mu }\) of a particle by non-homogeneous projective coordinates. The idea of noncommutative spacetime provides a solution to the problem of ultraviolet divergences in quantum field theory. However, this idea was abandoned due to the remarkable success of the theory of renormalization in QED.

The Snyder model is based on the algebra generated by the positions \(X_{\mu } \), momenta \(P_{\mu }\) and is given by

$$\begin{aligned} \left[ X_{\mu },P_{\nu }\right]& = {} i\hbar \left( \eta _{\mu \nu }+\beta P_{\mu }P_{\nu }\right) ;\nonumber \\ \left[ P_{\mu },P_{\nu }\right]& = {} 0;\left[ X_{\mu },X_{\nu }\right] =i\hbar \beta J_{\mu \nu }, \end{aligned}$$
(1)

where \(J_{\mu \nu }=X_{\mu }P_{\nu }-X_{\nu }P_{\mu }\) are the generators of the Lorentz symmetry and \(\beta \) is small positive parameter.

On the other hand, Einstein’s Special Relativity (E-SR) is one of the essential of modern physics, this theory assumes that there is one invariant universal constant c, i.e., speed of light and has global Poincaré–Minkowski spacetime symmetry. It is well known that the Poincaré group can be considered as a limit of the de Sitter group with the sphere radius \( R=\frac{1}{\sqrt{3\Lambda }}\rightarrow \infty ,\) where \(\Lambda \) is the cosmological constant [3]. The idea to introduce of the cosmological constant in equation of atomic physics goes back to Dirac [4]; he indicated that when \(\Lambda \ne 0\) the ordinary E-SR must be replaced by a de Sitter special relativity. Recently, it has been shown by Guo et al. [5] that there is correspondence between Snyder’s model in dS space of momenta and the dS-invariant special relativity on dS-spacetime as well as the minimum uncertainty-like relation, which means that the physics close the Planck length \(\ell _{p}\) and the radius R should be dual to each other. In addition, with a suitably chosen parametrization, Mignemi [6] showed that on the (A)dS background, the commutation relation should be modified by introducing a small correction \(\alpha \) with dimension of inverse length. On (anti-) de Sitter background, the extended commutation relations introduced by Mignemi is given by

$$\begin{aligned} \left[ X_{\mu },P_{\nu }\right]& = {} i\hbar \left( \eta _{\mu \nu }+\alpha X_{\mu }X_{\nu }\right) ; \left[ X_{\mu },X_{\nu }\right] =0;\nonumber \\ \left[ P_{\mu },P_{\nu }\right]& = {} i\hbar \alpha L_{\mu \nu }, \end{aligned}$$
(2)

where \(\mu ,\nu =0;1;2;3,\) \(L_{\mu \nu }=X_{\mu }P_{\nu }-X_{\nu }P_{\mu }\), and \(\alpha <0\) for de Sitter spacetime background and \(\alpha >0\) for anti-de Sitter spacetime background, which characterized by the presence of a nonzero minimum uncertainty in momentum. The introduction of this idea has drawn great attention in recent years, and many papers have been appeared in the literature to address the effects of the extended commutation relations on various quantum mechanical systems [7,8,9,10,11,12,13,14,15,16,17].

In this analysis, we examine problems of great importance known in quantum mechanics by various applications in the context of the extended uncertainty principle: the hydrogen atom and Kratzer molecular potentials. In order to achieve this, we consider the nonrelativistic version of extended commutation relations which is characterized by restricting the original model to its spatial part, while time is including as an external parameter and we treated the three-dimensional Schrödinger equation for the problems in question. The outline of this paper is as follows: in Sect. 2, we give a brief review of quantum mechanics with extended commutation relations. In Sect. 3, we obtain corrections to the spectrum of three-dimensional hydrogen atom using the perturbation theory. In Sect. 4, we perform the same study with the Kratzer molecular potential. Section 5 is left for concluding remarks.

