1 Introduction

The harmonic oscillator is undoubtedly the most prolific model in physics, from classical mechanics, electromagnetism, plasma, quantum mechanics, atomic physics, nuclear physics and elementary particles. In nuclear physics, the harmonic oscillator has been used in four, five and six dimensions to study the Coulomb problem, nuclear collective motions and a model of interacting bosons, respectively [1]. N-dimensional harmonic oscillator have been used in string theory [2, 3], and the four-dimensional harmonic oscillators have been used in D-brane space [4,5,6,7].

The quantum mechanical treatment of numerous three-dimensional problems has been the subject of intense and relevant investigation in physics, as an example, we may cite studies of specific heat, energy fluctuations and entropy of harmonic oscillators in one, two and three dimensions [8]. Recently, two-dimensional systems have attracted a great deal of attention due to the fact that technological advances now make it possible to design low-dimensional structures such as wells, wires and quantum dots [9,10,11,12]. The two-dimensional harmonic oscillator has been used by Ikhdair and Sever to study the relativistic harmonic oscillator plus Cornell potentials in an external magnetic field and of Aharonov–Bohm field [13]. The two-dimensional free hydrogen atom has been studied in a non-relativistic and relativistic quantum mechanics.

On the other hand, confined systems have been used as models in the study of a large variety of properties of matter under high pressure, embedded in cavities of different size and geometrical forms, [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]. The most studied confined systems are the one-dimensional [14, 22,23,24,25,26,27,28,29,30,31,32], three-dimensional [16, 33] confined harmonic oscillators (CHO) and the confined hydrogen atom (CHA) [14,15,16, 18,19,20,21,22], because these systems are quite simple and they have all the features normally found in more complicated confined systems. The one-dimensional CHO was used at the beginning of the 40’s by Kotari and Auluck [23], Auluck [24] and Chandrasekhar [25] as a model in the study of some properties of dense stars, white dwarfs and galactic clusters. The CHO was also used as a model to analyse certain electric and magnetic properties [28] and the specific heat of monocrystals [27].

Some examples of two-dimensional confined systems previously analysed are as follows: the computation of energy spectrum of a hydrogen atom confined in a circular impenetrable disc [34,35,36,37,38,39], the two-dimensional hydrogen atom in an external magnetic field, in empty space [40,41,42,43] or embedded in a plasma, as well as the effects arising on a helium atom adsorbed on the surface of graphite [44]. Other studies correspond to the description of impurity states and electron–hole junctions between insulators and semiconductors [26, 45], the two-dimensional free \(H_2^+\) and \(H_2\) molecules [46] and the two-dimensional molecular ion \(H_2^+\) confined in an ellipse [47].

Lumb et al. [48] studied the one-dimensional harmonic oscillator subjected to a constant electric field and in a time dependent laser field. Choluj and Bartkowiak [49] used the harmonic oscillator in two and three dimensions as models to study electrical properties of spatially confined LiH, LiF and HF molecules.

Information theory has been given a growing interest due to its applications. Some examples are as follows: chemical reactivity, structure of atoms and molecules, derivation of equations of physics, imagenology, seismology and quantum computation, the analysis of Dirac and non-relativistic oscillators among others [50,51,52,53,54,55,56,57,58]

Studies related to entropic measurements in quantum systems have been increasing in recent years. Shannon entropy and Fisher information have been employed as information theoretical measures in diatomic molecules with generalized Kratzer potential [59], in the study of correlation in the confined helium [60] ant two-electron systems [61], and many electron neutral atoms and ions [62]. Corzo et al. [63] studied the localization-delocalization of a free particle in a quantum corral using Shannon entropy and radial variance. The information entropy has been used in the study of atoms [64, 65] and in the analysis of avoided spectral crossings of atoms under external fields [66, 67].

However, little has comparatively been reported on the study of measures of theoretical–information in confined multidimensional systems, albeit some recent entropy and complexity information works on confined three-dimensional atoms and the confined two-dimensional hydrogen atom have been addressed. The aim of this report is to study the physical effects on an electron spatially confined in a two-dimensional harmonic oscillator inside an impenetrable circle and in the presence of a constant electric field by analysing the evolution of Shannon entropy, Fisher information and statistical variance in such a system.

This work is organized as follows: Sect. 2 describes the method of solution of the harmonic oscillator confined inside a circle subject to a constant electric field. In Sect. 3, we calculate the Shannon entropy and Fisher information as a function of the confinement radius and the electric field magnitude. In Sect. 4, we compute the radial standard deviation also as a function of the confinement radius and the electric field. In Sect. 5, we compare the results obtained for the three measures: Shannon entropy, Fisher information and variance. Finally, in Sect. 6 we give our conclusions.

