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Complexity study of q -deformed quantum harmonic oscillator
Physica A 533 (2019) 122041
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Physica A journal homepage: www.elsevier.com/locate/physa
Complexity study of q-deformed quantum harmonic oscillator ∗
Ferhat Nutku a , , K.D. Sen b,a , Ekrem Aydiner a a b
Department of Physics, Faculty of Science, İstanbul University, Vezneciler, İstanbul, 34134, Turkey School of Chemistry, University of Hyderabad, Hyderabad 500 046, India
highlights • • • •
q-deformed quantum harmonic oscillator is studied. Probability distribution changes with q-deformation. Probability distribution becomes localized with q-deformation. Complexity, disequilibrium, entropy changes with quantum number n and deformation q.
article
info
Article history: Received 22 February 2019 Received in revised form 16 May 2019 Available online 11 July 2019 Keywords: Complex systems Information theory Information entropy Complexity q-deformed harmonic oscillator q-deformation
a b s t r a c t The statistical complexity measure defined by López-Ruiz, Mancini, and Calbet is reported for a single particle under q-deformed quantum harmonic oscillator. It is found that q-deformation leads to typically a different behavior in complexity. An interesting variation appears in the exponential Shannon information entropy and disequilibrium. The product of these, which defines the statistical complexity are reported and discussed. It is shown that the deformation parameter q can be used to tune the statistical complexity of the quantum harmonic oscillator. © 2019 Published by Elsevier B.V.
1. Introduction Complexity measures are being increasingly employed in order to understand the behavior of systems encountered in several disciplines of scientific inquiry. Several different measures of complexity have been proposed in the literature [1]. A few examples of them are algorithmic complexity [2,3], a measure of the self-organization capacity of a system [4], Crutchfield and Young’s complexity [5] etc. For a finite many-particle system, complexity may be regarded as a measure of it’s internal order/disorder which can be represented in terms of the information entropy and the distance from equilibrium which is termed as disequilibrium. In the context of the electronic structure of atoms and molecules, a suitable statistical complexity measure CLMC has been proposed by López-Ruiz, Mancini and Calbet (LMC) [6,7]. This measure has been extensively employed in the literature to study the complexity of various quantum systems [7–9]. Indeed, CLMC allocates a multiplicative role to the measure of distance from equilibrium in conjunction with the information entropy to define a measure for the complexity of a finite system. According to the definition of CLMC , there are two extreme limits exist which correspond to totally ordered perfect crystal and totally disordered random gas. At these limits CLMC drops to zero [6]. ∗ Corresponding author. E-mail address: [email protected] (F. Nutku). https://doi.org/10.1016/j.physa.2019.122041 0378-4371/© 2019 Published by Elsevier B.V.
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F. Nutku, K.D. Sen and E. Aydiner / Physica A 533 (2019) 122041
In most of the previous studies focused on quantum systems, the wavefunction of a particle has been used to obtain quantities such as information entropy, disequilibrium and complexity. It would be interesting to study these physical quantities within various q-deformation types. Physically q-deformation can be thought as a separation from the ideal symmetric system. The q-deformed hyperbolic potentials have been introduced by Arai [10,11]. They are used as model potentials for describing electronic conductance in disordered metals and doped semiconductors [12], phonon spectrum in 4 He [13], oscillatory-rotational spectra of diatomic [14] and multi-atomic molecules [15]. Among the q-deformed systems, q-deformed quantum harmonic oscillator (q-oscillator) is the most studied one and it has several applications. For example, the spectrum of the ordinary anharmonic oscillator obtained by perturbation theory can also be represented by a q-deformed harmonic oscillator as well. On can also build a connection between the 3-dimensional q-oscillator and the nuclear shell model such as reviewed in Ref. [16]. Furthermore, the q-deformation of the Lie algebra of SU(2) can be represented by two independent q-deformed harmonic oscillators. This enables one to construct the quantum theory of angular momentum which is discussed in Ref. [17]. Besides, the q-deformation of the Morse potential has been investigated in Refs. [18–20]. Such deformations have been found to be useful in several applications in chemical physics. The aim of this letter is to present the results of our computations of statistical complexity and related measures such as Shannon information entropy and disequilibrium as a function of principal quantum number n and deformation parameter q within the q-oscillator. The outline of article is like the following: the definition and meaning of López-Ruiz, Mancini and Calbet complexity measure is presented in Sec. 2. In Sec. 3, CLMC is applied to the probability distribution of a single q-oscillator. Finally, a summary of the obtained results are listed in Sec. 4. 2. Complexity Measure of López-Ruiz, Mancini and Calbet The LMC complexity measure is defined for continuous systems as the following [7], CLMC = eSx Dx
(1)
where Sx and Dx are Shannon information entropy and disequilibrium, respectively. These are defined in one dimension as
∫
∞
ρ (x) ln ρ (x) dx
Sx = −
(2)
−∞ ∞
∫
ρ 2 (x) dx
Dx =
(3)
−∞
where ρ (x) is the probability density given in terms of the wavefunction as ρ (x) = |ψ (x)|2 . Above expressions are obtained by taking the continuous limit of the following expressions which are expressed in terms of a discrete probability distribution Sx = −kB
N ∑
pi (x) ln pi (x)
(4)
i=1
Dx =
N ( ∑ i=1
pi −
1
)2
N
(5)
where pi is the probability of occupying the state i, and N is the total number of accessible states in position space. It is generally assumed that CLMC in Eq. (1) is a good statistical measure of complexity for ergodic systems. The complexity measure in the momentum space can be similarly obtained by first taking the Fourier transformation of the wavefunction in the position space followed by the computation of probability distribution using the momentum space wavefunction resulting from the transformation. 3. Complexity study of q-deformed quantum harmonic oscillator In the literature q-deformation is applied to the harmonic oscillator in an algebraic way firstly by Arik and Coon [21]. The Hamiltonian of the q-deformed quantum oscillator is given as H=
ω 2
(a− a+ + a+ a− )
(6)
where ω is frequency of the oscillator, in our work h ¯ is set to 1, a+ and a− are creation and annihilation operators, respectively which satisfy the commutation relation
[a− , a+ ]q = a− a+ − qa+ a− = 1,
(7)
with the deformation parameter q taking values in the interval (0, 1). The effect of annihilation and creation operators to the states are given as below, a− | n⟩ =
√ √ [n]|n − 1⟩, a+ |n⟩ = [n + 1]|n + 1⟩,
(8)
F. Nutku, K.D. Sen and E. Aydiner / Physica A 533 (2019) 122041
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n
where [n]q =
1−q
which satisfies the limits, limq→1 [n]q = n and limq→0 [n]q = 1 [22]. 1−q − In q-oscillator, a and a+ operators are still defined in terms of xˆ and pˆ just like the quantum harmonic oscillator as the following,
) 1 ( a+ = √ xˆ − ipˆ . 2
) 1 ( a− = √ xˆ + ipˆ , 2
(9)
Coordinate and momentum operators for the q-oscillator are given by [23], xˆ =
a+ + a−
√
2
,
pˆ = i
a+ − a−
√
2
.