2 Quantum mechanics with extended Heisenberg relation

It is well known that (A)dS spacetime can be pictured as a four-dimensional pseudosphere embedded in a five-dimensional Minkowski space with metric signature \(\eta _{ab}=\mathrm{diag}\left( \begin{array}{ccccc} 1&-1&-1&-1&\pm 1 \end{array} \right) \). The 5-coordinates are denoted by \(\zeta ^{a}\) connected by the relation \(\zeta ^{a}\zeta ^{b}\eta _{ab}=R\) , when R \(\rightarrow \infty \) the (anti-) de Sitter invariant special relativity will be reduced to ordinary special relativity. In nonrelativistic limit, we exclude the time component and in this case the space of projective coordinates is four-dimensional [8, 18, 19].

The non-relativistic modified commutation relations leading to the extended commutation relations on anti-de Sitter space is given by [6]

$$\begin{aligned} \left[ X_{j},P_{k}\right]& = {} i\hbar \left( \delta _{jk}+\alpha X_{j}X_{k}\right) , \end{aligned}$$
(3)
$$\begin{aligned} \left[ X_{j},X_{k}\right]& = {} 0, \end{aligned}$$
(4)
$$\begin{aligned} \left[ P_{j},P_{k}\right]& = {} i\hbar \alpha L_{jk}, \end{aligned}$$
(5)

where \(j,k=1;2;3,\) \(L_{jk}=X_{j}P_{k}-X_{k}P_{j}\), and \(\alpha >0\) being the constant deformation parameter. As the case of ordinary quantum mechanics, the commutation relation (3) led to the following extended uncertainty principle (EUP)

$$\begin{aligned} \left( \Delta X_{i}\right) \left( \Delta P_{i}\right) \ge \frac{\hbar }{2} \left( 1+\alpha \left( \Delta X_{i}\right) ^{2}\right) , \end{aligned}$$
(6)

which implies the appearance of a nonzero minimal uncertainty in momentum (MUM). The minimization of (6) with respect to \(\left( \Delta X_{i}\right) \) gives

$$\begin{aligned} \left( \Delta P_{k}\right) _{\min }=\hbar \sqrt{\alpha }, \forall k, \end{aligned}$$
(7)

where the parameter is bounded as \(\sqrt{\alpha }\leqslant 10-10^{6}m^{-1}\) [13]. It is possible to define different representations of the algebra (3) on a Hilbert space, and they are usually given in a position representation. One possibility is

$$\begin{aligned} X_{k}=x_{k};P_{k}=\left( \delta _{kj}+\alpha x_{k}x_{j}\right) p_{j}. \end{aligned}$$
(8)

In this representation, the momentum operators are symmetric, i.e., \( \left\langle \psi \right| P_{k}\left| \phi \right\rangle =\left\langle \phi \right| P_{k}\left| \psi \right\rangle \), this occurs if one introduces a nontrivial integration measure in the x -space [20,21,22,23]

$$\begin{aligned} D^{3}x=\frac{d^{3}x}{\left( 1+\alpha r^{2}\right) ^{2}}, \end{aligned}$$
(9)

and the squared momentum operator in the three-dimensional case takes the form

$$\begin{aligned} \left\{ \begin{array}{c} P^{2}=\sum \limits _{j=1}^{3}P_{j}P_{j}=\sum \limits _{j=1}^{3}\left( p_{j}+\alpha x_{j}\sum \limits _{k=1}^{3}x_{k}p_{k}\right) \left( p_{j}+\alpha x_{j}\sum \limits _{s=1}^{3}x_{s}p_{s}\right) \\ =-\hbar ^{2}\left( 1+\alpha r^{2}\right) ^{2}\left( \frac{\partial ^{2}}{ \partial r^{2}}+\frac{2}{r}\frac{\partial }{\partial r}\right) +\frac{{\hat{L}} ^{2}}{r^{2}} \end{array} \right. . \end{aligned}$$
(10)

The other representation of the position and momentum operators obeying relation (3) is the following coordinate space representation [9] :

$$\begin{aligned} X_{k}& = {} \frac{x_{k}}{\sqrt{1-\alpha r^{2}}}, \end{aligned}$$
(11)
$$\begin{aligned} P_{k}& = {} \frac{\hbar }{i}\sqrt{1-\alpha r^{2}}\frac{\partial }{ \partial x_{k}}, \end{aligned}$$
(12)

with \(\ -\frac{1}{\sqrt{\alpha }}<r<\frac{1}{\sqrt{\alpha }}\). In this case, the measure for which the operators \(P_{k}\) are symmetric is given by [22]

