Vol. 70, No. 9, September 2020, pp. 778∼787 http://dx.doi.org/10.3938/NPSM.70.778
Quantum Information Entropy and Entropy Squeezing of Isospectral Modified Hylleraas Plus Exponential Rosen Morse Potential and Isospectral Eckart
Potential
Pooja Thakur
∗· Rama Gupta
†Department of Physics, D. A. V. University, Jalandhar-144 012, Punjab, India
Aarti Sharma
‡Department of Physics, SGGS Khalsa College (Panjab University), Mahilpur-146 105, Punjab, India
Anil Kumar
§Department of Physics, JC DAV College (Panjab University), Dasuya-144 205, Punjab, India (Received 30 March 2020 : revised 24 July 2020 : accepted 29 July 2020)
We have numerically evaluated the position space and momentum space information entropy of the isospectral Modified Hylleraas plus exponential Rosen Morse potential and established that each level can be re-arranged as a function of the deformation parameter. The information densities of this potential have been graphically demonstrated and their properties thoroughly analyzed. An asymmetric shape dependence on the values of quantum number (n, l) is observed for the position space information densities. The characteristic features of the information entropy in position and momentum space have been analyzed, and the lower bound of the sum of the entropies, expressed by using the Bialynicki-Birula and Mycielski inequality is satisfied. Compared to undeformed potential exhibiting squeezing phenomena in momentum space only the information entropy squeezing has been realized for position space, as well as momentum space, as a function of the deformation parameter with the choice of the same set of parameters. Interestingly, squeezed coherent states are obtained for the isospectral Eckart potential.
Keywords: Quantum information entropy, BBM inequality, Isospectral hamiltonian approach, Entropy squeezing
I. INTRODUCTION
Entropy plays a significant role in attaining knowledge about the information theory. The application of infor- mation theory appears in a variety of disciplines includ- ing solid state physics, nuclear physics, mathematical physics, chemical science, biology, statistical physics and computer science [1–4]. In 1948, a revolutionary paper,
∗E-mail: [email protected]
†E-mail: [email protected]
‡E-mail: [email protected]
§E-mail: Corresponding author:[email protected]
“A mathematical theory of communication” was pub- lished by CE Shannon [5] which discussed the quantifi- cation and retainment of digital data and also the study of communication of information. In recent times, a firm connection of Shannon information entropy has been es- tablished with various quantum mechanical systems such as cryptography [6–8], teleportations [9], modern com- munication technologies [10] and quantum entanglement [11–13] etc. It is the measure of the spatial spread of the wave functions for various states of a system. This has been interpreted as the uncertainty associated with position of the particle which relates to the degree of
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localization-delocalization. The single particle entropy [5] in the position space is given as:
Spos=−
∫
ρ(r) ln ρ(r)dr, (1) and in the corresponding momentum space, it is given as:
Smom=−
∫
ρ(p) ln ρ(p)dp, (2) where ρ(r) and ρ(p) denote the position and momentum space particle density, respectively.
These information entropies, in position space Spos
and momentum space Smom, connect information the- ory and quantum mechanics. The entropic uncertainty relation was investigated by Beckner et al., in 1975 [14].
It plays a significant role in numerous fields and has now become a universal concept in classical statistical physics [15–17]. From the general properties of Fourier trans- form, it was proved that for the wave functions normal- ized to unity, the entropic relation can be expressed as:
Spos+ Smom≥ d(1 + ln π), (3) where d is the spatial dimension [18]. The physical sig- nificance of the above expression lies in the fact that Spos
and Smom are individually unbounded, but their sum is bounded. In Eq. (3), the inequality comprehends that if the localization occurs in position space, there would be a corresponding delocalization in momentum space and vice versa. For the ground state of harmonic oscil- lator [19], a lower bound has been attained for the sum of information entropy.
The analytical determination of these entropies is quite difficult. Simple Harmonic oscillator [19], Eckart potential [20], asymmetric trignometric Rosen-Morse po- tential [21], a square tangent potential well [22], a hy- perbolic potential function [23], hyperbolic double-well potential [24], an infinite circular well [25] and a parity- restricted harmonic oscillator [26] are few quantum me- chanical systems for which analytical calculations are carried out and a saturated value of the Bialynicki-Birula and Mycielski (BBM) inequality is evaluated. The infor- mation entropic connection with various potentials, such as the static screened coulomb potential [27], an infi- nite spherical well [28], spherical confined hydrogen atom
[29], the Pöschl-Teller potential [30,31], isospectral po- tential [32], confined hydrogen atom [33], the Hylleraas plus exponential Rosen Morse potential [34], two electron atomic system [35] and the Hellmann potential [36] have also been obtained.
