In this work, we determine the Fisher and Shannon entropies, the expectation values and the squeeze state for a noncentral inversely quadratic plus exponential Mie-type potential analytically. The proposed potential is solved under the Schrödinger equation using a special Greene Aldrich approximation to the centrifugal term to obtain a normalised wave function within the framework of the Nikiforov–Uvarov method. Numerical results are obtained for different screening parameters: α = 0.1, 0.12 and 0.13 for varying real constant parameter (B). The numerical solutions are obtained only for ground state. The numerical results of Fisher entropy both for position and momentum spaces are in good agreement with existing literature. The normalisation constant, wave function, and probability density plots are carried out using a well designed Mathematica algorithm. The Fourier transform of position space entropy gives the momentum space entropy.
Ituen B Okon, Cecilia N Isonguyo, Akaninyene D Antia, Akpan N Ikot, Oyebola O Popoola. Fisher and Shannon information entropies for a noncentral inversely quadratic plus exponential Mie-type potential[J]. Communications in Theoretical Physics, 2020, 72(6): 065104. DOI: 10.1088/1572-9494/ab7ec9
1. Introduction
Entropic measures and information theory in general provides a clear understanding of quantum mechanical systems. In recent years, the study of bound state and scattering state solutions for both relativistic and non-relativistic wave equations has a gained wider interest because of their potential applications, particularly in the areas of information entropies and quantum technologies. Information entropy is a significant tool in studying the electronic structure of atoms and molecules. The Fisher information measurement is used as a tool for characterising complex signals of quantum mechanical systems with applications to biology, atomic physics and other related science disciplines [1–13]. Shannon, Fisher and other quantum information entropies usually measure the spread of probability distribution for allowed quantum mechanical states in a D-dimensional space [14–18]. Quantum information theory has a direct relationship with the Heisenberg uncertainty principle, which plays a very significant role in the simultaneous measurement of position and momentum of quantum mechanical particles. In 1948, a new uncertainty relation, based on Shannon entropy, was established as a basic tool for investigating the fundamental limit of signal processing [19, 20]. In this work, we developed a novel potential called a Noncentral Inversely Quadratic plus exponential Mie-type potential to study Fisher and and Shannon entropies, their expectation values, and their squeeze state, using suitable real constant parameters. This potential has applications in signal processing. The wave function and probability density plots obtained in this work reveal some significant properties and characteristics of quantum systems. We have discovered that all odd values of the wave function give the property of a wave function, while even values of the wave function plots give probability density. Larger values of the Shannon entropy are indicative of a more delocalised density while smaller values are associated with localised distribution. This means that Shannon entropy increases with an increase in uncertainty and vice versa [21]. There are other information entropies as investigated by various researchers. Recently Jen-Hao and Yew Ho carried out fantastic work on the benchmark calculations of Renyi, Tsallis and Onicescu information entropies for ground state helium using a correlated Hylleraas wave function [22]. The new Shannon entropy was first introduced by Beckner, Bialynicki-Birula and Mycieslki in 1975 as ${S}_{x}+{S}_{p}\geqslant D(1+{log}\pi )$ [23–25] where D represents the spatial dimension. However, the position Sx and momentum Sp is defined as [26–29].