$$\begin{aligned} D^{3}x=\frac{d^{3}x}{\sqrt{1-\alpha r^{2}}}, \end{aligned}$$
(13)

and the operator \(P^{2}\) takes the form

$$\begin{aligned} P^{2}={\hbar ^{2}\alpha r}\frac{\partial }{ \partial r}-\hbar ^{2}\left( 1-\alpha r^{2}\right) \left( \frac{\partial ^{2}}{\partial r^{2}}+\frac{2}{r}\frac{\partial }{\partial r}-\frac{{\hat{L}} ^{2}}{\hbar ^{2}r^{2}}\right) . \end{aligned}$$
(14)

In the next section, we will study the nonrelativistic hydrogen atom using representation (8), and we will be interested in particular to the effect of the MUM on the energy spectrum

3 Hydrogen atom

To study the eigenvalue problem for hydrogen atom in three-dimensional case, we start considering a standard Hamiltonian:

$$\begin{aligned} H=\frac{P^{2}}{2m}-\frac{e^{2}}{R}, \end{aligned}$$
(15)

where \(R=\sqrt{\sum _{j=1}^{3}X_{j}^{2}}\) . To our knowledge, the problem of the hydrogen atom in space with deformed Heisenberg algebra has been studied in several papers. In one dimension, it is possible to get the exact energy spectrum and eigenfunctions [24, 25], while in more than one-dimension case, only perturbative solutions have been found [9, 26,27,28,29,30].

At the first order in \(\alpha \), the squared momentum operator (10) can be written as follows:

$$\begin{aligned} P^{2}& = {} \left[ -\hbar ^{2}\left( \frac{\partial ^{2}}{\partial r^{2}}+\frac{2 }{r}\frac{\partial }{\partial r}\right) +\frac{{\hat{L}}^{2}}{r^{2}}\right] \nonumber \\&-2\alpha r^{2}\hbar ^{2}\left( \frac{\partial ^{2}}{\partial r^{2}}+\frac{2 }{r}\frac{\partial }{\partial r}\right) +\mathcal {O}\left( \alpha ^{2}\right) . \end{aligned}$$
(16)

With these considerations, taking into account the above expression of squared momentum operator, the Hamiltonian (15) can be split into two parts as

$$\begin{aligned} H^{(AdS)}=H_{0}+\alpha \Delta H, \end{aligned}$$
(17)

where \(H_{0}\) is the Hamiltonian of the hydrogen atom in the ordinary space, known as the Hamiltonian of the unperturbed system

$$\begin{aligned} H_{0}=\frac{1}{2m}\left[ -\hbar ^{2}\left( \frac{\partial ^{2}}{\partial r^{2}}+\frac{2}{r}\frac{\partial }{\partial r}\right) +\frac{{\hat{L}}^{2}}{ r^{2}}\right] -\frac{e^{2}}{r}, \end{aligned}$$
(18)

and the term \(\Delta H\)

$$\begin{aligned} \Delta H=-\frac{\hbar ^{2}r^{2}}{m}\left( \frac{\partial ^{2}}{\partial r^{2}}+\frac{2}{r}\frac{\partial }{\partial r}\right) , \end{aligned}$$
(19)

is called the perturbation.

We note, however, that in the ordinary case where the perturbation term \( \alpha \left( \Delta H\right) \) vanishes (\(\alpha \) \(\rightarrow 0\)), Eq. (15) becomes the problem of the hydrogen atom whose eigenvalues and the corresponding normalized eigenfunctions are given by:

$$\begin{aligned} E_{n}^{\alpha =0}& = {} -\frac{me^{4}}{2\hbar ^{2}n^{2}}, n=1;2;\ldots , \end{aligned}$$
(20)
$$\begin{aligned} \psi _{n,l}^{\alpha =0}\left( \mathbf {r}\right)& = {} \sqrt{\frac{2\left( n-l-1\right) !}{an^{5}\left[ \left( n+l\right) !\right] ^{3}}}\left( \frac{2r }{na}\right) ^{l}e^{-\frac{r}{na}}L_{n-l-1}^{2l+1}\nonumber \\&\left( \frac{2r}{na} \right) Y_{lM}\left( \theta ,\varphi \right) ,\text { where }l=0;1;\ldots ;n-1, \end{aligned}$$
(21)

where n and l are the principal quantum number and orbital quantum numbers, respectively, \(L_{n-l-1}^{2l+1}\left( r\right) \) are Laguerre polynomials and \(a=\frac{\hbar ^{2}}{me^{2}}\) is the atomic unit of length.