2 Energy eigenvalues and eigenfunctions

2.1 The two-dimensional harmonic oscillator confined in a circle

In this section, we give a brief description of the problem, where the details can be found in reference [68].

The Hamiltonian of the two-dimensional harmonic oscillator confined in a circle (2DHOC) in polar coordinates \((r,\phi )\) is given by:

$$\begin{aligned} H'= \frac{p'^2}{2m_e}+ \frac{1}{2}m_e\omega ^2r'^2 +V_c(r'), \end{aligned}$$
(1)

where \(m_e\), e and \(\omega \) are the electron mass, electron charge and the frequency, respectively. \(V_c (r')\) is the confinement potential, given by:

$$\begin{aligned} V_c(r')=\left\{ \begin{array}{rcl} 0 &{} \text{ si } &{} 0 \le r' \le r_{_0} \\ &{} &{} \\ \infty &{} \text{ si } &{} r'> r_{_0}.\\ \end{array} \right. \end{aligned}$$
(2)

The \('\) indicates variables with dimensions. In order to solve the problem it is convenient to write the Hamiltonian of Eq. (1) in dimensionless variables, for which we make the following change of variable

$$\begin{aligned} r'=r\sqrt{\frac{\hbar }{m_e\omega }}. \end{aligned}$$
(3)

In dimensionless variables, the Hamiltonian (Eq. (1)) is given by:

$$\begin{aligned} H= \frac{p^2}{2}+ \frac{1}{2}r^2 +V_c (r), \end{aligned}$$
(4)

which is the Hamiltonian of a two–dimensional harmonic oscillator confined in a circle of radius \(r_0\). The Schrödinger equation of the problem can be written as

$$\begin{aligned} H\psi (r,\phi )=E\psi (r,\phi ), \end{aligned}$$
(5)

where E is the energy, in units of \(\hbar \omega \).

The wave function must fulfil the Dirichlet boundary condition:

$$\begin{aligned} \psi (r=r_{_0},\phi )=0. \end{aligned}$$
(6)

The solution of the Schrödinger equation is found by proposing a separable function,

$$\begin{aligned} \psi (r,\phi )=R(r)\Phi (\phi ). \end{aligned}$$
(7)

The angular solutions are well-known eigenfunctions of the operator \(L_z\)

$$\begin{aligned} \Phi _{m}(\phi )=\frac{1}{\sqrt{2\pi }}e^{im\phi },\left| m\right| =0,1,2,3\ldots , \end{aligned}$$
(8)

whereas the radial function fulfils the following equation

$$\begin{aligned} -\frac{1}{2}\left[ \frac{{\textrm{d}}^{2}}{{\textrm{d}}r^{2}}+\frac{1}{r}\frac{{\textrm{d}}}{{\textrm{d}}r}-\frac{m^{2}}{r^{2}}\right] R+\frac{1}{2} r^{2}R=ER, \end{aligned}$$
(9)

with the boundary condition

$$\begin{aligned} R(r =r_{0})=0. \end{aligned}$$
(10)

The radial solutions are of the following form [68]:

$$\begin{aligned} R_{n,m}(r)= & {} N_{n,m} \quad r^{\left| m\right| } e^{-r^{2}/2} \nonumber \\{} & {} \times M\left[ \frac{1}{2}\left( \left| m\right| +1-E_{nm}\right) ;\left| m\right| +1;r^{2}\right] ,\nonumber \\ \end{aligned}$$
(11)

in which the function M is the confluent hypergeometric function [69], and \(N_{n,m}\) is the normalization constant.

The allowed energies are found by imposing the boundary condition given in Eq. (2) on the wave function R(r), which leads to

$$\begin{aligned} M\left[ \frac{1}{2}(\left| m\right| +1-E_{nm});\left| m\right| +1; r _{0}^{2}\right] =0 \end{aligned}$$
(12)

For a fixed value of m the successive roots are numbered as \(n=0,1,2,3\ldots \), where n is the radial quantum number. Those roots are the energy eigenvalues of the Schrödinger equation.

The eigenfunctions are given by

$$\begin{aligned} \psi _{n,m}= R_{n,m}(r) \frac{e^{i m \phi }}{\sqrt{2 \pi }}, \quad n{=}0,1,2,\ldots , |m|{=}0,1,2,\ldots n,\nonumber \\ \end{aligned}$$
(13)

where \(R_{nm}\) is given by Eq. (11).