where &PSgr;(x) is a normalised eigen function in a spatial coordinate and φ(x) is the normalised Fourier transform. In general, the momentum space entropy is obtained by taking the Fourier transform of position space in a spatial coordinate [30, 31]. Equations (1) and (2) give the measure of the spread of a single particle density of position and momentum space respectively. Equation (1) and (2) can also be expressed as
where $\rho (x)={\left|{\rm{\Psi }}(x)\right|}^{2}$ and $\rho (p)={\left|{\rm{\Psi }}(p)\right|}^{2}$ respectively. Equation (3) can further be expressed as
This work is divided into seven sections. Section 1 gives a brief introduction of the article. In section 2, we provide an overview of the generalised parametric Nikiforov–Uvarov method. In section 3 we solve analytically the radial solution of a Schrödinger wave equation to obtain a normalised wave function and the energy-eigen equation. In section 4 we apply the normalised wave function obtained in section 3 to analytically obtained position and momentum space Shannon entropy. In section 5, we obtain analytical solutions to Fisher position and momentum space entropies. We present the numerical solutions , their expectation values and the squeeze state in section 6. Results and discussion are presented in section 7 while section 8 gives the conclusion of the article. The adopted parameter values for both the wave function and probability density plots were all real constants. We discover that the proposed potential is most suitable for ground state energy. This is a further reason why our plots for wave functions and probability densities are all plotted for n, l = 0 using a well designed rigorous mathematica algorithm. We also discover that at higher state, that is for n, l > 0, The plots were not suitable in providing solutions to the information measures of Shannon and Fisher entropies for this particular potential. Therefore, this work is limited to ground state energy and wave function. The normalisation constant, the wave function plots, the probability density plots, and all numerical computations were carried out using Mathematica. The graph for the noncentral potential is given in figure 1, while that of the Greene Aldrich approximation and special Greene Aldrich approximation are presented in figures 1(a) and (b) respectively. The combination of the noncentral and exponential Mie-type potential is significant in the study of vibrational and rotational energies of diatomic molecules and their degeneracies for a particular quantum state [32]. In most cases, the topological properties of atoms of diatomic molecules, including their chemical functional groups, are evaluated using Density Functional theory(DFT) and pseudopotential formalism [33]. Inversely quadratic potential is a long range potential and in combination with other exponential type potentials may be used in finding the shape of organic molecules such as cyclic polyenes and benzene [22].
Figure 1. The graph of the Noncentral potential. (a) The centrifugal term $\tfrac{1}{{r}^{2}}$ for standard Greene Aldrich approximation with varying α = 0.1 to 0.5. (b) The centrifugal term $\tfrac{1}{{r}^{2}}$ for special Greene Aldrich approximation with varying α = 0.1 to 0.5. (c) The normalised wave function plot 1. (d) The normalised wave function plot 2. (e) The normalised wave function plot 3. (f) The probability density function plot 1. (g) The probability density function plot 2. (h) The probability density function plot 3. (i) The normalised wave function plot 1(a). (j) The normalised wave function plot 2(a). (k) The normalised wave function plot 3(a). (l) The probability density function plot 1(b). (m) The probability density function plot 2(b). (n) The probability density function plot 3(b).
2. The generalised parametric Nikiforov–Uvarov method
The NU method was presented by Nikiforov and Uvarov [31] and has been employed to solve second order differential equations such as Schrödinger wave equations (SWE), Klein–Gordon equations (KGE), Dirac equations (DE), etc. The Schrödinger wave equation is given as
correspondingly, for the purpose of this work, we shall be considering the exact solution of a Schrödinger equation where the orbital angular quantum number l = 0, hence equation (9) reduces to
Equation (10) can be solved by transforming it into an hypergeometric type equation using the transformation s = s(x) and its resulting equation is expressed as:
where σ (s) and $\tilde{\sigma }(s)$ must be polynomials of at most second degree, $\tilde{\tau }(s)$ is the first degree polynomial, and ψ (s) is a function of the hypergeometric type. The parametric generalisation of the NU method is given by the generalised hypergeometric type equation 12.
Equation (12) has been applied to provide bound state solutions to both relativistic and non-relativistic wave equations with considerable potential such as Hulthen, Eckart, Coulomb, Pseudoharmonic and many others [34–41]. Equation (12) is solved by comparing with equation (11) and the following polynomials are obtained.
where A is the potential depth in electron volts, B, η and C are real constant parameters. α is the screening parameter. The graph of this potential is shown in figure 1.
To solve equation (18) analytically, we treat sin $\alpha $ as a constant trignometric function that tends to 1 and define special Greene Aldrich approximation to the centrifugal term as
By comparing the graph of a standard Greene Aldrich approximation in figure 1(a) to a special Greene Aldrich approximation in figure 1(b), it can be observed that figure 1(a) converges asymptotically for different values of the screening parameter α = 0.1 to 0.5. The same thing is also applicable to figure 1(b). This signifies that the special Greene Aldrich approximation could be approximated to the standard Greene Aldrich approximation; this serves as a good approximation to the proposed potential. Substituting equation (19) into 18, and making use of equation (20), equation (18) is reduced to a hypergeometric type equation:
The normalised wave function and probability density curves for the first set of real constant parameters are given below.
The graph for the second set of adopted parameters, with specific adjustable screening parameters, were also plotted for wave function and probability density for the sake of comparison, as shown in the figures below.