Now, to study the influence of the MUM on the energy levels of the hydrogen atom, we will consider the term \(\alpha \left( \Delta H\right) \) in Eq. (17) as perturbation in ordinary Schrodinger equation. To find the first correction in the energy levels, we take the expectation value of the perturbation operator by using eigenfunctions (21)

$$\begin{aligned} \Delta E_{n,l}=\alpha \frac{\left\langle \psi _{n,l}^{\alpha =0}\right| \Delta H\left| \psi _{n,l}^{\alpha =0}\right\rangle }{\left\langle \psi _{n,l}^{\alpha =0}\right| \psi _{n,l}^{\alpha =0}\rangle }. \end{aligned}$$
(22)

So, the energy eigenvalues in AdS space can then be written as

$$\begin{aligned} E_{n,l}^{(AdS)}=E_{n}^{\alpha =0}+\Delta E_{n,l}. \end{aligned}$$
(23)

To perform this calculation, we rewrite the Hamiltonian \(\Delta H\) (19 ) as follows

$$\begin{aligned} \left\{ \begin{array}{c} \Delta H=2r^{2}\left\{ \frac{1}{2m}\left[ -\hbar ^{2}\left( \frac{ \partial ^{2}}{\partial r^{2}}+\frac{2}{r}\frac{\partial }{\partial r} \right) +\frac{{\hat{L}}^{2}}{r^{2}}-\frac{{\hat{L}}^{2}}{r^{2}}\right] -\frac{ e^{2}}{r}+\frac{e^{2}}{r}\right\} \\ =2r^{2}\left[ H_{0}-\frac{1}{2m}\frac{{\hat{L}}^{2}}{r^{2}}+\frac{e^{2}}{r} \right] =2H_{0}r^{2}+2e^{2}r-\frac{{\hat{L}}^{2}}{m} \end{array} \right. . \end{aligned}$$
(24)

Now, let us calculate the corrections to the energy levels of the hydrogen atom caused by the MUM \(\Delta E_{n,l}\) at the first order in \(\alpha \), we introduce the new form (24), the expression of the matrix elements for a hydrogen atom can be expressed as follow:

$$\begin{aligned} \Delta E_{n,l}=2\alpha E_{n}^{\alpha =0}\left\langle r^{2}\right\rangle +2\alpha e^{2}\left\langle r\right\rangle -\frac{\alpha }{m}\left\langle {\hat{L}}^{2}\right\rangle . \end{aligned}$$
(25)

Thus, to calculate the MUM effects, one has to evaluate the matrix elements

$$\begin{aligned} \left\langle r^{p}\right\rangle& = {} \frac{2\left( n-l-1\right) !}{a^{3}n^{5} \left[ \left( n+l\right) !\right] ^{3}}\int r^{2+p}\nonumber \\&\left( \frac{2r}{na} \right) ^{2l}e^{-\frac{2r}{na}}\left[ L_{n-l-1}^{2l+1}\left( \frac{2r}{na} \right) \right] ^{2}dr. \end{aligned}$$
(26)

Using Kramer’s recursive relations [31]:

$$\begin{aligned} \left\langle r\right\rangle& = {} \frac{a}{2}\left[ 3n^{2}-l\left( l+1\right) \right] , \end{aligned}$$
(27)
$$\begin{aligned} \left\langle r^{2}\right\rangle& = {} \frac{1}{2}\left[ 5n^{2}+1-3l\left( l+1\right) \right] n^{2}a^{2}, \end{aligned}$$
(28)

and the matrix elements of the operator \({\hat{L}}^{2}\)

$$\begin{aligned} \left\langle {\hat{L}}^{2}\right\rangle =\hbar ^{2}l\left( l+1\right) , \end{aligned}$$
(29)

one gets

$$\begin{aligned} \Delta E_{n,l}=\frac{1}{2m}\left( \Delta P\right) _{\min }^{2}\left[ n^{2}-l\left( l+1\right) -1\right] . \end{aligned}$$
(30)