2.2 The two-dimensional harmonic oscillator in a circle subjected to a constant electric field

The Hamiltonian H of an electron subjected to a harmonic oscillator potential (HO) plus a stationary electric field of magnitude \(f^{\prime }\) pointing in the direction of the x-axis, in a circular region can be written as

$$\begin{aligned} H^{\prime }=\frac{p^{\prime 2}}{2m_e}+\frac{1}{2}m_e\omega ^{2}r^{\prime 2}+ef^{\prime }x^{\prime }+V_{c}^{\prime }, \end{aligned}$$
(14)

where the confining potential \(V_c ^{\prime }\) is given by Eq. (2).

In order to solve the problem, it is convenient to write the Hamiltonian in dimensionless variables, r in units of \(\sqrt{\frac{\hbar }{m_e \omega }}\) and the energy in units of \(\hbar \omega \).

$$\begin{aligned} H=\frac{p^{2}}{2}+\frac{1}{2}r^{2}+f r\cos {\phi } +V_{c}, \end{aligned}$$
(15)

where

$$\begin{aligned} f=\frac{ef^{\prime }}{\sqrt{\hbar m_e\omega ^{3}}}. \end{aligned}$$
(16)

The Hamiltonian H can be written as

$$\begin{aligned} H=H^{(0)}+H^{(1) }, \end{aligned}$$
(17)

in which \(H^{(0)}\) is the Hamiltonian corresponding to an electron subjected to a HO potential confined in a circle of radius \(r_{0}\) (Eq. (4)), discussed in the previous subsection,

$$\begin{aligned} H^{(0)}=\frac{p^{2}}{2}+\frac{1}{2} r^{2}+V_{c}, \end{aligned}$$
(18)

and \(H^{(1) }\) is given by

$$\begin{aligned} H^{(1) }=f r\cos {\phi } \end{aligned}$$
(19)

The Schrödinger equation to solve is given by

$$\begin{aligned} H\Psi =E \Psi . \end{aligned}$$
(20)
Table 1 Energy eigenvalues of the ground state and the first excited states of the 2DHOC as a function of the electric field f, for \(r_{0}=1.0, 2.0, 3.0\)
Full size table

Note that \(L_z\) does not commute with H, so m is no longer a good quantum number. To find the energy eigenvalues and eigenfunctions of the Hamiltonian H (Eq. (15)), we use the linear variational method. The eigenfunctions \(\Psi \) of Hamiltonian H are expanded in terms of the eigenfunctions \(\psi _{n,m}\) of the Hamiltonian \(H^{(0)}\) of the 2DCHO (previous subsection), by summing over the quantum number m for a fixed value of the radial quantum number n.

The approximate wave function \(\Psi (r,\phi )\) can be written as:

$$\begin{aligned} \Psi (r,\phi )=\sum _{m=-M}^{M} c_m \psi _{n,m} (r,\phi ), \end{aligned}$$
(21)

where \(\psi _{n,m} (r,\phi )\) are the eigenfunctions of \(H^{(0)}\) given by Eq. (13), \(M \ge 0 \) is the maximum angular momentum considered in the calculations and the \(\{ c_m \}\)’s are linear variational parameters. \(k=2\,M+1\) is the number of functions of the basis in the summation.

According to the linear variational method, solving the Schrödinger equation is equivalent to finding the solution of the eigenvalue problem

$$\begin{aligned} (\textbf{H}-E \textbf{I}) \textbf{c}=0, \end{aligned}$$
(22)

where E is the energy eigenvalue, \(\textbf{I}\) is the identity matrix, \(\textbf{c}=(c_{-M}, c_{-M+1},c_{-M+2}, \ldots , c_M)\) is the coefficient vector and \(\textbf{H}\) is the Hamiltonian matrix whose elements are given by

$$\begin{aligned} H_{k,l}= H^{(0)}_{k,l}+H^{(1)}_{k,l}, \end{aligned}$$
(23)

where

$$\begin{aligned} H^{(0)}_{k,l}=E^{(0)}_{n,l} \delta _{k,l}, \end{aligned}$$
(24)

and the \(E^{(0)}_{n,l}\)’s are the energy eigenvalues of 2DHOC of Sect. 2.1.

The non-diagonal matrix elements are given by

$$\begin{aligned} H^{(1)}_{k,l}=\frac{f}{2} \langle R_{n,k}|r|R_{n,l} \rangle (\delta _{l,k-1}+\delta _{l,k+1}). \end{aligned}$$
(25)

By solving the eigenvalue problem (Eq. (22)) the energies and their corresponding coefficient vectors are obtained where the wave functions are constructed by means of Eq. (21). After having performed the numerical calculations we found those for the ground state, second and fourth excited states \(c_k=c_{-k}\), therefore, Eq. (21) is of the following form:

$$\begin{aligned} \Psi (r,\phi )=\frac{c_0}{\sqrt{2 \pi }} R_{n,0}(r)+\sum _{k=1}^{M} \frac{2 c_k}{\sqrt{2 \pi }} R_{n,k}(r) \cos (k \phi ), \end{aligned}$$
(26)

whereas for the first and third excited state we found \(c_k=-c_{-k}\), their wave functions Eq. (21) can be written as:

$$\begin{aligned} \Psi (r,\phi )=\sum _{k=1}^{M} \frac{2 c_k}{\sqrt{2 \pi }} R_{n,k}(r) \sin (k \phi ). \end{aligned}$$
(27)

The wave functions that will be used in the following sections to calculate the entropic measurements are those given by the real equations (26) and (27).