(10)
On the other hand explicit definition of the a− and a+ operators of the q-oscillator are quite different than the quantum harmonic oscillator and given in Ref. [23]. At this step, it would be instructive to study the canonical commutation relationship between xˆ and pˆ . By using the definitions of xˆ and pˆ in Eq. (10) and operators a− and a+ defined in Eq. (8), one can prove that by considering the omitted h ¯ in equations, the commutation relation between xˆ and pˆ is [ˆx, pˆ ] = ih¯ qn . This result is different than the result of [ˆx, pˆ ] = ih ¯ obtained for the quantum harmonic oscillator and it is deformation dependent. A q-oscillator has more complex wavefunctions than a simple quantum harmonic oscillator. If the calculation steps in Ref. [23] are repeated, the normalized wavefunctions can be obtained as 2
|n⟩ = Ψn (x) =
exp(− x2 + 32 iα x) n√ 1 π 4 in (1 − exp(−2α 2 )) 2 [n]q !
n ∑ (−1)k [n]q ! exp(2iα (n − k)x − kα 2 ) [k]q ![n − k]q !
(11)
k=0
√
where α = i (log q)/2. In the above formula [n]q ! is the q-factorial and defined as
[n]q ! =
n ∏
1 − qn 1 − qn−1
[k]q =
1−q
k=1
1−q
···
1 − q2 1 − q 1−q 1−q
=
(q; q)n (1 − q)n
.
(12)
where (q; q)n is the q-Pochhammer symbol. On the other hand, energy eigenvalues for q-oscillator are given by [23], En =
( ) ( ) ) ω( qn ω 2 − (1 + q)qn [n]q + [n + 1]q = ω [n]q + = . 2 2 2 1−q
(13)
We note that, in the limit of q → 1, energy eigenvalues reduce to En = ω n + 21 of the ordinary quantum harmonic oscillator. However, under a small perturbation from unity (q = 1 − ε ), the energy spectrum becomes quadratic and energy eigenvalues can be approximated as the following,
(
(
En = ω n +
1 2
−
n2 2
) ε + O(ε2 ) .
)
(14)
For ϵ = 0, Eq. (14) reduces to energy eigenvalues of the quantum harmonic oscillator as well. At this point it is useful to check the equipartition theorem by calculating the expectation value of the potential energy of the nth state. This can be done by using Eqs. in (8) and ⟨V ⟩ is found as
⏐ ⟩ ⟨ ⏐ ( ) ⏐ 1 2 2⏐ ) ω 2 − (1 + q)qn ω( ⏐ ⟨V ⟩ = ⟨n|V |n⟩ = n⏐ ω xˆ ⏐⏐n = [n]q + [n + 1]q = 2 4 4 1−q
(15)
which is plotted for several n values as presented in Fig. 1. We note that as the q-deformation increases where q approaches to zero, q dependence of ⟨V ⟩ becomes saturated into a single value. In q-deformed oscillator, as one anticipates that for a given energy level n, expectation value of the potential energy is equal to the half of the total energy. This result is consistent with the equipartition theorem. We note that the q-oscillator has a nonlinear spectrum and it satisfies the Heisenberg uncertainty as [23]
∆x∆p =
En
ω
( =
2 − (1 + q)qn 2(1 − q)
)
( ) 1 ≤ n+ 2
(16)
where q changes in the interval of (0, 1). For all states, the uncertainty of q-oscillator is less than the uncertainty of quantum harmonic oscillator. In order to be more quantitative, q dependence of the uncertainty is plotted for several energy levels as seen in Fig. 2. q-deformation decreases the uncertainty in all excited states and it is more effective on high energy levels. Applying a simple numerical procedure to the related equations, with the help of Eq. (11), we have computed the probability distribution ρ (x) of q-oscillator for different eigenstates by setting m, ω and h ¯ to 1. Probability distributions of excited states (n = 1, 2, 5, 6) for different q values are given in Fig. 3a–d. As it can be seen from figures, q-oscillator shows very interesting behavior depending on both principal quantum number and q-parameter. The shape of the probability distribution dependence upon n is generally known for the quantum harmonic oscillator i.e, q = 1. Here we note that
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F. Nutku, K.D. Sen and E. Aydiner / Physica A 533 (2019) 122041
Fig. 1. Effect of q-deformation on the expectation value of the potential energy for several energy levels n, where q = 1 corresponds to the quantum harmonic oscillator case.