With using Eqs. (20) and (30) and substituting in Eq. (23), the complete energy spectrum of the Hydrogen atom in the presence of a minimal uncertainty in momentum can be written as follows:

$$\begin{aligned} E_{n,l}^{(AdS)}=E_{n}^{\alpha =0}+\frac{1}{2m}\left( \Delta P\right) _{\min }^{2}\left[ n^{2}-l\left( l+1\right) -1\right] . \end{aligned}$$
(31)

The correction to the energy spectrum \(\Delta E_{n,l}\) depends on two quantum numbers \(\left( n,l\right) \) and the degeneracy of energy levels is lifted. Also, this correction is equal to zero \(\Delta E_{n,l}=0,\) for \(n=1\) and the maximal contribution is obtained for \(n>1\), \(l=0\).

In [27], Brau studied the hydrogen atom in deformed space with minimal length, and he found that the corrections to the energy spectrum are always positive. In the presence of MUM, the corrections to the energy spectrum are qualitatively similar to those associated with the minimum position uncertainty discussed in [27].

Finally, we can consider constraints on the deformation parameter. As it was mentioned in [28,29,30], the best estimation of the deformation parameter can be obtained by including the contributions of the MUM effects in the Lamb shift. Thus, from Eq. (31), the difference \(\left( L_{2S-2P}^{ex}-L_{2S-2P}^{th}\right) \) can be taken as

$$\begin{aligned} \Delta E_{2S-2P,l=0}=h\left( L_{2S-2P}^{ex}-L_{2S-2P}^{th}\right) , \end{aligned}$$
(32)

where \(L_{2S-2P}^{ex}=1057.845(9)MHz\) is the experimental Lamb shift for the \(2S-2P\) level and \(L_{2S-2P}^{th}=1057.830(6)MHz\) is the theoretical value [32]. So, the upper bound of the EUP parameter is

$$\begin{aligned} \sqrt{\alpha }\le 4.\, 292\times 10^{3}\mathrm{m}^{-1}. \end{aligned}$$
(33)

Now, let us focus on the de Sitter space. In this case, the position and momentum operators are

$$\begin{aligned} X_{k}=x_{k};P_{k}=\frac{\hbar }{i}\left( \delta _{kj}-\alpha x_{k}x_{j}\right) \frac{\partial }{\partial x_{j}}, \end{aligned}$$
(34)

and the Hamiltonian of the hydrogen atom in de Sitter space at the first order in \(\alpha \) can be obtained by replacing \(\alpha \rightarrow -\alpha \) in (17)

$$\begin{aligned} H^{(dS)}=H_{0}-\alpha \Delta H. \end{aligned}$$
(35)

Then, the energy eigenvalues of the hydrogen atom in de Sitter space are

$$\begin{aligned} E_{n,l}^{(dS)}=E_{n}^{\alpha =0}-\frac{\hbar ^{2}\alpha }{2m}\left[ n^{2}-l\left( l+1\right) -1\right] . \end{aligned}$$
(36)

In Fig. 1, we notice that the report \(E=\frac{E_{n,l}^{\alpha \left( AdS,dS\right) }}{E_{n}^{\alpha =0}}\) is presented as a function of the deformation parameter in units of the Bohr radius for values of \(\left( n \text { , }l\right) \) energetic sub-levels:

  • For the anti-de Sitter space, we clearly see that this report \(E^{(AdS)}\) is a decreasing monotonous function with the increase in the quantum fluctuations due to the extended uncertainty principle, which explains the spectrum energy on anti-de Sitter is bigger than the energy in ordinary case.

  • On the other hand for the de Sitter space, we find that this report \( E^{(dS)}\) is a increasing monotonous function with the increase in the quantum fluctuations, which explains the spectrum energy on the de Sitter is smaller than the energy in ordinary case.