To compute the wave functions in momentum space, we have performed the Fourier integral transform of the position wave functions (Eq. (21)), obtaining

$$\begin{aligned} \Phi (\textbf{p})=\frac{1}{2 \pi } \int \Psi (r,\phi ) e^{-i \textbf{p} \cdot \textbf{r} } {\textrm{d}}\textbf{r} \end{aligned}$$
(28)

which can be rewritten as

$$\begin{aligned} \Phi (p, \theta )=\sum _{m=-M}^{M} \frac{c_m i^m e^{i m \theta } }{\sqrt{2 \pi }} \int _{0}^{r_0} R_{n,m}(r) J_m(pr) r{\textrm{d}}r, \end{aligned}$$
(29)

where we used the following identity [70],

$$\begin{aligned} \int _{0}^{2\pi }e^{-ikr\cos {(\varphi -t)}}\left\{ \begin{array}{c} \cos {(mt)} \\ \sin {(mt)} \end{array} \right\} dt{} & {} \nonumber \\ = 2\pi i^{m}\left\{ \begin{array}{c} \cos {(mt)} \\ \sin {(mt)} \end{array} \right\} J_{m}(kr).{} & {} \end{aligned}$$
(30)

\(J_m\) being the Bessel function of the first kind and order m.

Hereafter we devote our attention to the study of states with radial quantum number \(n=0\). Table 1 reports the energy eigenvalues of the ground and lowest four excited states of 2DHOC as a function of the electric field f for three different values of the confinement radius \(r_0\). The number of functions of the basis set was increased until energy convergence was reached with a previously fixed error. To achieve the energy accuracy shown in Table 1, only 17 basis functions were sufficient. In Fig. 1, we show the variation of the energy eigenvalues as a function of the electric field f. The energy of all states decreases with increasing electric field value. When the electric field becomes zero, the first and second excited states become degenerate, which is to be expected since these are the 2DHOC states with quantum numbers: \(n=0, m=-1\) and \(n=0, m=1\), respectively. The same trend occurs for the third and fourth excited states.

Fig. 1
figure 1

Energy eigenvalues of the ground and some excited states as a function of the magnitude of the electric field f

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3 Entropic measures

In this section, we investigate the effect of box size and magnitude of the electric field on the Shannon entropy and Fisher information of the ground and four excited states of the 2DHOC, in both position and momentum spaces. We also study the corresponding uncertainty relations.

The corresponding position and momentum probability densities for this system are as follows:

$$\begin{aligned} \rho (r,\phi )=|\Psi (r,\phi )|^2, \end{aligned}$$
(31)

and

$$\begin{aligned} \gamma (p,\theta )=|\Phi (p,\theta )|^2, \end{aligned}$$
(32)

which are the basic variables of the information theory for the two-dimensional harmonic oscillator confined in an impenetrable circular box embedded in a constant electric field. In the next sections, those probability densities will be used to compute the Shannon entropy and Fisher information of the system, as a function of the box radius \(r_0\) and the electric field f.

In Fig. 2, we show the probability density \(\rho (r,\phi )\) (Eq. (31)) for ground state and the first four excited states, for a fixed value of \(r_0=2\), for selected values of the electric field strength. The electric field is applied along the x-axis direction. As the electric field increases, the probability density moves in the opposite direction of the electric field. As the electric field grows, the electron density regroups on the left side of the circular well (localization). A similar behaviour is found for a circular box of radius \(r_0=1\), but it is necessary to apply electric fields of higher intensity to polarize the electron cloud. On the other hand, for a circular box of radius \(r_0=3\), the electric fields that polarize the electronic cloud are smaller than for a radius \(r_0=2\).