Fig. 2. q dependence of uncertainty plot for ground and excited states of q-oscillator.
the number of peaks in the probability distribution increases with n. In the case of q = 1, the probability distribution at the origin is very different from the distribution for the classical harmonic oscillator. Additionally, probability distribution is dependent on the principal quantum number n. In particular, ρ (x) at point x = 0 becomes maximum or minimum depending on the even or odd state n, respectively. As expected, with the increase in quantum number n, the probability distribution becomes more like that of the classical oscillator. In the course of our numerical study, we have ascertained that for q = 1, the solution gives the standard ρ (x) of the quantum harmonic oscillator. The system behaves completely dispersed. With the variation of q parameter from 1 to 0, indicating the increase of deformation, ρ (x) becomes more localized as seen in Fig. 3a–d. It is evident from Fig. 3a and b that the shape of the probability function has a similar characteristic form for some q values. However, for small q values this behavior of the ρ (x) changes dramatically. For example, at q = 0.001, probability function becomes independent of the even or odd n and stays localized at point x = 0. It is then inferred that the qdeformation changes the characteristic of probability distribution from dispersed one to the localized Gaussian one, which might be due to the broken symmetry of the q-oscillator with the increasing q-deformation parameter. This behavior has also been pointed in Ref. [23] and it is stated that, in the limit q → 0, the probability density of the q-oscillator for any n, degenerates or in other words collapses and becomes localized to the ground state probability density, according to the following expression, lim |Ψn (x)|2 =
q→0
1
π 1/2
2
e−x .
(17)
A similar localization behavior in the case of higher excited states for example n = 5 and 6 is evident in Fig. 3c and d. Here, depending on q values, the oscillating form of the ρ (x) changes quite significantly. As q decreases from 1 to
F. Nutku, K.D. Sen and E. Aydiner / Physica A 533 (2019) 122041
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Fig. 3. q dependence of the probability distribution of a q-oscillator for energy levels n = 1, 2, 5, 6. The arrow indicates increment direction of q.
0.1, the form gradually becomes more narrow and the oscillating behavior of the ρ (x) between two edges decreases and vanishes so that oscillating form evolves into a straight oscillating plateau. We note that the q dependence of ρ (x) is a very interesting result, which emerges from the extent of q-deformation. After having discussed eigenenergies, uncertainty and probability distribution of q-oscillator, now we present our results on the Shannon information entropy, disequilibrium and complexity measure of q-oscillator. Based on Eqs. (1)– (3), these quantities can be obtained numerically for ground and several excited states depending on q. As it can be seen from Fig. 4a that information entropy of the ground state is constant and independent from q. However, this behavior changes for higher n values. Information entropy increases as q increases and after a critical saturation value, it starts decreasing between the values of q = 0.8–1.0. This behavior significantly depends on the n value. The maxima of the entropy in Fig. 4a depend on n and their change is indicated by an arrow. It is seen from Fig. 3a–d, that as q decreases a smooth transition occurs in the probability distribution from dispersed one to localized one, which affects the behavior of the information entropy. In Fig. 4b, q dependence of the disequilibrium is given for different excited states. Disequilibrium of ground state is independent from q and has the largest value compared to excited states. However, disequilibrium of q-oscillator increases with decreasing q and decreasing n values, which means that the system passes from dispersed to localized behavior. On the other hand, after attaining a certain minimum value in the region q = 0.6–0.9, the disequilibrium is found to be slightly increasing as q increases especially at states with larger n. The minima of the disequilibrium in Fig. 4b depend on n and their change is indicated by an arrow.
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F. Nutku, K.D. Sen and E. Aydiner / Physica A 533 (2019) 122041
Fig. 4. q dependence of the Shannon information entropy (a), disequilibrium (b) and complexity (c) for q-deformed harmonic oscillator for energy levels changing in [0,10] where n = 7, 9 have been omitted for preventing messiness.