Fig. 1
figure 1

Plot of the energy eigenvalues \(E=\frac{E_{n,l}^{\left( AdS,dS\right) }}{E_{n}^{\alpha =0}}\) of the Hydrogen atom in (A)dS space as a function of the deformation parameter in units of the Bohr radius \(\beta =\alpha a,\) for principal quantum numbers \(n=2\) and \(l=0,1.\)

Full size image

4 Kratzer potential

The modified Kratzer molecular potential has played a crucial role in the quantum chemistry, and it is used to describe the interactions of molecular structure in quantum mechanics. This potential appears in many fields of physics and chemistry such as molecular physics [33], nuclear physics [34], chemical physics [35], and quantum chemistry [36]. The modified Kratzer potential is defined as [37]:

$$\begin{aligned} V\left( r\right) =D_{e}\left( \frac{r-r_{e}}{r}\right) ^{2}, \end{aligned}$$
(37)

where \(D_{e}\) is the dissociation energy and \(r_{e}\) is the equilibrium internuclear separation. The Schrödinger equation for potential (37 ) is similar to that of a radial Coulomb problem with an effective value of rotational angular momentum [38, 39]. In the presence MUM, the radial part of the Schrödinger equation at the first order in \(\alpha \) for the modified Kratzer potential takes the following form:

$$\begin{aligned} \left[ H_{0}^\mathrm{Kratzer}+\alpha \left( \Delta H\right) ^\mathrm{Kratzer}\right] \psi _{n,l}^{\alpha }=E_{n,l}^{\alpha }\psi _{n,l}^{\alpha }, \end{aligned}$$
(38)

where

$$\begin{aligned} H_{0}^\mathrm{Kratzer}& = {} -\frac{\hbar ^{2}}{2m}\left[ \frac{\partial ^{2}}{\partial r^{2}}+\frac{2}{r}\frac{\partial }{\partial r}\right] \nonumber \\&+\frac{1}{2m}\frac{ {\hat{L}}^{2}+2mD_{e}r_{e}^{2}}{r^{2}}-\frac{2D_{e}r_{e}}{r}+D_{e}, \end{aligned}$$
(39)

is the unperturbed Hamiltonian and

$$\begin{aligned} \left( \Delta H\right) ^\mathrm{Kratzer}=-\frac{r^{2}\hbar ^{2}}{m}\left( \frac{ \partial ^{2}}{\partial r^{2}}+\frac{2}{r}\frac{\partial }{\partial r} \right) . \end{aligned}$$
(40)

The energy spectrum and the corresponding normalized eigenfunctions of the unperturbed Kratzer molecular potential are given by [37]

$$\begin{aligned} E_{n,l,\alpha =0}^\mathrm{Kratzer}& = {} D_{e}-\frac{2mD_{e}^{2}r_{e}^{2}}{\hbar ^{2}\left( n+\frac{1}{2}+\sqrt{\frac{2mD_{e}r_{e}^{2}}{\hbar ^{2}}+\left( l+\frac{1}{2}\right) ^{2}}\right) ^{2}}, \end{aligned}$$
(41)
$$\begin{aligned} \psi _{n,l,\alpha =0}^\mathrm{Kratzer}& = {} \frac{\left( \frac{4mD_{e}r_{e}^{2}}{\hbar ^{2}\left( \lambda +n\right) }\right) ^{\lambda +\frac{1}{2}}}{\Gamma \left( 2\lambda \right) }\sqrt{\frac{\Gamma \left( 2\lambda +n\right) }{ r_{e}^{3}2n!\left( \lambda +n\right) }}e^{-\frac{2mD_{e}r_{e}}{\hbar ^{2}\left( \lambda +n\right) }r}\left( \frac{r}{r_{e}}\right) ^{\lambda -1}\nonumber \\&F\left( -n,2\lambda ,\frac{4mD_{e}r_{e}}{\hbar ^{2}\left( \lambda +n\right) }r\right) Y_{l\mu }\left( \varphi ,\theta \right) , \end{aligned}$$
(42)

where, \(n=0;1;2;\ldots \), \(l=0;1;2;\ldots ;n\), \(F\left( a,b,r\right) \) is a confluent hypergeometric function and

$$\begin{aligned} \lambda =\frac{1}{2}+\sqrt{\frac{2mD_{e}r_{e}^{2}}{\hbar ^{2}}+\left( l+ \frac{1}{2}\right) ^{2}}. \end{aligned}$$
(43)