Fig. 2
figure 2

The probability density \(\rho (r,\phi )=\Psi ^*(r,\phi ) \Psi (r,\phi )\) of ground state (a), first excited state (b), second excited state (c), third excited state (d) and fourth excited state (e), for a fixed confinement radius \(r_{_0}=2\) and few selected values of the electric field f

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3.1 Shannon entropy

The Shannon entropy of the two-dimensional harmonic oscillator in a constant electric field confined in a circle of radius \(r_0\), in configuration and momentum spaces are given by [71,72,73,74]

$$\begin{aligned} S_{\rho }=-\int _{r=0}^{r_0} \int _{\phi =0}^{2 \pi } \rho (r,\phi ) ln[\rho (r,\phi )] r{\textrm{d}}r {\textrm{d}}\phi \end{aligned}$$
(33)

and

$$\begin{aligned} S_{\gamma }=-\int _{p=0}^{\infty } \int _{\theta =0}^{2 \pi } \gamma (p, \theta ) ln[\gamma (p, \theta )] p \textrm{d}p \textrm{d} \theta , \end{aligned}$$
(34)

where \(\rho (r, \phi )\) and \(\gamma (p, \theta )\), defined previously, are the position and momentum probabilities, respectively. In practice, the calculation of the integral of \(S_{\gamma }\) over p is performed numerically from 0 to \(p_{\max }\). The value of \(p_{\max }\) is obtained by requiring that the normalization of the function \(\Phi (p, \theta )\) to be \(1 \pm \epsilon \), where \(\epsilon \) is at most of the order of \(1 \times 10^{-5}\). Both position and momentum global quantities fulfil the well-known uncertainty entropy relation BBM [75]

$$\begin{aligned} S_T=S_{\rho }+S_{\gamma } \ge 2 (1+ ln(\pi )) \simeq 4.29 \end{aligned}$$
(35)

The Shannon entropy constitutes a measure of spreading of the probability density distribution. In configuration space, a large value of the Shannon entropy \(S_{\rho }\) means that the uncertainty in the particle position is also large and thus delocalized. A small value of the Shannon entropy \(S_{\rho }\) means that the probability density is more concentrated and the position uncertainty is small, and therefore, the particle is more localized. The interpretation of Shannon entropy in the momentum space, \(S_{\gamma }\), although is completely analogous, one should bear in mind that when analysing a simultaneous evolution of both entropies, localization in configuration space is accompanied by delocalization in momentum space, and vice versa.

Fig. 3
figure 3

Behaviour of the Shannon entropy in the configuration space \(S_{\rho }\) as a function of the electric field

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Fig. 4
figure 4

Behaviour of the Shannon entropy in the momentum space \(S_{\gamma }\) as a function of the electric field

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Fig. 5
figure 5

Behaviour of the Shannon entropy in the configuration space \(S_{\rho }\), momentum space \(S_{\gamma }\) and the total entropy \(S_T\) as a function of the electric field for the ground state and the first four excited states

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In Figs. 3 and 4 it is illustrated the behaviour of Shannon entropy for \(r_0\) = 1.0, 2.0 and 3.0 as a function of the electric field.

Configuration space entropy

In Fig. 3, we show how the Shannon entropies \(S_{\rho }\) evolve for the ground and lowest four excited states as a function of the electric field f. At box radius \(r_0=1\), it decreases (localization) for the ground, first and third excited states as a function of f. On the other hand, for the second state it grows (delocalization) in the electric field interval [0,6], whereas for the third and fourth excited states it remains almost constant as a function of f. At box radius \(r_0=2\), it also decreases for the ground, first and third excited states, whereas increasing for the second and fourth excited states and attaining a maximum value to then go down as the electric field f becomes more intense. At box radius \(r_0=3.0\), the behaviour of the Shannon entropy is similar to that in the previous cavity, albeit the maxima for the second and fourth excited states are attained at comparatively smaller values of the electric field f. It should be noted that \(S_{\rho }\) always decreases for all states as a function of the electric field since physically the particle will tend to be more localized for increasingly higher values of f. The apparent Shannon entropy increase for the second and fourth excited states simply reflects the symmetry breaking between the first and second as well as the third and fourth states, albeit all of them should decrease as a function of the electric field, i.e. the combined effect of symmetry breaking and decrease of \(S_{\rho }\) for higher values of f forces one of the originally degenerate states to go up (the second and fourth) although eventually they also decrease for sufficiently intense magnitudes of the electric field. Such combined effect is more pronounced for stronger confinement regimes (\(r_0\) = 1 and 2) and lower state energies.