There is an intimate link between the information en- tropy and squeezing phenomenon, where the entropic measure plays a prominent role. Information entropy is found to squeeze in position and momentum space if there is squeezing property in either of the quadratures for all quantum states [37,38]. These states are quite different from the well-known squeezed states in quan- tum optics which are obtained from the action of the squeezing operator on the vacuum of the field. In con- text with the connection between BBM inequality and the Heisenberg uncertainty relationship, the variances of the two quadratures of the field associated with the Mod- ified Hylleraas plus exponential Rosen Morse potential [34] and the Eckart potential [20] are calculated. The squeezing in one of the quadratures leads to compression in information entropy. But it has been expected that quadrature squeezing is not observed for some specific states that possess entropic grip in position [39].
In the present work, using supersymmetric quantum mechanics techniques, the isospectral Hamiltonian ap- proach has been utilized to obtain the isospectral eigen- functions [40]. The Hamiltonians are exactly isospec- tral when they have same eigenvalue spectrum but dif- ferent eigenfunctions and hence the dependent quanti- ties. A first order differential equation, admitting a free parameter (deformation parameter), is involved for the re-introduction of the state that one gets after the dele- tion of a bound state of the given potential V (q). With the change in deformation parameter, we obtain differ- ent eigenfunctions without any change in the eigenvalue spectrum. Implementation of this technique has been extensively performed in many quantum mechanical sys- tems, soliton solutions of nonlinear differential equations, the Korteweg-de Vries equation, the sine-Gordon equa- tion or the nonlinear Schrödinger equation and the time dependent Dyson equation [41–45]. In this paper, inter- esting entropy squeezing has been found for isospectral Modified Hylleraas plus exponential Rosen Morse po- tential and isospectral Eckart potential. The methodol- ogy to obtain isospectral hamiltonian approach has been
briefly described in section 2. Section 3 presents a com- plete analysis of the graphical plots of the information densities for the selected values of deformation parame- ters. The position and momentum space wave functions of the potential in ground state have been analytically obtained. Using the isospectral approach, numerical re- sults of information entropy, in position and momentum space for various values of the deformation parameter, are obtained in section 4. Different information theo- retic concepts have been analyzed as a function of defor- mation parameter. In section 5, the position space and momentum space wave function in the ground state of isospectral Eckart potential have been analytically eval- uated. Using the isospectral formulation, we have ob- tained the entropy squeezing in position space in addition to momentum space for same set of parameters. Further, the numerical squeezed coherent state [37,46] is observed for particular value of the deformation parameter which have not been reported earlier in the literature to the best of our knowledge. The squeezed coherent states are the most general set of states saturating the Heisenberg un- certainty principle. The existence of entropy squeezing has also been analyzed for isospectral Modified Hylleraas plus exponential Rosen Morse potential. Theoretical dis- cussion and the conclusion of the current work are given in section 6.
II. Extraction of Mathematical Formulism of Isospectral Hamiltonian Approach
The information entropy has been calculated in position space and momentum space, for the potentials and re- arranged as a function of deformation parameter, using supersymmetric quantum mechanics. In supersymmet- ric quantum mechanics, the superpotential W (q) deter- mines the two partner potentials,
V1,2(q) = W2(q)±dW
dq . (4)
Let us assume that there exists a more general superpo- tential ˆW (q) which satisfies,
V1(q) = ˆW2(q) + ˆW′(q). (5)
Here ˆW (q) = W (q) is one of the solution. To find the most general solution, it is expressed as,
W (q) = W (q) + ϕ(q).ˆ (6) It is found that Y = 1/ϕ(q) satisfies the Bernoulli equa- tion,
Y′(q) = 1 + 2W Y, (7) whose solution is given by,
1
Y (q) = ϕ(q) = d
dqln [I(q) + K] , (8) where I(q) = ∫q
−∞ψ02(q′)dq′, K is a constant of inte- gration and ψ0(q) is the normalized ground state wave function of V2(q). Thus, from Eq.(6), it is calculated as,
W (q) = W (q) +ˆ d
dqln [I(q) + K] . (9) Consequently, there exist a one parameter family of po- tentials ˆV2(q, K) given by,
Vˆ2(q, K) = V2(q)− 2 d2
dq2(ln(I(q) + K)), (10) and V1(q) is the SUSY partner of this family of poten- tials. Here, for large values of K, it get the original potential i.e. for K → ±∞, the potential ˆV2(q, K) → V2(q). The normalized ground state wave function cor- responding to the potential ˆV2(q, K) is,
ψˆ0(q, K) =
√K(1 + K)ψ0(q)
I(q) + K , (11)
where K̸∈ (0, −1). If the potential V(q) is exactly solv- able (i.e. En, ψn(q) are all known) then ˆV (q, K) is also exactly solvable with En = ˆEn. It has been written ψˆ0(q, K) in terms of ψ0(q), while ˆψn+1(q, K) can also be written in terms of normalized eigenfunctions ψn+1(q).