With reference to the ground state, for a given excited state, complexity of the q-oscillator is displayed in Fig. 4c. For all excited states complexity decreases depending on the increment in q from zero, however, it starts to increase above a critical minimum q value. The LMC complexity being a product of disequilibrium and exponential Shannon entropy, both of which depend upon the eigenstate, the net product shows the minimum at a certain q which is different for different states. This feature of the complexity results from the interplay between the variation of disequilibrium and information entropy values. We can conclude that q-oscillator has a critical transition in complexity. Around the value of q in the neighborhood of 0.9 the statistical complexity attains an approximately similar value for all excited states. Above this bunching point of statistical complexity, for the excited states, the CLMC versus q curves are found to cross over with the inversion of relative ordering of the quantum number n of the excited states. Below this bunching point, as q decreases, for each excited state the statistical complexity goes through a minimum at a certain q. The depth of the minimum value is found to be decreasing with the decreasing quantum number of the excited state. With further decrease in q beyond the minimum, the CLMC values are found to be increasing, finally, culminating at a common value at the lowest q. In order to comprehend the numerical change, we present variation of data in magnitude of Shannon information entropy,disequilibrium and complexity as obtained by using Eqs. (1)–(3) in Table 1. 4. Conclusions and remarks In this letter, we have considered a single particle q-deformed quantum harmonic oscillator with the aim of investigating the effect of deformation on the statistical complexity and defining the constituent quantities such as Shannon information entropy and disequilibrium. We have presented accurate numerical estimates of these quantities as a function of the eigenstate defined by the quantum number n and the deformation parameter q. The numerical values of statistical complexity, entropy and disequilibrium are found to be sensitively depending on the deformation parameter q. In our opinion entropy increases due to the increment of the standard deviation ∆x with q and n. Moreover, it can be seen from Fig. 4a that for all excited states as the q-parameter increases, the entropy of q-oscillator increases, which is an
F. Nutku, K.D. Sen and E. Aydiner / Physica A 533 (2019) 122041
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Table 1 Energy level and q-deformation dependence of Shannon information entropy, disequilibrium and complexity for q-oscillator. n
q
S
D
C = eS D
0
0.001 0.4 1 0.001 0.4 1 0.001 0.4 1
1.07236 1.07236 1.07236 1.07829 1.59322 1.76806 1.07829 1.61096 2.01018
0.39894 0.39894 0.39894 0.39232 0.21132 0.19666 0.39232 0.20561 0.15668
1.16582 1.16582 1.16582 1.15329 1.03962 1.15235 1.15329 1.02962 1.16957
5
10
expected result. Because in the limit of q → 0, the probability density of the q-oscillator has only one peak, however, when q → 1, probability density has a number of peaks depending on the quantum number n. This interesting result shows that q can play an important role, which drives the system from dispersed one to localized one. For n = 1 state, while q goes to 1, these peaks become enlarged and the well between the peaks goes to a minimum. Whereas, for n ≥ 2 intermediate peaks occur for large values of q, however, a plateau shape appears between the main probability density peaks at small values of q. For all cases, the probability distribution spreads in the position space and gives a big contribution to the Shannon entropy integral depending on the logarithmic factor. As it can be seen from Fig. 4b, unlike entropy, disequilibrium in q-oscillator decreases with increasing q for all quantum states as a general trend. We remark that a very interesting disequilibrium behavior in q-oscillator appears for large n values where it starts increasing after q reaches to a critical value. An opposite behavior is seen in the entropy for particularly large n values. Similarly, these behaviors of the disequilibrium can be explained depending on the behavior of the probability density distributions. In q-oscillator, for small n values the contribution of Gaussian peak to disequilibrium integral is greater than that of the small peaks which appears at large q values. However, in certain q intervals, plateau shape of the distribution gives almost constant contributions to the disequilibrium. However, when n increases this contribution increases due to appearing peaks in the distribution. Complexity measure of q-oscillator reveals a quite puzzling picture which depends on both entropy and disequilibrium. It is seen that q-deformation plays an important role on the complexity for states n ≥ 1. Indeed, complexity at first decreases with increasing q value, however it smoothly increases up to a critical q value. After the critical q value, complexity behavior completely changes. Over the whole extent, the understanding of the complexity of quantum systems under different environments is a rapidly emerging area of scientific inquiry. Scaling of the information theoretical measures in general with the key parameters of the quantum potentials have lead to an interesting new understanding of the structure and stability of such interacting systems. For example, the studies which include the relativistic effects, and the effects due to the spatial confinements have revealed novel characteristic features of complexity and other information theoretical measures [24– 28]. We believe that the q-deformed potentials extend, similarly, the scope of modeling the quantum systems under the influence of novel and diverse environments. It would be interesting to extend the present study to include the effect of spatial confinement on the quantum systems under various q-deformation types. Acknowledgments Authors are grateful to Professor Metin Arik for a valuable discussion and suggestions to clarify q-deformed quantum harmonic oscillator. KDS is grateful to The Scientific and Technological Research Council of Turkey (TÜBİTAK) for a visiting scientist award under its 2221-program, grant number 1059B211601794 and acknowledges with thanks the support received under the Senior Scientist scheme, I.N.S.A. New Delhi. References [1] S. Lloyd, Measures of complexity: a nonexhaustive list, IEEE Control Syst. Mag. 21 (4) (2001) 7–8, http://dx.doi.org/10.1109/MCS.2001.939938. [2] G.J. Chaitin, On the length of programs for computing finite binary sequences, J. ACM 13 (4) (1966) 547–569, http://dx.doi.org/10.1145/321356. 321363. [3] D.W. Miller, Selected Translations in Mathematical Statistics and Probability, Vol. 7, JSTOR, 1969. [4] G.Y. Georgiev, K. Henry, T. Bates, E. Gombos, A. Casey, M. Daly, A. Vinod, H. Lee, Mechanism of organization increase in complex systems, Complexity 21 (2) (2015) 18–28, http://dx.doi.org/10.1002/cplx.21574. [5] J.P. Crutchfield, K. Young, Inferring statistical complexity, Phys. Rev. Lett. 63 (2) (1989) 105–108, http://dx.doi.org/10.1103/PhysRevLett.63.105. [6] R. López-Ruiz, H.L. Mancini, X. Calbet, A statistical measure of complexity, Phys. Lett. A 209 (5–6) (1995) 321–326, http://dx.doi.org/10.1016/ 0375-9601(95)00867-5. [7] R. López-Ruiz, Shannon information, LMC complexity and Rényi entropies: A straightforward approach, Biophys. Chem. 115 (2-3 SPEC. ISS.) (2005) 215–218, http://dx.doi.org/10.1016/j.bpc.2004.12.035. [8] K. Sen (Ed.), Statistical Complexity: Applications in Electronic Structure, Springer, New York, 2012.