Figures 1(c), (d) and (e) are wave function plots for the first set of real constant parameters for &PSgr;, ${\left|{\rm{\Psi }}\right|}^{2}$, and ${\left|{\rm{\Psi }}\right|}^{3}$, respectively, while figures 1(i), (j) and (k) represent the wave function plots &PSgr;, ${\left|{\rm{\Psi }}\right|}^{2}$, and ${\left|{\rm{\Psi }}\right|}^{3}$, respectively, for the second set of real constant parameters. However,by observation, the wave function plots for these two sets of real parameters are almost the same, and this enables us to propose a generalised conclusion that odd value wave functions have the same properties, especially when the wave function is continuous, having continuous partial derivatives. A careful observation of the probability density in figures 1(f), (g) and (h) for the first set of parameter and (l), (m) and (n) for the second set of parameter shows that even values of the wave function represent the probability density for a continuous wave function having continuous partial derivatives.
4. Position and momentum space for Shannon entropy
In order to determine both position and momentum space for Shannon entropy, one needs to calculate the probability density from the wave function. Meanwhile, the wave function must be normalised. First, we simplify equation (25) for easy normalisation. Assuming that ${\sigma }_{1}=\sqrt{{\varepsilon }^{2}+{\chi }_{3}}$ and $\ {\sigma }_{2}=\sqrt{\tfrac{1}{4}+{\chi }_{1}+2{\chi }_{2}}$ Then equation (25) reduces to
From (20), the integration boundaries from $(-\infty ,+\infty )$ in r-dimension change to (0,1) in s-dimension and by making use of equation (3), the Shannon entropy for position space for the noncentral potential is given as
The corresponding normalised wave function in momentum space is derived by taking the Fourier transform of the position space wave function. The Fourier transform is given as
Substituting equation (43) into equation (4) gives the Shannon momentum entropy, which gives a highly complicated integral of a regularised confluent hypergeometric type as shown in equation (44)
From (20), if α is very small such that $\sin \alpha \to 1$, then by expressing equation (45) in terms of (20), the position space information entropy reduces to
In order to carry out the derivative of equation (26), which involves the Jacobi polynomial, it should be noted that the derivative of classical orthogonal polynomials of the same family but with different parameters can be done using orthogonality Pearson’s relations. This is applicable to hypergeometric functions of Jacobi, Laguerre and Hermite polynomials as [44]
where ${P}_{n}^{\left(\alpha ,\beta \right)}\left(x\right)$, ${L}_{n}^{\left(\alpha \right)}\left(x\right)$ and ${H}_{n}\left(x\right)$ are Jacobi, Laguerre and Hermite polynomials, respectively. Substituting equation (48) into (46) gives the Fisher entropy for position space as
Equation (49) is a complicated integral. However, in this article we evaluate only for the Fisher and Shannon information entropies for the ground state, that is for n = 0. The Fisher information at ground state is given as
The integral of equation (50) is carried using mathematica. Hence, the solution to equation (50) expressed in terms of regularised hypergeometric function is given as
Equation (52) is the position space Fisher entropy for the proposed noncentral potential. However, taking the Fourier transform of the position space Fisher entropy gives the corresponding momentum space Fisher entropy. This is obtained by substituting equation (43) into (45) which gives a more complicated integral.
6. Numerical computations for Fisher information entropy
In this section, we carry out numerical computations for both position and momentum Fisher entropies. we calculated the expectation values of $\langle r\rangle $, $\langle {r}^{2}\rangle $, $\langle p\rangle $, and $\langle {p}^{2}\rangle $. We calculated the Heinsenberg uncertainties in position and momentum using the variance relations $\langle r\rangle =\sqrt{\langle {r}^{2}\rangle -\langle r{\rangle }^{2}}$ and $\langle p\rangle \,=\sqrt{\langle {p}^{2}\rangle -\langle p{\rangle }^{2}}$. The Heinsenberg uncertainty principle is the product of uncertainties of position and momentum, which is expressed in terms of Planck’s constant as ${\rm{\Delta }}(r){\rm{\Delta }}(p)\geqslant \tfrac{{\hslash }}{2\pi }$. Meanwhile the expectation values are calculated numerically using the following expressions:
The adopted real numerical constants parameters are: $A=0.5\,\mathrm{eV}$ representing the potential depth, B varies infinitesimally from 0.001 to 0.03 in the step of 0.001, η = 0.01 and ℏ = 1.