We return now to Eq. (38), in the deformed case (\(\alpha \ne 0\)), we can consider the term \(\alpha \left( \Delta H\right) ^\mathrm{Kratzer}\), as a perturbation to the ordinary Schrodinger equation. By using Eq. (39), we can then rewrite the Hamiltonian \(\left( \Delta H\right) ^\mathrm{Kratzer}\) in terms of \(H_{0}^\mathrm{Kratzer}\) as

$$\begin{aligned} \left( \Delta H\right) ^\mathrm{Kratzer}& = {} 2r^{2}\left[ H_{0}^\mathrm{Kratzer}-\frac{1}{2m} \frac{{\hat{L}}^{2}+2mD_{e}r_{e}^{2}}{r^{2}}\right. \nonumber \\&\left. +\frac{2D_{e}r_{e}}{r}-D_{e}\right] . \end{aligned}$$
(44)

According to the perturbation theory, at the first order in \(\alpha \) we have

$$\begin{aligned}&\Delta E_{n,l}^\mathrm{Kratzer}=2\alpha \left( E_{n,l}^{\alpha =0}-D_{e}\right) \left\langle \psi _{n,l}^{0}\right| r^{2}\left| \psi _{n,l}^{0}\right\rangle \nonumber \\&\quad +4\alpha D_{e}r_{e}\left\langle \psi _{n,l}^{0}\right| r\left| \psi _{n,l}^{0}\right\rangle -\frac{ \alpha }{m}\left\langle \psi _{n,l}^{0}\right| \left( {\hat{L}} ^{2}+2mD_{e}r_{e}^{2}\right) \left| \psi _{n,l}^{0}\right\rangle . \end{aligned}$$
(45)

By inserting the values of the matrix elements (27), (28), (29) and the expression of \(E_{n,l}^{\alpha =0}\) given by Eq. (41) into Eq. (45), we get

$$\begin{aligned} \Delta E_{n,l}^\mathrm{Kratzer}& = {} \frac{\hbar ^{2}\alpha }{2m}\left[ \left( n+\frac{ 1}{2}+\sqrt{\frac{2mD_{e}r_{e}^{2}}{\hbar ^{2}}+\left( l+\frac{1}{2} \right) ^{2}}\right) ^{2}\right. \nonumber \\&\left. -1-l\left( l+1\right) -\frac{2mD_{e}r_{e}^{2}}{ \hbar ^{2}}\right] , \end{aligned}$$
(46)

the above equation shows the influence of MUM on the energy levels of Kratzer molecular potential, we remark that the correction to the spectrum energy \( \Delta E_{n,l}^\mathrm{Kratzer}\) due to the MUM are qualitatively different to those associated with the minimal length uncertainty [40]. Finally, the total energy spectrum of the Kratzer molecular potential in the presence of MUM can be written as follows:

$$\begin{aligned} E_{n,l}^{\alpha }& = {} D_{e}-\frac{2mD_{e}r_{e}^{2}}{\hbar ^{2}}\frac{D_{e}}{ \left( n+\frac{1}{2}+\sqrt{\frac{2mD_{e}r_{e}^{2}}{\hbar ^{2}}+\left( l+ \frac{1}{2}\right) ^{2}}\right) ^{2}} \nonumber \\&\quad +\frac{\alpha \hbar ^{2}}{2m}\left[ \left( n+\frac{1}{2}+\sqrt{\frac{ 2mD_{e}r_{e}^{2}}{\hbar ^{2}}+\left( l+\frac{1}{2}\right) ^{2}}\right) ^{2}\right. \nonumber \\&\qquad \left. -1-l\left( l+1\right) -\frac{2mD_{e}r_{e}^{2}}{\hbar ^{2}}\right] . \end{aligned}$$
(47)

In order to provide a better description of the energy spectrum \( E_{n,l}^{\alpha }\) of a diatomic molecule, we use the most common equation for the vibrational energy levels of a diatomic molecule [41,42,43]:

$$\begin{aligned} E_{n,l}^{\alpha }& = {} \omega _{e}\left( n+\frac{1}{2}\right) -\omega _{e}\varkappa _{e}\left( n+\frac{1}{2}\right) ^{2}\nonumber \\&+\omega _{e}y_{e}\left( n+ \frac{1}{2}\right) ^{3}+B_{e}l\left( l+1\right) \nonumber \\&-\alpha _{e}\left( n+\frac{1}{2}\right) l\left( l+1\right) +Y_{00}, \end{aligned}$$
(48)