The normalized excited state eigenfunctions for the po- tential ˆV (q, K) are given by,
ψˆn+1(q, K) =ψn+1(q) + 1 En+1× (
I′(q) I(q) + K
) ( d
dq + W (q) )
ψn+1(q), (12) where W (q) =−dqd[ln ψ0(q)].
Eq.(10), Eq.(11) and Eq.(12) depict the isospectral po- tentials and the corresponding ground and excited state wave functions, which have been in the analysis of infor- mation entropy and squeezing phenomenon.
III. Information Density For Isospectral Modified Hylleraas Plus Exponential Rosen
Morse Potential
The amount of information that can be stored in the compact form is well examined by the information den- sity. In the analysis of this formulation, we get an ab- stract of information in usual manner. A great deal of interest has been raised for one of the most useful po- tential models to describe the interaction between two atoms in diatomic molecules, known as Morse potential, which is reduced to the Hylleraas potential. This poten- tial is expressed as:
V (r) =−V0
b
[a + e−2α(r−rc) 1 + e−2α(r−rc) ]
− 4V1e−2α(r−rc) (1 + e−2α(r−rc))2 + V2
(1− e−2α(r−rc) 1 + e−2α(r−rc)
) ,
(13) where V0, V1, V2, α and rc are the potential depths, ad- justable parameter and the distance from the equilibrium position, a and b are the Hylleraas parameters, respec- tively. The Schrödinger equation for the potential is,
d2Ψnl dr2 + 2µ
[
Enl− V (r) −ℓ(ℓ + 1) 2µr2
]
Ψnl= 0. (14) The analytical expression of radial wave function for the potential is expressed as [47]
Rnl(q) = Nnl(1− q)1+ε2 qϑ2Pnϑ,ε(1− 2q), (15) using the transformation q =−e−2α(r−rc)and
ϑ = 2
√
− µE 2α2¯h2 − G,
E = µ
2α2¯h2b(V0+ V2b)−ℓ(ℓ + 1)a1
4α2 ,
F =− µV0
2α2¯h2b(a + 1)−2µV1
α2¯h2 +ℓ(ℓ + 1)
2α2 (a1+a2
2 ),
G = µ
2α2¯h2b(V0a− V2b)−ℓ(ℓ + 1)
4α2 (a1+ a2+ a3).
where ϵ, and υ are the adjustable potential parameters.