7. Results and discussion
First, we demonstrated from figures 1(c) to (n) that in general, odd powers of wave function give the property of a wave function while even powers of wave function represent the probability density for a continuous wave function with continuous partial derivatives. The wave function and probability density curves were plotted for two set of parameters: (A = 0.01, B = 0.02), (A = 0.02, B = −0.04), (A = 0.04, B = −0.08), (A = 0.08, B = 0.01) and (A = 0.01, B = 0.02), (A = 0.02, B = −0.04), (A = 0.05, B = −0.07), (A = −0.002, B = −0.1).
The numerical computations for position and momentum Fisher information for the three different values of α=0.1, 0.12 and 0.13 give perfect results for values of variation parameter B, because it is expected that ${I}_{r}.{I}_{p}\geqslant 36.0$ which is shown in tables 1, 2 and 3. The computation of various expectation values for different values of α also gave a good result, which is in agreement with existing literature. Here, it is expected that the Heinsenberg uncertainty relation of the product of position and momentum entropies should be given as as ${\rm{\Delta }}(r){\rm{\Delta }}(p)\geqslant \tfrac{{\hslash }}{2\pi }$ such that the squeeze state of ${\rm{\Delta }}{(r)}^{2}{\rm{\Delta }}{(p)}^{2}$ should be a minimum of 0.250 00. Tables 4, 5 and 6 gives the various expectation values for different values of α and their corresponding squeeze state. This table is in agreement with existing literature with a squeeze state value of more than 0.250 00. The expectation values of $\langle r\rangle $, $\langle {r}^{2}\rangle $ decrease with an increase in the variation parameter (B), while $\langle {p}^{2}\rangle $ and Δ (p) increase with an increase in the variation parameter (B) for α = 0.1, 0.12, and 0.13, respectively, as shown in tables 4, 5 and 6. The graph of position entropy in figure 2 for different values of α increases exponentially. However, the graph of momentum entropy as presented in figure 3 shows a clear distinction and quantisation of momentum theory for different values of α. Figure 4 is the graph of the squeeze state which is the product of position and momentum entropy with variation parameter (B) for different values of α. This graph gives a perfect curve with distinct quantisation for different values of α.
Figure 4. The product of position and momentum entropy for n = 0.
Table 1. Numerical results for Fisher information for n = 0, with various values of B, with A = 0.5 eV for the noncentral potential (NCP) for α = 0.1.
B
Ir
Ip
${I}_{r}{I}_{p}\geqslant 36.0$
min(${I}_{r}{I}_{p}$)
0.001
0.96355
97.0367
93.4997
36.0
0.002
1.01546
96.2569
97.7455
36.0
0.003
1.06619
95.4925
101.814
36.0
0.004
1.11572
94.7429
105.707
36.0
0.005
1.16403
94.0077
109.428
36.0
0.006
1.21111
93.2865
112.981
36.0
0.007
1.25695
92.5789
116.367
36.0
0.008
1.30155
91.8843
119.592
36.0
0.009
1.34489
91.2025
122.657
36.0
0.010
1.38696
90.5330
125.566
36.0
0.011
1.46732
89.8755
128.322
36.0
0.012
1.42777
89.2295
130.928
36.0
0.013
1.50559
88.5949
133.387
36.0
0.014
1.54259
87.9711
135.703
36.0
0.015
1.57832
87.3580
137.879
36.0
0.016
1.61279
86.7553
139.918
36.0
0.017
1.64598
86.1625
141.822
36.0
0.018
1.67791
85.5795
143.595
36.0
0.019
1.70858
85.0061
145.240
36.0
0.020
1.73800
84.4418
146.760
36.0
0.021
1.76617
83.8865
148.157
36.0
0.022
1.79309
83.3400
149.436
36.0
0.023
1.81877
82.8020
150.598
36.0
0.024
1.84322
82.2723
151.646
36.0
0.025
1.86645
81.7507
152.584
36.0
0.026
1.86647
81.2369
153.413
36.0
0.027
1.90928
80.7309
154.138
36.0
0.028
1.92889
80.2323
154.759
36.0
0.029
1.94732
79.7410
155.281
36.0
0.030
1.96456
79.2569
155.705
36.0
Table 2. Numerical results for Fisher information for n = 0, with various values of B, with A = 0.5 eV for the noncentral potential (NCP) for α = 0.12.