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Physica A journal homepage: www.elsevier.com/locate/physa
Complexity study of q-deformed quantum harmonic oscillator ∗
Ferhat Nutku a , , K.D. Sen b,a , Ekrem Aydiner a a b
Department of Physics, Faculty of Science, İstanbul University, Vezneciler, İstanbul, 34134, Turkey School of Chemistry, University of Hyderabad, Hyderabad 500 046, India
highlights • • • •
q-deformed quantum harmonic oscillator is studied. Probability distribution changes with q-deformation. Probability distribution becomes localized with q-deformation. Complexity, disequilibrium, entropy changes with quantum number n and deformation q.
article
info
Article history: Received 22 February 2019 Received in revised form 16 May 2019 Available online 11 July 2019 Keywords: Complex systems Information theory Information entropy Complexity q-deformed harmonic oscillator q-deformation
a b s t r a c t The statistical complexity measure defined by López-Ruiz, Mancini, and Calbet is reported for a single particle under q-deformed quantum harmonic oscillator. It is found that q-deformation leads to typically a different behavior in complexity. An interesting variation appears in the exponential Shannon information entropy and disequilibrium. The product of these, which defines the statistical complexity are reported and discussed. It is shown that the deformation parameter q can be used to tune the statistical complexity of the quantum harmonic oscillator. © 2019 Published by Elsevier B.V.
1. Introduction Complexity measures are being increasingly employed in order to understand the behavior of systems encountered in several disciplines of scientific inquiry. Several different measures of complexity have been proposed in the literature [1]. A few examples of them are algorithmic complexity [2,3], a measure of the self-organization capacity of a system [4], Crutchfield and Young’s complexity [5] etc. For a finite many-particle system, complexity may be regarded as a measure of it’s internal order/disorder which can be represented in terms of the information entropy and the distance from equilibrium which is termed as disequilibrium. In the context of the electronic structure of atoms and molecules, a suitable statistical complexity measure CLMC has been proposed by López-Ruiz, Mancini and Calbet (LMC) [6,7]. This measure has been extensively employed in the literature to study the complexity of various quantum systems [7–9]. Indeed, CLMC allocates a multiplicative role to the measure of distance from equilibrium in conjunction with the information entropy to define a measure for the complexity of a finite system. According to the definition of CLMC , there are two extreme limits exist which correspond to totally ordered perfect crystal and totally disordered random gas. At these limits CLMC drops to zero [6]. ∗ Corresponding author. E-mail address: [email protected] (F. Nutku). https://doi.org/10.1016/j.physa.2019.122041 0378-4371/© 2019 Published by Elsevier B.V.
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In most of the previous studies focused on quantum systems, the wavefunction of a particle has been used to obtain quantities such as information entropy, disequilibrium and complexity. It would be interesting to study these physical quantities within various q-deformation types. Physically q-deformation can be thought as a separation from the ideal symmetric system. The q-deformed hyperbolic potentials have been introduced by Arai [10,11]. They are used as model potentials for describing electronic conductance in disordered metals and doped semiconductors [12], phonon spectrum in 4 He [13], oscillatory-rotational spectra of diatomic [14] and multi-atomic molecules [15]. Among the q-deformed systems, q-deformed quantum harmonic oscillator (q-oscillator) is the most studied one and it has several applications. For example, the spectrum of the ordinary anharmonic oscillator obtained by perturbation theory can also be represented by a q-deformed harmonic oscillator as well. On can also build a connection between the 3-dimensional q-oscillator and the nuclear shell model such as reviewed in Ref. [16]. Furthermore, the q-deformation of the Lie algebra of SU(2) can be represented by two independent q-deformed harmonic oscillators. This enables one to construct the quantum theory of angular momentum which is discussed in Ref. [17]. Besides, the q-deformation of the Morse potential has been investigated in Refs. [18–20]. Such deformations have been found to be useful in several applications in chemical physics. The aim of this letter is to present the results of our computations of statistical complexity and related measures such as Shannon information entropy and disequilibrium as a function of principal quantum number n and deformation parameter q within the q-oscillator. The outline of article is like the following: the definition and meaning of López-Ruiz, Mancini and Calbet complexity measure is presented in Sec. 2. In Sec. 3, CLMC is applied to the probability distribution of a single q-oscillator. Finally, a summary of the obtained results are listed in Sec. 4. 2. Complexity Measure of López-Ruiz, Mancini and Calbet The LMC complexity measure is defined for continuous systems as the following [7], CLMC = eSx Dx
(1)
where Sx and Dx are Shannon information entropy and disequilibrium, respectively. These are defined in one dimension as
∫
∞
ρ (x) ln ρ (x) dx
Sx = −
(2)
−∞ ∞
∫
ρ 2 (x) dx
Dx =
(3)
−∞
where ρ (x) is the probability density given in terms of the wavefunction as ρ (x) = |ψ (x)|2 . Above expressions are obtained by taking the continuous limit of the following expressions which are expressed in terms of a discrete probability distribution Sx = −kB
N ∑
pi (x) ln pi (x)
(4)
i=1
Dx =
N ( ∑ i=1
pi −
1
)2
N
(5)
where pi is the probability of occupying the state i, and N is the total number of accessible states in position space. It is generally assumed that CLMC in Eq. (1) is a good statistical measure of complexity for ergodic systems. The complexity measure in the momentum space can be similarly obtained by first taking the Fourier transformation of the wavefunction in the position space followed by the computation of probability distribution using the momentum space wavefunction resulting from the transformation. 3. Complexity study of q-deformed quantum harmonic oscillator In the literature q-deformation is applied to the harmonic oscillator in an algebraic way firstly by Arik and Coon [21]. The Hamiltonian of the q-deformed quantum oscillator is given as H=
ω 2
(a− a+ + a+ a− )
(6)
where ω is frequency of the oscillator, in our work h ¯ is set to 1, a+ and a− are creation and annihilation operators, respectively which satisfy the commutation relation
[a− , a+ ]q = a− a+ − qa+ a− = 1,
(7)
with the deformation parameter q taking values in the interval (0, 1). The effect of annihilation and creation operators to the states are given as below, a− | n⟩ =
√ √ [n]|n − 1⟩, a+ |n⟩ = [n + 1]|n + 1⟩,
(8)
F. Nutku, K.D. Sen and E. Aydiner / Physica A 533 (2019) 122041
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n
where [n]q =
1−q
which satisfies the limits, limq→1 [n]q = n and limq→0 [n]q = 1 [22]. 1−q − In q-oscillator, a and a+ operators are still defined in terms of xˆ and pˆ just like the quantum harmonic oscillator as the following,
) 1 ( a+ = √ xˆ − ipˆ . 2
) 1 ( a− = √ xˆ + ipˆ , 2
(9)
Coordinate and momentum operators for the q-oscillator are given by [23], xˆ =
a+ + a−
√
2
,
pˆ = i
a+ − a−
√
2
.