Here Pnϑ,ε(q) and Nnl are the Jacobi polynomials and
normalization constant, respectively. The connection be- tween wave function ψ(q, θ, ϕ) and the radial wave func- tion is established using the following relation
ψ(q, θ, ϕ) = R(q)
q Ylm(θ, ϕ), (16) Using equation Eq.(11), the normalized ground state wave function ˆΨ0(q, K) for the one dimensional potential is obtained as,
Ψˆ0(q, K) =
(1− q)1+υ2 qϵ2√
K(1 + K)
√ 1
B[1+ϵ,2+υ]
m +q√(1+ϵ)[p1−qp2]
1 B[1+ϵ,2+υ]
,
(17) where
p1=2F1
[1 + ϵ,−υ, 2 + ϵ, q 1 + ϵ
] ,
p2=2F1
[2 + ϵ,−υ, 3 + ϵ, q 2 + ϵ
] ,
where B is the beta function. Information density, for the one-dimensional potential of the momentum space, is obtained using Fourier transform of the correspond- ing position space wave function. For ground state, the momentum space wave function has been evaluated after some calculations as,
Ψˆ0(p, K) =
√K(1 + K)√
1
B[1+ϵ,2+ϵ]Γ[1 +ϵ2]Γ[3+υ2 ]s1
√2π[
K−2πB[1+ϵ,2+ϵ]1 iΓ[1 +2ϵ]2Γ[3+υ2 ]2(s2+ s3)− s4
] ,
(18)
where
s1=1F1
[2 + ϵ
2 ,(5 + ϵ + υ) 2 ,−ip
] ,
s2=wFz [
1,1 2 +ϵ
4, 1 + ϵ 4,5
4+ ϵ 4 +υ
4,7 4 +ϵ
4 +υ 4, 1
] ,
s3= (2 + ϵ)wFz
[1, 1 +ϵ4, 1 +4ϵ,74+4ϵ +υ4,94 +4ϵ+υ4, 1]
(5 + ϵ + υ) ,
s4= iB[2+ϵ2 ,3+υ2 ]2Γ[(5+ϵ+υ)2 ]3(−1 + Γ[(3+ϵ+υ)2 ]s5) π(1 + υ)B[1 + ϵ, 2 + υ]Γ[(3+ϵ+υ)2 ] ,
s5=wFz [1 + υ
2 ,(3 + ϵ + υ) 2 , ip
] ,
where 2F1, 1F1 and wFz are the hypergeometric func- tions, regularized confluent hypergeometric function and regularized generalized hypergeometric function, respec- tively. For some levels of the potential, information den- sity has been graphically plotted in Fig. 1 and Fig. 2 for position space and momentum space, respectively.
The sharp distant peaks are observed for the momentum probability density functions of the isospectral Modified Hylleraas plus exponential Rosen Morse potential. The dip in the position and momentum space density func- tions varies with the deformation parameter (K). As observed in Fig. 1, the position space entropy density exhibits asymmetric shape depending on the values of deformation parameter. The figure clearly indicates a strong dependance of the number of minima along with their corresponding depth on K. The solid thick line in Fig. 1 and Fig. 2, representing the information density corresponding to undeformed potential, is approached for higher values of deformation parameter.
IV. Information Entropy
For higher excited states, the momentum space wave functions become quite large and are used for numeri- cal calculations. Due to complications in the integral, the analytical expressions of the information entropy are quite difficult, especially in the momentum space.
The information entropy, in position and momentum space, is calculated and the BBM inequality has been satisfied. Figure 3,4 and 5 present the information en- tropy values in position space, momentum space and en- tropy sum for arbitrary values of the deformation pa- rameter. Using the isospectral wavefunction, position and momentum space entropy for first few levels is also calculated. For the case of undeformed potential, Spos and Smom take the values 0.2203 and 2.4799 for ground state, 0.4508 and 2.9946 for the first excited state, 0.5274 and 3.2524 for the second excited state, respectively.
But when it is evaluated using isospectral wave func- tions ˆΨ0(q, K), ˆΨ1(q, K) and ˆΨ2(q, K), the information entropy contents are observed to reduce substantially, with the change in deformation parameter. We have Spos = 2.3443 for K = 0.001 and this value reduces to 0.2206 with increasing K, shown in Fig.3.
Initially, for smaller values of deformation paramenter, the total information entropy shows an increment for the choice of a set of parameters. For higher values of K, the information entropy attains a constant value coin- ciding the undeformed potential. For the selected value of K, there is reduction in total information entropy, as is clear from Fig.4and Fig.5. In the excited state, there is change in position and momentum space information entropy but the sum is bounded above inequality. It has been observed that sum of the entropies approach the BBM bound for the specific values of K. A class of Gaussian wave packets is used to reach the lower bound of the BBM inequality. A delocalized distribution in the position space corresponds to a strong localized distribu- tion in the momentum space.
V. The Entropy Squeezing For Isospectral Eckart potential and Isospectral Modified
Hylleraas Plus Exponential Rosen Morse potential
For the measure of quantum uncertainty, information entropy is preferred over variance. In order to define the important quantum mechanical effect such as squeezing of quantum fluctuations, variance is a vital entity. These states can be expressed as the minimum uncertainty states in position and momentum observables [37]. A squeezed state based on Heisenberg uncertainty relation is the one with⟨∆q⟩2< 0.5 or⟨∆p⟩2< 0.5, but it is not possible to squeeze both the quadratures simultaneously by uncertainty principle. A good measure of quantum fluctuation can be done with the entropy of a single par- ticle over variance. The domain for the formation of these states is not limited to Gaussian cases only.