B
Ir
Ip
${I}_{r}{I}_{p}\geqslant 36.0$
min(${I}_{r}{I}_{p}$)
0.001
1.04867
64.3148
67.4451
36.0
0.002
1.09086
63.8761
69.6802
36.0
0.003
1.13221
63.4448
71.8329
36.0
0.004
1.17271
63.0204
73.9046
36.0
0.005
1.21235
62.6029
75.8964
36.0
0.006
1.25112
62.1922
77.8097
36.0
0.007
1.28902
61.7879
79.6457
36.0
0.008
1.32604
61.3899
81.4057
36.0
0.009
1.36219
60.9981
83.0911
36.0
0.010
1.39746
60.6124
84.7031
36.0
0.011
1.43183
60.2324
86.2429
36.0
0.012
1.46533
59.8582
87.7119
36.0
0.013
1.49793
59.4896
89.1114
36.0
0.014
1.52965
59.1264
90.4427
36.0
0.015
1.56048
58.7685
91.7069
36.0
0.016
1.59042
58.4158
92.9054
36.0
0.017
1.61947
58.0681
94.0393
36.0
0.018
1.64763
57.7253
95.1100
36.0
0.019
1.67491
57.3873
96.1187
36.0
0.020
1.70131
57.0540
97.0666
36.0
0.021
1.72682
56.7254
97.9548
36.0
0.022
1.75146
56.4012
98.7846
36.0
0.023
1.77523
56.0814
99.5571
36.0
0.024
1.79812
55.7658
100.274
36.0
0.025
1.82014
55.4545
100.935
36.0
0.026
1.84130
55.1472
101.543
36.0
0.027
1.86160
54.8440
102.098
36.0
0.028
1.88105
54.5446
102.601
36.0
0.029
1.89964
54.2491
103.054
36.0
0.030
1.91739
53.9574
103.458
36.0
Table 3. Numerical results for Fisher information for n = 0, with various values of B, with A = 0.5 eV for the noncentral potential (NCP) for α = 0.13.
B
Ir
Ip
${I}_{r}{I}_{p}\geqslant 36.0$
min(${I}_{r}{I}_{p}$)
0.001
1.08189
53.5829
57.9710
36.0
0.002
1.09086
53.2419
59.6565
36.0
0.003
1.13221
52.9061
61.2833
36.0
0.004
1.17271
52.5754
62.8524
36.0
0.005
1.21235
52.2496
64.3646
36.0
0.006
1.25112
51.9287
65.8206
36.0
0.007
1.28902
51.6126
67.2214
36.0
0.008
1.32604
51.3010
68.5679
36.0
0.009
1.36219
50.9939
69.8610
36.0
0.010
1.39746
50.6912
71.1014
36.0
0.011
1.43183
50.3928
72.2901
36.0
0.012
1.46533
50.0986
73.4279
36.0
0.013
1.49793
49.8084
74.5157
36.0
0.014
1.52965
49.5223
75.5544
36.0
0.015
1.56048
49.2400
76.5447
36.0
0.016
1.59042
48.9616
77.4876
36.0
0.017
1.61947
48.6869
78.3838
36.0
0.018
1.64763
48.4159
79.2343
36.0
0.019
1.67491
48.1484
80.0398
36.0
0.020
1.70131
47.8843
80.8011
36.0
0.021
1.72682
47.6237
81.5191
36.0
0.022
1.75146
47.3665
82.1946
36.0
0.023
1.77523
47.1125
82.8283
36.0
0.024
1.79812
46.8617
83.4210
36.0
0.025
1.82014
46.6140
83.9736
36.0
0.026
1.84130
46.3694
84.4868
36.0
0.027
1.86160
46.1277
84.9613
36.0
0.028
1.88105
45.8890
85.3979
36.0
0.029
1.89964
45.6533
85.7974
36.0
0.030
1.91739
45.4203
86.1604
36.0
Table 4. Numerical results for uncertainty relation in the ground eigenstates for various values of B, with n, l = 0, A = 0.5 eV α = 0.1.