Momentum space entropy

In Fig. 4, it is displayed the Shannon entropy \(S_{\gamma }\) evolution for the ground and lowest four excited states as a function of the electric field f, for the same box radii considered when analysing \(S_{\rho }\): \(r_0=\) 1.0, 2.0 and 3.0. At \(r_0=1.0\) it grows (delocalization) for all states as a function of the electric field f. At \(r_0=2.0\) they have the same value for the first and second excited state at \(f=0\) and they separate as the field increases, whereas for the third and fourth excited states the \(S_{\gamma }\) values are very similar at the electric field range \(f < 2\) and then they separate to become again very similar at \(f=6\). At box radius \(r_0=3.0\) the Shannon entropies for the first and second excited states clearly separate as the field increases. Likewise, for the third and fourth excited states they also separate where the former gets higher in the range \(1.5< f < 3\), whereas in \(3< f < 6\) such behaviour is reversed. The evolution of \(S_{\gamma }\) for all states can be understood in a similar way as explained for \(S_{\rho }\) in Fig. 3 having in mind that both quantities behave in exactly opposite manner with respect to each other, i.e. when one decreases (\(S_{\rho }\)) the other increases (\(S_{\gamma }\)) as a function of the electric field, and vice versa. Notice however that the behaviour for \(S_{\gamma }\) is less pronounced in comparison to \(S_{\rho }\) in terms of the electric field.

Total entropy \(S_T\)

The BBM relation (Eq. (35)) is fulfilled for all states and all values of f and box radii \(r_0\) here considered. In Fig. 5, the evolution of \(S_{\rho }, S_{\gamma }\) and \(S_T\) is shown for the ground state and for the first four excited states as a function of the electric field f, at box radii \(r_0=1.0\) (Fig. 5a, d, g, j, m), \(r_0=2.0\) (Fig. 5b, e, h, k, n) and \(r_0=3.0\) (Fig. 5c, f, i, l, o). The curves for the configuration and momentum spaces appear to proceed in a symmetric fashion—one above (\(S_{\gamma }\)) and the other below (\(S_{\rho }\))—with respect to the total entropy \(S_T\) which remains almost unchanged, all three as a function of the electric field f. The main difference for the three states among the various confinement regimes consists of an almost constant evolution of the total entropy \(S_T\) at box radius \(r_0=1\) whereas comparatively it slightly increases at \(r_0=2\) and a bit more at \(r_0=3\), as a function of the electric field f. The flattening of the \(S_T\) curve appears to occur more rapidly the higher the state energy and the higher the confinement regime (i.e. the smaller the box radius) because of Heisenberg’s Uncertainty Principle reduced box radii lead to an increased energy for each state.

Fig. 6
figure 6

Behaviour of the Fisher information in the configuration space \(F_{\rho }\) as a function of the electric field f for three values of \(r_0\)

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Fig. 7
figure 7

Behaviour of the Fisher information in the momentum space \(F_{\gamma }\) as a function of the electric field f for three values of \(r_0\)

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3.2 Fisher information

The Fisher information for the position probability density \(\rho (r, \phi )\) is given by:

$$\begin{aligned} F_{\rho }=\int _{r=0}^{r_0} \int _{\phi =0}^{2 \pi } \frac{\mid \nabla \rho (r, \phi ) \mid ^2 }{\rho (r, \phi )} r {\textrm{d}}r {\textrm{d}} \phi , \end{aligned}$$
(36)

this information measure quantifies the narrowness or concentration of the probability density [76,77,78,79] and it is very sensitive to the density oscillations of the system.

The momentum Fisher information is defined as:

$$\begin{aligned} F_{\gamma }=\int _{p=0}^{\infty } \int _{\theta =0}^{2 \pi } \frac{\mid \nabla \gamma (p, \theta ) \mid ^2 }{\gamma (p, \theta )} p \textrm{d}p \textrm{d} \theta , \end{aligned}$$
(37)

where the momentum density \(\gamma (p, \theta )\) is given by Equation (32). As mentioned in Sect. 3.1, in practice the integral over p is performed from 0 to \(p_{\max }\).

Configuration space

In Fig. 6, we show the Fisher information \(F_{\rho }\) as a function of f for three values of \(r_0\). For a box radius \(r_0=1\) it increases (localization) for the ground and first excited states whereas for the second excited state it slowly diminishes (delocalization). On the other hand, the third and fourth excited states have almost the same value of \(F_{\rho }\) which remains virtually constant as we proceed along the electric field. For a box radius \(r_0=2\) it grows (localization) for ground and all excited states. An analogous behaviour of the lowest five states is found for \(r_0=3\). Notice that here it is also manifested the same trend for degenerate states where symmetry breaking and growing effect on the Fisher information compete with each other as we increase the electric field. However, such combined effects are less pronounced in comparison to the evolution of the Shannon entropy in configuration space for all states.