(10)
On the other hand explicit definition of the a− and a+ operators of the q-oscillator are quite different than the quantum harmonic oscillator and given in Ref. [23]. At this step, it would be instructive to study the canonical commutation relationship between xˆ and pˆ . By using the definitions of xˆ and pˆ in Eq. (10) and operators a− and a+ defined in Eq. (8), one can prove that by considering the omitted h ¯ in equations, the commutation relation between xˆ and pˆ is [ˆx, pˆ ] = ih¯ qn . This result is different than the result of [ˆx, pˆ ] = ih ¯ obtained for the quantum harmonic oscillator and it is deformation dependent. A q-oscillator has more complex wavefunctions than a simple quantum harmonic oscillator. If the calculation steps in Ref. [23] are repeated, the normalized wavefunctions can be obtained as 2
|n⟩ = Ψn (x) =
exp(− x2 + 32 iα x) n√ 1 π 4 in (1 − exp(−2α 2 )) 2 [n]q !
n ∑ (−1)k [n]q ! exp(2iα (n − k)x − kα 2 ) [k]q ![n − k]q !
(11)
k=0
√
where α = i (log q)/2. In the above formula [n]q ! is the q-factorial and defined as
[n]q ! =
n ∏
1 − qn 1 − qn−1
[k]q =
1−q
k=1
1−q
···
1 − q2 1 − q 1−q 1−q
=
(q; q)n (1 − q)n
.
(12)
where (q; q)n is the q-Pochhammer symbol. On the other hand, energy eigenvalues for q-oscillator are given by [23], En =
( ) ( ) ) ω( qn ω 2 − (1 + q)qn [n]q + [n + 1]q = ω [n]q + = . 2 2 2 1−q
(13)
We note that, in the limit of q → 1, energy eigenvalues reduce to En = ω n + 21 of the ordinary quantum harmonic oscillator. However, under a small perturbation from unity (q = 1 − ε ), the energy spectrum becomes quadratic and energy eigenvalues can be approximated as the following,
(
(
En = ω n +
1 2
−
n2 2
) ε + O(ε2 ) .
)
(14)
For ϵ = 0, Eq. (14) reduces to energy eigenvalues of the quantum harmonic oscillator as well. At this point it is useful to check the equipartition theorem by calculating the expectation value of the potential energy of the nth state. This can be done by using Eqs. in (8) and ⟨V ⟩ is found as
⏐ ⟩ ⟨ ⏐ ( ) ⏐ 1 2 2⏐ ) ω 2 − (1 + q)qn ω( ⏐ ⟨V ⟩ = ⟨n|V |n⟩ = n⏐ ω xˆ ⏐⏐n = [n]q + [n + 1]q = 2 4 4 1−q
(15)
which is plotted for several n values as presented in Fig. 1. We note that as the q-deformation increases where q approaches to zero, q dependence of ⟨V ⟩ becomes saturated into a single value. In q-deformed oscillator, as one anticipates that for a given energy level n, expectation value of the potential energy is equal to the half of the total energy. This result is consistent with the equipartition theorem. We note that the q-oscillator has a nonlinear spectrum and it satisfies the Heisenberg uncertainty as [23]
∆x∆p =
En
ω
( =
2 − (1 + q)qn 2(1 − q)
)
( ) 1 ≤ n+ 2
(16)
where q changes in the interval of (0, 1). For all states, the uncertainty of q-oscillator is less than the uncertainty of quantum harmonic oscillator. In order to be more quantitative, q dependence of the uncertainty is plotted for several energy levels as seen in Fig. 2. q-deformation decreases the uncertainty in all excited states and it is more effective on high energy levels. Applying a simple numerical procedure to the related equations, with the help of Eq. (11), we have computed the probability distribution ρ (x) of q-oscillator for different eigenstates by setting m, ω and h ¯ to 1. Probability distributions of excited states (n = 1, 2, 5, 6) for different q values are given in Fig. 3a–d. As it can be seen from figures, q-oscillator shows very interesting behavior depending on both principal quantum number and q-parameter. The shape of the probability distribution dependence upon n is generally known for the quantum harmonic oscillator i.e, q = 1. Here we note that
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F. Nutku, K.D. Sen and E. Aydiner / Physica A 533 (2019) 122041
Fig. 1. Effect of q-deformation on the expectation value of the potential energy for several energy levels n, where q = 1 corresponds to the quantum harmonic oscillator case.