min $\left\{{({\rm{\Delta }}r)}^{2}{({\rm{\Delta }}p)}^{2}\right\}$
0.001
24.2592
3.40874
3.55523
2.90712
0.96354
0.490803
1.74492
3.04474
0.250000
0.002
24.0642
3.39462
3.54130
2.89748
1.01546
0.503851
1.78429
3.18368
0.250000
0.003
23.8731
3.38073
3.52757
2.88797
1.06619
0.516283
1.82123
3.31687
0.250000
0.004
23.6857
3.36707
3.51406
2.87857
1.11572
0.528139
1.85591
3.44440
0.250000
0.005
23.5019
3.35362
3.50074
2.86929
1.16403
0.539452
1.88848
3.56635
0.250000
0.006
23.3216
3.34038
3.48762
2.86013
1.21111
0.550253
1.91907
3.68284
0.250000
0.007
23.1447
3.32735
3.47469
2.85107
1.25695
0.560570
1.94781
3.79395
0.250000
0.008
22.9711
3.31451
3.46195
2.84213
1.30155
0.570427
1.97479
3.89979
0.250000
0.009
22.8006
3.30187
3.44939
2.83329
1.34488
0.579846
2.00011
4.00046
0.250000
0.010
22.6332
3.28941
3.43700
2.82456
1.38696
0.588847
2.02387
4.09604
0.250000
0.011
22.4689
3.27714
3.42479
2.81593
1.42777
0.597447
2.04613
4.18666
0.250000
0.012
22.3074
3.26505
3.41275
2.80740
1.46731
0.605664
2.06698
4.27240
0.250000
0.013
22.1487
3.25312
3.40087
2.28090
1.50558
0.613512
2.08647
4.35337
0.250000
0.014
21.9928
3.24137
3.38914
2.79063
1.54259
0.621005
2.10468
4.42967
0.250000
0.015
21.8395
3.22978
3.37758
2.78239
1.57832
0.628157
2.12165
4.50139
0.250000
0.016
21.6888
3.21835
3.36616
2.77424
1.61278
0.634977
2.13744
4.56864
0.250000
0.017
21.5406
3.20707
3.35495
2.76618
1.64598
0.641479
2.15209
4.63151
0.250000
0.018
21.3949
3.19595
3.34377
2.75821
1.67791
0.647671
2.16567
4.69011
0.250000
0.019
21.2515
3.18497
3.33279
2.75032
1.70858
0.653564
2.17819
4.74452
0.250000
0.020
21.1105
3.17414
3.32195
2.74252
1.73799
0.659166
2.18971
4.79485
0.250000
0.021
20.9716
3.16344
3.31123
2.73480
1.76616
0.664486
2.20027
4.84118
0.250000
0.022
20.8350
3.15288
3.30066
2.72716
1.79308
0.669531
2.20989
4.88362
0.250000
0.023
20.7005
3.14246
3.29020
2.71960
1.81877
0.674309
2.21861
4.92225
0.250000
0.024
20.5681
3.13217
3.27987
2.71212
1.84322
0.678827
2.22647
4.95716
0.250000
0.025
20.4377
3.12200
3.26967
2.70472
1.86645
0.683091
2.23348
4.98845
0.250000
0.026
20.3092
3.11196
3.25958
2.69739
1.88846
0.687108
2.23969
5.01621
0.250000
0.027
20.1827
3.10204
3.24962
2.69013
1.90927
0.690883
2.24511
5.04052
0.250000
0.028
20.0581
3.09224
3.23976
2.68295
1.92889
0.694423
2.24977
5.06146
0.250000
0.029
19.9353
3.08256
3.23003
2.67584
1.94732
0.697731
2.25369
5.07913
0.250000
0.030
19.8142
3.07299
3.22039
2.66879
1.96456
0.700815
2.25690
5.09361
0.250000
Table 5. Numerical results for uncertainty relation in the ground eigenstates for various values of B, with n, l = 0, A = 0.5 eV α = 0.12.