Momentum space

In Fig. 7, we show the Fisher information \(F_{\gamma }\) in the momentum space as a function of f. As can be seen, at the box radii here considered \(F_{\gamma }\) decreases for all states, thus indicating delocalization as we move forward in the electric field f. At \(r_0=1\) it decreases almost linearly for all states and at a more rapid rate for larger box radii, as a function of f. Here, similarly to the behaviour of Shannon entropies \(S_{\rho }\) and \(S_{\gamma }\) when compared to each other, the evolution of \(F_{\gamma }\) for all states can also be understood along the same lines as explained for \(F_{\rho }\) in Fig. 6, having in mind that both quantities behave in exactly opposite manner with respect to each other, i.e. when one increases (\(F_{\rho }\)) as a function of the electric field, the other decreases (\(F_{\gamma }\)) and vice versa.

The Fisher information \(F_{\rho }\) and \(F_{\gamma }\) are local quantities fulfilling the uncertainty relation:

$$\begin{aligned} F_{\rho }\times F_{\gamma } \ge 4 \times 2^2=16, \end{aligned}$$
(38)

for the real wave functions of the system [80].

Fig. 8
figure 8

The radial probability \( r \rho _r(r)\) of ground state (a), first excited state (b), second excited state (c), third excited state (d) and fourth excited state (e), for a fixed value of the confinement radius \(r_{_0}=2\) and the selected values of the electric field

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Fig. 9
figure 9

Behaviour of the standard deviation \(\sigma _r\) as a function of the electric field for three different confinement radii

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4 The radial standard deviation

A measure of location used in the quantum corral problem [63] and of an electron within a circular region with constant magnetic field [81] is the radial standard deviation.

The radial density is defined as

$$\begin{aligned} \rho _r(r)=\int _{0}^{2 \pi } \psi ^{*}(r,\phi ) \psi (r,\phi ) {\textrm{d}}\phi \end{aligned}$$
(39)

The radial probability distribution \(\rho _r(r) r {\textrm{d}}r\) is the probability of finding the electron between the radial distances r and \(r+{\textrm{d}}r\). Fig. 8 shows the radial probability distribution for the first five states for \(r_0=2\) and for three values of the electric field f = 1, 3 and 6.

The radial standard deviation is given by [63]

$$\begin{aligned} \sigma _r=\left[ \int _0^{r_0} r^2 \rho _r(r) r{\textrm{d}}r- \left( \int _0^{r_0} r \rho _r(r) r{\textrm{d}}r\right) ^{2} \right] ^{1/2} \end{aligned}$$
(40)

The standard deviation \(\sigma _r\) is associated with dispersion around the mean value of radial distance r and in the present context it is used as a measure of localization– delocalization of a particle: as the standard deviation diminishes the particle is more localized, and vice versa, as it grows, the particle becomes delocalized. We computed the standard deviation of r for the ground and lowest four excited states at three different values of box size \(r_0\), as a function of the electric field f.

In Fig. 9, we show the standard deviation \(\sigma _r\) as a function of the electric field f for three different box sizes: \(r_0=1.0\), 2.0 and 3.0. For the first excited state, it decreases (localize) very slightly at \(r_0=1.0\), whereas growing (delocalize) at a very low rate for the other states. The small variation in \(\sigma _r\) is due to the fact that for this box size the electric field polarizes the electron at a very limited extent. For electric field magnitudes \(f<1\), the first and second excited states possess the same value of \(\sigma _r\), while splitting for field values \(f>1\). On the other hand, the same values of \(\sigma _r\) occur for the third and fourth excited states as a function of f, at least at the range of values shown in the figure. At \(r_0=2.0\), the standard deviation for the ground state increases (delocalize), attaining a maximum around \(f=2.5\) and slowly decreasing (localize) thereafter. For the first excited state the standard deviation decreases (localize) slowly as the electric field f increases, whereas for the second excited state it increases (delocalize), also up to a maximum value, to then slowly decrease (localize). The third and fourth excited states have the same value of the standard deviation, which increases (delocalize) up to a maximum value to then very slowly decrease (localize). For \(r_0=3.0\) the values of \(\sigma _r\) increase (delocalize) for all states, reaching a maximum to then decrease (localize). Here we should note that the third and fourth excited states have the same value of \(\sigma _r\) as in the previous box radii, \(r_0=\) 1.0 and 2.0.

5 Discussion

The behaviour of Shannon entropy, Fisher information and \(\sigma _r\) as a function of f is different for some states and box sizes, which can be seen by comparing \(S_r\), \(F_{\rho }\) and \(\sigma _r\) in Figs. 3, 6 and 9, respectively. For the ground state, the Shannon entropy decreases (localization) and the Fisher information increases (localization) as a function of f for the three box sizes, thus, both quantities lead to a probability density that is more localized as the electric field increases. On the other hand, at \(r_0=1\), \(\sigma _r\) increases slowly with f, indicating an increment in the probability density dispersion (delocalization). For \(r_0=2\), the \(\sigma _r\) value increases (delocalization) with the field, and around \(f=2.5\), it starts to decrease (localization). Thus, we see that Shannon entropy and Fisher information behave in a consistent manner, whereas \(\sigma _r\) apparently evolves in an opposite fashion along certain intervals of f.