Fig. 2. q dependence of uncertainty plot for ground and excited states of q-oscillator.
the number of peaks in the probability distribution increases with n. In the case of q = 1, the probability distribution at the origin is very different from the distribution for the classical harmonic oscillator. Additionally, probability distribution is dependent on the principal quantum number n. In particular, ρ (x) at point x = 0 becomes maximum or minimum depending on the even or odd state n, respectively. As expected, with the increase in quantum number n, the probability distribution becomes more like that of the classical oscillator. In the course of our numerical study, we have ascertained that for q = 1, the solution gives the standard ρ (x) of the quantum harmonic oscillator. The system behaves completely dispersed. With the variation of q parameter from 1 to 0, indicating the increase of deformation, ρ (x) becomes more localized as seen in Fig. 3a–d. It is evident from Fig. 3a and b that the shape of the probability function has a similar characteristic form for some q values. However, for small q values this behavior of the ρ (x) changes dramatically. For example, at q = 0.001, probability function becomes independent of the even or odd n and stays localized at point x = 0. It is then inferred that the qdeformation changes the characteristic of probability distribution from dispersed one to the localized Gaussian one, which might be due to the broken symmetry of the q-oscillator with the increasing q-deformation parameter. This behavior has also been pointed in Ref. [23] and it is stated that, in the limit q → 0, the probability density of the q-oscillator for any n, degenerates or in other words collapses and becomes localized to the ground state probability density, according to the following expression, lim |Ψn (x)|2 =
q→0
1
π 1/2
2
e−x .
(17)
A similar localization behavior in the case of higher excited states for example n = 5 and 6 is evident in Fig. 3c and d. Here, depending on q values, the oscillating form of the ρ (x) changes quite significantly. As q decreases from 1 to
F. Nutku, K.D. Sen and E. Aydiner / Physica A 533 (2019) 122041
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Fig. 3. q dependence of the probability distribution of a q-oscillator for energy levels n = 1, 2, 5, 6. The arrow indicates increment direction of q.
0.1, the form gradually becomes more narrow and the oscillating behavior of the ρ (x) between two edges decreases and vanishes so that oscillating form evolves into a straight oscillating plateau. We note that the q dependence of ρ (x) is a very interesting result, which emerges from the extent of q-deformation. After having discussed eigenenergies, uncertainty and probability distribution of q-oscillator, now we present our results on the Shannon information entropy, disequilibrium and complexity measure of q-oscillator. Based on Eqs. (1)– (3), these quantities can be obtained numerically for ground and several excited states depending on q. As it can be seen from Fig. 4a that information entropy of the ground state is constant and independent from q. However, this behavior changes for higher n values. Information entropy increases as q increases and after a critical saturation value, it starts decreasing between the values of q = 0.8–1.0. This behavior significantly depends on the n value. The maxima of the entropy in Fig. 4a depend on n and their change is indicated by an arrow. It is seen from Fig. 3a–d, that as q decreases a smooth transition occurs in the probability distribution from dispersed one to localized one, which affects the behavior of the information entropy. In Fig. 4b, q dependence of the disequilibrium is given for different excited states. Disequilibrium of ground state is independent from q and has the largest value compared to excited states. However, disequilibrium of q-oscillator increases with decreasing q and decreasing n values, which means that the system passes from dispersed to localized behavior. On the other hand, after attaining a certain minimum value in the region q = 0.6–0.9, the disequilibrium is found to be slightly increasing as q increases especially at states with larger n. The minima of the disequilibrium in Fig. 4b depend on n and their change is indicated by an arrow.
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F. Nutku, K.D. Sen and E. Aydiner / Physica A 533 (2019) 122041
Fig. 4. q dependence of the Shannon information entropy (a), disequilibrium (b) and complexity (c) for q-deformed harmonic oscillator for energy levels changing in [0,10] where n = 7, 9 have been omitted for preventing messiness.