Now, the entropy squeezing of isospectral Eckart poten- tial has been analyzed [48]. The potential reads
V (q) =−α e−qa
1− e−qa + β e−qa
(1− e−qa )2, (19) where α, β and a are the depth and range of the poten- tial, respectively.
The eigen function for the potential is expressed as Unl(q) = Nnle−qρa
(
1− e−qa )σ
F (n, n+2(σ+ρ), 1+2ρ; e−qa ), (20)
Fig. 1. (Color online) Position space information densities of the potential for α = 0.1, a = 1.0, b = 2.0 and (a) n = 0, l = 0 and K = 0.1 (dotdash line), K = 1.0 (dashed line), (b) n = 1, l = 0 and K = 10.0 (dotdash line), K = 200.0 (dashed line), (c) n = 2, l = 0 and K = 100.0 (dotdash line), K = 300.0 (dashed line),(d) n = 3, l = 0, K = 100.0 (dotdash line), K = 300.0 (dashed line), (e) n = 4, l = 0, K = 100.0 (dotdash line), K = 200.0 (dashed line), (f) n = 5, l = 0, K = 50.0 (dotdash line), K = 100.0 (dashed line), and in all the plots, the solid thick line describes the undeformed information densities.
Fig. 2. (Color online) Momentum space information densities of the potential for α = 0.1, a = 1.0, b = 2.0 and (a) n = 0, l = 0, and K = 0.1 (dotdash line), K = 3.0 (dashed line),(b) n = 1, l = 0, K = 150.0 (dotdash line), K = 400.0 (dashed line),(c) n = 2, l = 0, K = 100.0 (dotdash line), K = 400.0 (dashed line), (d) n = 3, l = 0, K = 100.0 (dotdash line), K = 400.0 (dashed line), (e) n = 4, l = 0, K = 100.0 (dotdash line), K = 400.0 (dashed line), (f) n = 5, l = 0, K = 100.0 (dotdash line), K = 500.0 (dashed line), and in all the plots, the solid thick line describes the undeformed information densities.
where
σ =1 2
( 1 +√
1 + 4λγ + 4Θ )
,
ρ =
√
−2µa2E
¯
h2 , Θ = 2µa2β
¯ h2 ,
Fig. 3. (Color online) Position space (a), momentum space (b) and total information entropy (c) as a function of deformation parameter K for ground state of the isospectral Modified Hylleraas plus exponential Rosen Morse potential for α = 0.1, a = 1, b = 2, V0= 0.0007, V1=−0.0006, V2= 0.006. The solid line in (c) shows the lower bound (1 + ln π).
Fig. 4. (Color online) Position space (a), momentum space (b) and total information entropy (c) as a function of deformation parameter K for first excited state of the potential for n = 1, l = 0, α = 0.1, a = 1, b = 2, V0 = 0.0007, V1=−0.0006, V2= 0.006. The solid line Smin in (c) shows the lower bound (1 + ln π).
Fig. 5. (Color online) Position space (a), momentum space (b) and total information entropy (c) as a function of deformation parameter K for the second excited state of the potential for n = 2, l = 0, α = 0.1, a = 1, b = 2, V0 = 0.0007, V1=−0.0006, V2= 0.006. The solid line Smin in (c) shows the lower bound (1 + ln π).
and
γ = ℓ(ℓ + 1).
Where F (n, n + 2(σ + ρ), 1 + 2ρ; e−qa ) and Nnl, are the hypergeometric function and normalization constant, re- spectively.
The position and momentum space wave function for the
ground state of the isospectral potential is obtained after
some lengthy calculation as,
Ψˆ0(q, K) = e−qρa
(
1− e−qa )σ√
K(1 + K)
√ 1
aB[2ρ,1+2σ]
K− t1+
e−qρa
(
1−e−qa )σ
t2
2(ρ+σ)B[2ρ,1+2σ]
,
(21) where
t1= Γ[2ρ]Γ[1 + 2σ]
B[2ρ, 1 + 2σ]Γ[1 + 2ρ + 2σ],
t2=2F1
[
1, 1− 2ρ, 1 − 2ρ − 2σ, eqa] .