min $\left\{{({\rm{\Delta }}r)}^{2}{({\rm{\Delta }}p)}^{2}\right\}$
0.001
16.0787
2.76177
2.90712
8.45134
0.262168
0.512023
1.48851
2.21567
0.250000
0.002
15.9690
2.75202
2.89748
8.39540
0.272716
0.522222
1.51313
2.28956
0.250000
0.003
15.7551
2.74242
2.88797
8.34035
0.283053
0.532027
1.53648
2.36076
0.250000
0.004
15.6507
2.73294
2.87857
8.28616
0.293177
0.541458
1.55863
2.42931
0.250000
0.005
15.6507
2.72358
2.86929
8.23283
0.303086
0.550532
1.57964
2.49526
0.250000
0.006
15.5480
2.71435
2.86013
8.18032
0.312779
0.559266
1.59957
2.55864
0.250000
0.007
15.4477
2.70524
2.85107
8.12862
0.322255
0.567674
1.61848
2.61948
0.250000
0.008
15.3475
2.69625
2.84213
8.07770
0.331511
0.575769
1.63641
2.67785
0.250000
0.009
15.2495
2.68737
2.83329
8.02755
0.340548
0.583564
1.65341
2.73376
0.250000
0.010
15.1531
2.67861
2.82456
7.97815
0.349364
0.591070
1.66951
2.78728
0.250000
0.011
15.1531
2.66995
2.81593
7.92947
0.357959
0.598296
1.68476
2.83842
0.250000
0.012
14.9646
2.66140
2.81593
7.92947
0.366332
0.605253
1.69919
2.88725
0.250000
0.013
12.2407
2.65296
2.81593
5.20251
0.374483
0.611950
1.39580
1.94825
0.250000
0.014
14.7816
2.64461
2.79063
7.78763
0.382412
0.618395
1.72571
2.97809
0.250000
0.015
14.6921
2.63637
2.78239
7.74170
0.390119
0.624595
1.73787
3.02019
0.250000
0.016
14.6039
2.62822
2.77423
7.69641
0.397604
0.630558
1.74932
3.06012
0.250000
0.017
14.5176
2.62017
2.76617
7.65175
0.404867
0.636291
1.76010
3.09794
0.250000
0.018
14.4313
2.61221
2.75821
7.60771
0.411908
0.641800
1.77022
3.13367
0.250000
0.019
14.3468
2.60434
2.75032
7.56426
0.418728
0.647091
1.77971
3.16737
0.250000
0.020
14.2635
2.59656
2.74252
7.52140
0.425327
0.652171
1.78859
3.19906
0.250000
0.021
14.1813
2.58887
2.73479
7.47912
0.431706
0.657043
1.79688
3.22878
0.250000
0.022
14.1003
2.58126
2.72715
7.43740
0.437866
0.661714
1.80460
3.25658
0.250000
0.023
14.0203
2.57374
2.71960
7.39623
0.443806
0.666187
1.81177
3.28249
0.250000
0.024
13.9415
2.56629
2.71212
7.35560
0.449529
0.670469
1.81839
3.30656
0.250000
0.025
13.8636
2.55893
2.70471
7.31549
0.455035
0.674563
1.82450
3.32881
0.250000
0.026
13.7868
2.55165
2.69738
7.27590
0.460325
0.678472
1.83010
3.34928
0.250000
0.027
13.7110
2.54444
2.69013
7.23681
0.465401
0.682202
1.83522
3.36801
0.250000
0.028
13.6362
2.53731
2.68294
7.19821
0.470262
0.685756
1.83985
3.38505
0.250000
0.029
13.5623
2.53025
2.67583
7.16010
0.474911
0.689137
1.84402
3.40041
0.250000
0.030
13.4893
2.52327
2.66879
7.12245
0.479349
0.692350
1.84774
3.41414
0.250000
Table 6. Numerical results for uncertainty relation in the ground eigenstates for various values of B, with n, l = 0, A = 0.5 eV α = 0.13.