Regarding the first excited state, the Shannon entropy decreases (localization) and the Fisher information increases (localization) as a function of f, for the three values of \(r_0\). On the other hand, for the same state, \(\sigma _r\) decreases (localization) as a function of f for \(r_0=1\) and \(r_0=2\), whereas for \(r_0=3\), it increases (delocalization), attaining a maximum, to then decrease (localization) thereafter.

Analysing the other excited states, as seen in the same figures, it is concluded that the Shannon entropy and Fisher information behave consistently whereas the standard deviation \(\sigma _r\) apparently proceeds in a slightly anomalous way at certain intervals of f and some values of \(r_0\).

Although the evolution of the standard deviation is apparently inconsistent in comparison to Shannon entropy and Fisher information as a function of the electric field in some intervals, its increase for the second and fourth excited states reflects symmetry breaking between the first and second although not for the third and fourth states (whose \(\sigma _r\) values are equal). So, even though \(\sigma _r\) would in general be expected to decrease, the combined effect of symmetry breaking (only for the first and second states) and different degrees of dispersion for the various excited states leads it to go up for one of the originally degenerate states although eventually it decreases for sufficiently intense magnitudes of the electric field. Such combined effect is more pronounced for weaker confinement regimes (\(r_0\) = 2 and 3) and higher state energies.

The fact that symmetry breaking is only observed for the first (\(\sigma _r\) goes down) and second (\(\sigma _r\) goes up) states, whereas not so for the third and fourth states (\(\sigma _r\) goes up for both and with the same value) may be a consequence of m not being a good quantum number since \(L_z\) does not commute with H, as mentioned above. This would explain the anomalous behaviour of the standard deviation as compared with that of \(S_r\) and \(F_{\rho }\), which evolve in a consistent fashion as a function of the electric field.

The Shannon entropy in configuration and momentum spaces satisfy the BBM (Eq. (35)) inequality for the values of f and \(r_0\) used in the present work. Likewise, the Fisher information in configuration and the momentum spaces fulfil the inequality product (Eq. (38)) for all values of f and \(r_0\) here considered.

6 Conclusions

We have obtained energies and wave functions of an electron subjected to a two-dimensional harmonic oscillator potential confined in a circular well in the presence of an external constant electric field. In the absence of the latter, all states with nonzero angular momentum are doubly degenerate, where such degeneracy is removed when introducing an electric field.

The electric field polarizes the electron density and shifts the probability density maxima from the positions originally held in its absence. In order to quantify how the probability density is modified in the presence of an electric field, Shannon entropy, Fisher information and radial standard deviation were here analysed. Our findings indicate that the overall effect of the electric field on a charged particle under the influence of a 2-D harmonic oscillator and confined by hard circular barriers is a spatial localization of the electron density.

The variation in Shannon entropy and Fisher information, as a function of the electric field, shows that these measures are more sensitive to changes in the probability density caused by the presence of the electric field.

The total Shannon entropy of the ground state increases slightly as a function of the electric field for boxes of \(r_0=1, 2\), and more quickly for \(r_0=3\). For the first excited state, the total Shannon entropy is almost constant as a function of the electric field, for \(r_0=1, 2\) and 3. On the order hand, for the second excited state the total Shannon entropy increases, slightly for \(r_0=1\) and more quickly for \(r_0=2, 3\), as the electric field grows. The third excited state has a behaviour analogous to that of the ground state and first excited state, while the fourth excited state behaves analogously to the second excited state. This behaviour is analogous to that found in the problem of a particle inside an impenetrable sphere in the presence of a constant electric field [82].

Fisher information in configuration space shows an increase as a function of electric field, while in momentum space it shows a decrease, as expected.

We find that the Shannon entropy and Fisher information better describe the electron localization-delocalization than the radial standard deviation, which agrees with the results found in the study of a particle confined in a circle in the presence of a constant magnetic field [81].

Shannon entropy and Fisher information explicitly incorporate the probability distribution of electron positions, whereas radial standard deviation only considers the spread of positions without regard to their probabilities. This makes Shannon entropy and Fisher information more sensitive to the actual distribution of electrons, capturing nuances in localization and delocalization. Also, Shannon entropy and Fisher information are more sensitive to changes in the shape and spread of probability distributions. They can distinguish between different types of distributions, such as highly localized versus delocalized electron distributions, which may have similar radial standard deviations but different information content.