Ψˆ0(p, K) =
2a(1− e−ipa )σ(1− eipa)σ√
K(1 + K) πB[2ρ, 1 + 2σ]
[
K− t3+2ia2(1−e
ip
a)σ(1−eipa)1+σt4 πσB[2ρ,1+2σ]
],
(22) where
t3= a2Γ[1− 2ρ − 2σ]Γ[1 + 2σ]
π(ρ + σ)B[2ρ, 1 + 2σ]Γ[1− 2ρ],
t4=2F1
[
1, 1 + σ, 1− σ, eipa] .
where2F1and B are the hypergeometric function and beta function, respectively. The entropy squeezing has been analyzed for the different values of deformation pa- rameter, in position space and momentum space, corre- sponding to ground state of isospectral Eckart potential.
In Table1, for the values of K, 0.001 to 0.53, only posi- tion space is squeezed whereas momentum space squeez- ing phenomenon is attained for the values of K, 0.53 to 1.0, respectively. It is an interesting result as the squeezed coherent states have been numerically achieved for the value of deformation parameter ′K = 0.53′, in Table1. In the corresponding undeformed potential, the position and momentum space uncertainty acquire val- ues ⟨∆q⟩2 = 1.3755 and ⟨∆p⟩2 = 0.2281, respectively, which exhibits squeezing in the momentum space only.
The different parameters used for the calculations are a = 1, ξ = 1.0, λ = 0.98, µ = 1, β = 0.001, α = 1a. Us- ing the isospectral formulism, it is possible to obtain the
squeezing in position space in addition to momentum space with choice of the same set of parameters. The most important result has been obtained for this poten- tial, as the existence of squeezed coherent states shows the minimum quantum noise.
For different values of deformation parameter, the un- certainty in position and momentum observables is cal- culated in the ground state of the isospectral Modified Hylleraas plus exponential Rosen Morse potential and the squeezing phenomenon has been obtained for posi- tion observable. For the analysis of this formulation in this potential, wave functions and their Fourier trans- form have been used. Existence of the entropy squeezing in position space of the potential are tabulated in Ta- ble 2. The results obtained hold immense importance, as the uncertainty product approaches the undeformed potential, with the increasing value of K during the po- sition space squeezing. For the measure of quantum un- certainty, the information entropy is preferred over vari- ance.
VI. Conclusion
In this work, we have explored the position and mo- mentum eigen states for the ground and first few ex- cited states of isospectral Modified Hylleraas plus expo- nential Rosen Morse potential. The graphical represen- tation of the information density and their properties have been deeply analyzed. The number of minima and their depth are the function of the deformation param- eter. Further, it is shown that for higher values of K, the position and momentum space information densities approach corresponding undeformed potential densities.
For some particular values of K, there is reduction in the value of information entropy. The accurate prediction of the localization of the particle becomes higher for lower values of information entropy. Squeezing phenomenon has been observed in position space information entropy for different values of K in the ground states of the po- tential. Some interesting features of squeezing formula- tion have been established for both quadratures of the isospectral Eckart potential. The numerical squeezed co- herent state for this potential has been achieved for the specific value of K. Using the isospectral formulism, the
Table 1. Numerical results for calculating the Heisen- berg uncertainty relationship ⟨∆q⟩2⟨∆p⟩2 ≥ 0.25 of the isospectral Eckart potential for a = 1, ξ = 1.0, λ = 0.98, µ = 1, β = 0.001, α = 1a.