With reference to the ground state, for a given excited state, complexity of the q-oscillator is displayed in Fig. 4c. For all excited states complexity decreases depending on the increment in q from zero, however, it starts to increase above a critical minimum q value. The LMC complexity being a product of disequilibrium and exponential Shannon entropy, both of which depend upon the eigenstate, the net product shows the minimum at a certain q which is different for different states. This feature of the complexity results from the interplay between the variation of disequilibrium and information entropy values. We can conclude that q-oscillator has a critical transition in complexity. Around the value of q in the neighborhood of 0.9 the statistical complexity attains an approximately similar value for all excited states. Above this bunching point of statistical complexity, for the excited states, the CLMC versus q curves are found to cross over with the inversion of relative ordering of the quantum number n of the excited states. Below this bunching point, as q decreases, for each excited state the statistical complexity goes through a minimum at a certain q. The depth of the minimum value is found to be decreasing with the decreasing quantum number of the excited state. With further decrease in q beyond the minimum, the CLMC values are found to be increasing, finally, culminating at a common value at the lowest q. In order to comprehend the numerical change, we present variation of data in magnitude of Shannon information entropy,disequilibrium and complexity as obtained by using Eqs. (1)–(3) in Table 1. 4. Conclusions and remarks In this letter, we have considered a single particle q-deformed quantum harmonic oscillator with the aim of investigating the effect of deformation on the statistical complexity and defining the constituent quantities such as Shannon information entropy and disequilibrium. We have presented accurate numerical estimates of these quantities as a function of the eigenstate defined by the quantum number n and the deformation parameter q. The numerical values of statistical complexity, entropy and disequilibrium are found to be sensitively depending on the deformation parameter q. In our opinion entropy increases due to the increment of the standard deviation ∆x with q and n. Moreover, it can be seen from Fig. 4a that for all excited states as the q-parameter increases, the entropy of q-oscillator increases, which is an
F. Nutku, K.D. Sen and E. Aydiner / Physica A 533 (2019) 122041
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Table 1 Energy level and q-deformation dependence of Shannon information entropy, disequilibrium and complexity for q-oscillator. n
q
S
D
C = eS D
0
0.001 0.4 1 0.001 0.4 1 0.001 0.4 1
1.07236 1.07236 1.07236 1.07829 1.59322 1.76806 1.07829 1.61096 2.01018
0.39894 0.39894 0.39894 0.39232 0.21132 0.19666 0.39232 0.20561 0.15668
1.16582 1.16582 1.16582 1.15329 1.03962 1.15235 1.15329 1.02962 1.16957
5
10
expected result. Because in the limit of q → 0, the probability density of the q-oscillator has only one peak, however, when q → 1, probability density has a number of peaks depending on the quantum number n. This interesting result shows that q can play an important role, which drives the system from dispersed one to localized one. For n = 1 state, while q goes to 1, these peaks become enlarged and the well between the peaks goes to a minimum. Whereas, for n ≥ 2 intermediate peaks occur for large values of q, however, a plateau shape appears between the main probability density peaks at small values of q. For all cases, the probability distribution spreads in the position space and gives a big contribution to the Shannon entropy integral depending on the logarithmic factor. As it can be seen from Fig. 4b, unlike entropy, disequilibrium in q-oscillator decreases with increasing q for all quantum states as a general trend. We remark that a very interesting disequilibrium behavior in q-oscillator appears for large n values where it starts increasing after q reaches to a critical value. An opposite behavior is seen in the entropy for particularly large n values. Similarly, these behaviors of the disequilibrium can be explained depending on the behavior of the probability density distributions. In q-oscillator, for small n values the contribution of Gaussian peak to disequilibrium integral is greater than that of the small peaks which appears at large q values. However, in certain q intervals, plateau shape of the distribution gives almost constant contributions to the disequilibrium. However, when n increases this contribution increases due to appearing peaks in the distribution. Complexity measure of q-oscillator reveals a quite puzzling picture which depends on both entropy and disequilibrium. It is seen that q-deformation plays an important role on the complexity for states n ≥ 1. Indeed, complexity at first decreases with increasing q value, however it smoothly increases up to a critical q value. After the critical q value, complexity behavior completely changes. Over the whole extent, the understanding of the complexity of quantum systems under different environments is a rapidly emerging area of scientific inquiry. Scaling of the information theoretical measures in general with the key parameters of the quantum potentials have lead to an interesting new understanding of the structure and stability of such interacting systems. For example, the studies which include the relativistic effects, and the effects due to the spatial confinements have revealed novel characteristic features of complexity and other information theoretical measures [24– 28]. We believe that the q-deformed potentials extend, similarly, the scope of modeling the quantum systems under the influence of novel and diverse environments. It would be interesting to extend the present study to include the effect of spatial confinement on the quantum systems under various q-deformation types. Acknowledgments Authors are grateful to Professor Metin Arik for a valuable discussion and suggestions to clarify q-deformed quantum harmonic oscillator. KDS is grateful to The Scientific and Technological Research Council of Turkey (TÜBİTAK) for a visiting scientist award under its 2221-program, grant number 1059B211601794 and acknowledges with thanks the support received under the Senior Scientist scheme, I.N.S.A. New Delhi. References [1] S. Lloyd, Measures of complexity: a nonexhaustive list, IEEE Control Syst. Mag. 21 (4) (2001) 7–8, http://dx.doi.org/10.1109/MCS.2001.939938. [2] G.J. Chaitin, On the length of programs for computing finite binary sequences, J. ACM 13 (4) (1966) 547–569, http://dx.doi.org/10.1145/321356. 321363. [3] D.W. Miller, Selected Translations in Mathematical Statistics and Probability, Vol. 7, JSTOR, 1969. [4] G.Y. Georgiev, K. Henry, T. Bates, E. Gombos, A. Casey, M. Daly, A. Vinod, H. Lee, Mechanism of organization increase in complex systems, Complexity 21 (2) (2015) 18–28, http://dx.doi.org/10.1002/cplx.21574. [5] J.P. Crutchfield, K. Young, Inferring statistical complexity, Phys. Rev. Lett. 63 (2) (1989) 105–108, http://dx.doi.org/10.1103/PhysRevLett.63.105. [6] R. López-Ruiz, H.L. Mancini, X. Calbet, A statistical measure of complexity, Phys. Lett. A 209 (5–6) (1995) 321–326, http://dx.doi.org/10.1016/ 0375-9601(95)00867-5. [7] R. López-Ruiz, Shannon information, LMC complexity and Rényi entropies: A straightforward approach, Biophys. Chem. 115 (2-3 SPEC. ISS.) (2005) 215–218, http://dx.doi.org/10.1016/j.bpc.2004.12.035. [8] K. Sen (Ed.), Statistical Complexity: Applications in Electronic Structure, Springer, New York, 2012.
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