min $\left\{{({\rm{\Delta }}r)}^{2}{({\rm{\Delta }}p)}^{2}\right\}$
0.001
13.3957
2.51467
2.65936
7.07218
0.270473
0.520070
1.38305
1.91283
0.250000
0.002
13.3105
2.50638
2.65114
7.02855
0.280120
0.529264
1.40315
1.96884
0.250000
0.003
13.2265
2.49819
2.64302
6.98557
0.283053
0.538131
1.42229
2.02292
0.250000
0.004
13.1438
2.49011
2.63500
6.94322
0.298868
0.546688
1.44052
2.07511
0.250000
0.005
13.0624
2.48212
2.62707
6.90148
0.307966
0.554947
1.45788
2.12543
0.250000
0.006
12.9822
2.47423
2.61923
6.86035
0.316879
0.562920
1.47442
2.17390
0.250000
0.007
12.9031
2.46644
2.61147
6.81979
0.325606
0.570619
1.49016
2.22057
0.250000
0.008
12.8252
2.45875
2.60381
6.77981
0.334145
0.578053
1.50514
2.26544
0.250000
0.009
12.7485
2.45114
2.59623
6.74039
0.342497
0.585232
1.51940
2.30856
0.250000
0.010
12.6728
2.44362
2.58873
6.70151
0.350660
0.592165
1.53295
2.34995
0.250000
0.011
12.5982
2.43619
2.58131
6.66317
0.358633
0.598860
1.54584
2.38963
0.250000
0.012
12.5246
2.42885
2.57398
6.62535
0.366417
0.605324
1.55809
2.42764
0.250000
0.013
12.4521
2.42158
2.56672
6.58804
0.374012
0.611565
1.56971
2.46400
0.250000
0.014
12.3806
2.41441
2.55954
6.55122
0.381416
0.617589
1.58074
2.49874
0.250000
0.015
12.3100
2.40731
2.55243
6.51489
0.388630
0.623402
1.59119
2.53188
0.250000
0.016
12.2404
2.40028
2.54540
6.47904
0.395655
0.629011
1.60108
2.56346
0.250000
0.017
12.1717
2.39334
2.53844
6.44365
0.402489
0.634420
1.61044
2.59350
0.250000
0.018
12.1040
2.38647
2.53155
6.40872
0.409134
0.639636
1.61927
2.62203
0.250000
0.019
12.0371
2.37967
2.52473
6.37424
0.415589
0.644662
1.62760
2.64907
0.250000
0.020
11.9711
2.37295
2.51797
6.34019
0.421856
0.649504
1.63543
2.67465
0.250000
0.021
11.9059
2.36630
2.51129
6.30657
0.427933
0.654166
1.64280
2.69879
0.250000
0.022
11.8416
2.35971
2.50467
6.27337
0.433822
0.658652
1.64971
2.72153
0.250000
0.023
11.7781
2.35320
2.49812
6.24058
0.439524
0.662966
1.65617
2.74289
0.250000
0.024
11.7154
2.34675
2.49162
6.20819
0.445039
0.667112
1.66219
2.76289
0.250000
0.025
11.6535
2.34036
2.48519
6.17619
0.450367
0.671094
1.66780
2.78155
0.250000
0.026
11.5923
2.33404
2.47883
6.14458
0.455510
0.674915
1.67300
2.79892
0.250000
0.027
11.5319
2.32779
2.47252
6.11335
0.460467
0.678577
1.67780
2.8150
0.250000
0.028
11.4723
2.32159
2.46627
6.08248
0.465241
0.682086
1.68221
2.82982
0.250000
0.029
11.4133
2.31546
2.46008
6.05198
0.469831
0.685442
1.68624
2.84341
0.250000
0.030
11.3551
2.30938
2.45394
6.02184
0.474240
0.688651
1.68991
2.85579
0.250000
8. Conclusion
In this work, we develop a class of noncentral potential (noncentral inversely quadratic plus exponential Mie-type potential) to investigate the quantum information of Fisher and Shannon entropies in a Schrödinger equation. We obtained the normalised wave function and energy-eigen equations by solving the radial solution of the Schrödinger equation via the Nikiforov–Uvarov method. We analytically obtain position and momentum space for both Fisher and Shannon entropies and calculation were carried out with ground state wave function where the principal quantum number n and orbital quantum number l equals zero. The numerical computations were only carried out for Fisher information in position and momentum space, since that of Shannon was a great deal more complicated due to the nature of the potential. All computations were carried out for α = 0.1, 0.12 and 0.13 because these are the only set of values that give the expected result. However, the wave function plots and probability density curves were plotted only for α = 0.1 because this is the best value of α that gave the best plot. The normalisation constants were obtained using confluent hypergeometric functions with the help of Mathematica. The expectation values obtained with respect to position entropy decreases with an increase in variation parameter (B), whereas that of momentum increases with an increase in this parameter. All the numerical results obtained are in agreement with existing literature, including Heinsenberg uncertainties for position and momentum. The study of these entropies is very significant in terms of their potential applications in signal processing and examining the electronic structures of atoms.
The authors are very grateful to the editorial team, and especially to the reviewers for their useful comments and corrections, which have significantly helped improve the quality of this article.