K ⟨∆q⟩2 ⟨∆p⟩2 ⟨∆q⟩2⟨∆p⟩2 min⟨∆q⟩2⟨∆p⟩2 0.001 0.0005 998.7910 0.4994 0.2500 0.002 0.0009 896.1171 0.8065 0.2500 0.003 0.0014 793.4435 1.1108 0.2500 0.004 0.0019 690.7500 1.3124 0.2500 0.005 0.0024 587.0561 1.4089 0.2500 0.006 0.0028 484.6922 1.3571 0.2500 0.007 0.0033 379.0183 1.2508 0.2500 0.008 0.0038 275.3844 1.0465 0.2500 0.009 0.0043 172.7105 0.7427 0.2500 0.01 0.0047 100.7105 0.4733 0.2500
0.02 0.0096 39.1997 0.3763 0.2500
0.03 0.0147 20.0358 0.2945 0.2500
0.04 0.0199 14.6719 0.2919 0.2500
0.05 0.0252 11.3080 0.2850 0.2500
0.06 0.0307 8.9411 0.2745 0.2500
0.07 0.0364 6.9346 0.2524 0.2500
0.08 0.0422 5.9533 0.2512 0.2500
0.09 0.0482 5.2058 0.2509 0.2500
0.1 0.0544 4.6138 0.2509 0.2500
0.5 0.4625 0.5425 0.2509 0.2500
0.51 0.4774 0.5256 0.2509 0.2500
0.52 0.4844 0.5179 0.2509 0.2500
0.53 0.5007 0.5010 0.2509 0.2500
0.54 0.5236 0.4792 0.2509 0.2500
0.55 0.5400 0.4647 0.2509 0.2500
0.6 0.6230 0.4028 0.2509 0.2500
0.7 0.8099 0.3098 0.2509 0.2500
0.8 1.0224 0.2454 0.2509 0.2500
0.9 1.2572 0.1996 0.2509 0.2500
1.0 1.5059 0.1666 0.2509 0.2500
entropy squeezing has been achieved, in position space as well as momentum space, with choice of the same set of parameters. It has been shown that squeezed coher- ent states, for this potential, have minimum uncertainty with most stable state. The entropy squeezing has great significance in the field of quantum computing, quantum entanglement and quantum optical systems. The preci- sion measurement of quantities in quantum experiments is mostly restricted by quantum noises of the quantum mechanical systems [49,50]. The entropy squeezing is helpful in the suppression of quantum noise in these sys- tems.
ACKNOWLEDGMENTS
The authors are thankful to the referees for suggestions to improve the manuscript and Dr. C.N. Kumar, De-
Table 2. Numerical results for calculating the Heisen- berg uncertainty relationship ⟨∆q⟩2⟨∆p⟩2 ≥ 0.25 of the isospectral Modified Hylleraas plus exponential Rosen Morse potential for α = 0.1, a = 1, b = 2, V0 = 0.0007, V1=−0.0006, V2= 0.006.
K ⟨∆q⟩2 ⟨∆p⟩2 ⟨∆q⟩2⟨∆p⟩2 min⟨∆q⟩2⟨∆p⟩2
0.01 0.0079 57.4872 0.4541 0.2500
0.02 0.0117 41.0915 0.4808 0.2500
0.03 0.0144 40.3265 0.5807 0.2500
0.04 0.0165 42.8026 0.7062 0.2500
0.05 0.0183 37.4119 0.6846 0.2500
0.06 0.0198 33.6566 0.6664 0.2500
0.07 0.0211 30.6131 0.6459 0.2500
0.08 0.0222 28.4876 0.6324 0.2500
0.09 0.0232 29.4915 0.6842 0.2500
0.1 0.0241 30.119 0.7258 0.2500
0.2 0.0297 20.8084 0.6180 0.2500
0.3 0.0325 26.0923 0.8479 0.2500
0.4 0.0342 26.1513 0.8944 0.2500
0.5 0.0354 26.1023 0.9240 0.2500
0.6 0.0362 26.0769 0.9439 0.2500
0.7 0.0367 26.0289 0.9553 0.2500
0.8 0.0372 25.5286 0.9497 0.2500
0.9 0.0375 25.7753 0.9666 0.2500
1.0 0.0378 25.5040 0.9641 0.2500
1.5 0.0386 22.7168 0.8769 0.2500
2.0 0.0389 12.2018 0.4747 0.2500
5.0 0.0396 9.7037 0.3843 0.2500
6.0 0.0396 11.0968 0.4394 0.2500
7.0 0.0396 11.6561 0.4616 0.2500
8.0 0.0397 11.6457 0.4623 0.2500
9.0 0.0397 11.7421 0.4662 0.2500
10.0 0.0397 11.7734 0.4674 0.2500
15.0 0.0397 11.6843 0.4639 0.2500
20.0 0.0397 11.5668 0.4592 0.2500
25.0 0.0398 11.4809 0.4569 0.2500
50.0 0.0398 11.2847 0.4491 0.2500
partment of Physics, Panjab University, Chandigarh for many useful discussions. The financial support from De- partment of Science and Technology, New Delhi through Women Scientist Scheme-A project (Ref. No. SR/ WOS- A/ PM-109/2017 G) is gratefully acknowledged by Pooja Thakur.
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