where \(\omega _{e}\) is the harmonic frequency, \(\omega _{e}\varkappa _{e}\) and \(\omega _{e}y_{e}\) are the anharmonicity constants, \(B_{e}\) is the rotational constant, and \(\alpha _{e}\) is the rotational-vibrational constant. According to the [43], the parameter \(\theta ^{2}=\frac{ 2mD_{e}r_{e}^{2}}{\hbar ^{2}}\) is very large for the majority of molecules. Expanding Eq. (47) for powers of \(\frac{1}{\theta }\), we get

$$\begin{aligned} E_{n,l}^{\alpha }& = \frac{2D_{e}}{\theta }\left[ 1+\frac{\alpha \hbar ^{2} }{32D_{e}m}\right] \left( n+\frac{1}{2}\right) \nonumber \\&\quad -\frac{3D_{e}}{\theta ^{2}} \left[ 1-\frac{\alpha \hbar ^{2}\theta ^{2}}{6D_{e}m}\right] \left( n+ \frac{1}{2}\right) ^{2}+\frac{4D_{e}}{\theta ^{3}}\left( n+\frac{1}{2} \right) ^{3} \nonumber \\&\quad +\frac{D_{e}}{\theta ^{2}}\left[ 1-\frac{\alpha \hbar ^{2}\theta ^{2}}{ 2D_{e}m}\right] l\left( l+1\right) \nonumber \\&\quad -\frac{3D_{e}}{\theta ^{3}}\left[ 1-\frac{ \alpha \hbar ^{2}\theta ^{2}}{12mD_{e}}\right] \left( n+\frac{1}{2}\right) l\left( l+1\right) \nonumber \\&\quad +\frac{D_{e}}{4\theta ^{2}}-\frac{\alpha \hbar ^{2}}{2m}. \end{aligned}$$
(49)

The identification of Eqs. (49) and (48) conduct to the following spectroscopic constants:

$$\begin{aligned} Y_{00}& = {} \frac{D_{e}}{4\theta ^{2}}-\frac{\alpha \hbar ^{2}}{2m}; \omega _{e}=\frac{2D_{e}}{\theta }\left[ 1+\frac{\alpha \hbar ^{2}}{ 32D_{e}m}\right] ;\nonumber \\&\omega _{e}\varkappa _{e}=\frac{3D_{e}}{\theta ^{2}}\left[ 1-\frac{\alpha \hbar ^{2}\theta ^{2}}{6D_{e}m}\right] \nonumber \\ B_{e}& = {} \frac{D_{e}}{\theta ^{2}}\left[ 1-\frac{\alpha \hbar ^{2}\theta ^{2}}{2D_{e}m}\right] ;\alpha _{e}=\frac{3D_{e}}{\theta ^{3}}\left[ 1-\frac{\alpha \hbar ^{2}\theta ^{2} }{12mD_{e}}\right] . \end{aligned}$$
(50)

Equation (49) shows the effect of the MUM on the energy spectrum of the Kratzer potential interaction. We can see that the MUM correction carries new terms in the rotational constant \(B_{e}\), which do not exist in the minimal length case [36]. Finally, Eq. (47) can be seen as a formula of a 3-parameter potential \(D_{e}\), \(r_{e}\), and \(\alpha \) and it can be used to compute the values of \(\alpha \) for any molecule.

5 Conclusion

We have studied the hydrogen atom and the molecular potential of Kratzer in space with the deformed Heisenberg algebra leading to a nonzero minimum uncertainty in momentum. For the hydrogen atom, we use the perturbation theory to calculate the correction of the energy spectrum due to EUP in a coordinate space and we find that:

  • the energy level corrections are in agreement with those associated with the minimum position uncertainty

  • the energy levels corrections decrease well the depth of the potential of the hydrogen atom

  • the upper limit of the deformation parameter is of the order \(\sqrt{ \alpha }\sim 10^{3}m^{-1}\).

For the modified Kratzer molecular potential, the energy spectrum due to the existence of MUM is derived, by assuming that the parameter \(\theta \) is very large, the spectroscopic constants are obtained. Contrary to [40], the rotation constant depends on the deformation parameter \(\alpha \) and has nonzero values.