Momentum Probability Densities Research Articles

Quantum Information Entropy for Another Class of New Proposed Hyperbolic Potentials.

In this work, we investigate the Shannon entropy of four recently proposed hyperbolic potentials through studying position and momentum entropies. Our analysis reveals that the wave functions of the single-well potentials U0,3 exhibit greater localization compared to the double-well potentials U1,2. This difference in localization arises from the depths of the single- and double-well potentials. Specifically, we observe that the position entropy density shows higher localization for the single-well potentials, while their momentum probability density becomes more delocalized. Conversely, the double-well potentials demonstrate the opposite behavior, with position entropy density being less localized and momentum probability density showing increased localization. Notably, our study also involves examining the Bialynicki-Birula and Mycielski (BBM) inequality, where we find that the Shannon entropies still satisfy this inequality for varying depths u¯. An intriguing observation is that the sum of position and momentum entropies increases with the variable u¯ for potentials U1,2,3, while for U0, the sum decreases with u¯. Additionally, the sum of the cases U0 and U3 almost remains constant within the relative value 0.01 as u¯ increases. Our study provides valuable insights into the Shannon entropy behavior for these hyperbolic potentials, shedding light on their localization characteristics and their relation to the potential depths. Finally, we extend our analysis to the Fisher entropy F¯x and find that it increases with the depth u¯ of the potential wells but F¯p decreases with the depth.

Relativistic energies and information entropy of the inversely quadratic Hellmann potential

The solutions to the Dirac equation are obtained in the spin and pseudospin symmetry limits are presented using the parametric Nikiforov-Uvarov (pNU) method with the inversely quadratic Hellman (IQH) potential. In the exact non-relativistic spin symmetry limit, the energy and wave function of the IQH potential are obtained and used to investigate the Shannon information entropy of the system. Numerical results of the relativistic energies of the spin and pseudospin symmetry limits of the Dirac equation with the IQH potential are presented and observed to exhibit degeneracy. Also, the results of the Shannon entropy for six states (n = 0, 1, 2, 3, 4, 5) show that the momentum space wave function and probability density are better localized than the position space wave function. Also, the Bialynicki-Birula and Mycielski (BBM) inequality is verified for the system. Our results are found to be consistent with those previously reported in the literature.

Quantum information of the modified Mobius squared plus Eckart potential

AbstractThe quantum information measures and complexity of the modified Mobius squared plus Eckart (MMSE) potential are presented in this paper. First, the energy eigenvalues and wave function of the system are obtained from the approximate solutions of the Schrödinger equation via the parametric Nikiforov‐Uvarov (pNU) method. Using the wave function, the Shannon entropy, Onicescu information energy and Fisher information of the system are examined for two low‐lying states along with the modified Lopez‐Ruiz‐Mancini‐Calbet (LMC) complexity and Heisenberg uncertainty relation. The results of the work point to the fact that the radial (momentum) probability density peak shifts to lower (higher) values with increase in the screening parameter. Furthermore, the Bialynicki‐Birula and Mycielski (BBM) inequality, the lower bound of the modified LMC complexity, the Fisher information sum inequality and the Stam‐Cramer‐Rao inequality are verified for the system. Also, the Heisenberg uncertainty principle is verified for the MMSE potential and the existence of squeezed states is observed for both position and momentum states.

Quantum Information Entropy of Hyperbolic Potentials in Fractional Schrödinger Equation.

In this work we have studied the Shannon information entropy for two hyperbolic single-well potentials in the fractional Schrödinger equation (the fractional derivative number (0<n≤2) by calculating position and momentum entropy. We find that the wave function will move towards the origin as the fractional derivative number n decreases and the position entropy density becomes more severely localized in more fractional system, i.e., for smaller values of n, but the momentum probability density becomes more delocalized. And then we study the Beckner Bialynicki-Birula−Mycieslki (BBM) inequality and notice that the Shannon entropies still satisfy this inequality for different depth u even though this inequality decreases (or increases) gradually as the depth u of the hyperbolic potential U1 (or U2) increases. Finally, we also carry out the Fisher entropy and observe that the Fisher entropy increases as the depth u of the potential wells increases, while the fractional derivative number n decreases.

Spherical-Symmetry and Spin Effects on the Uncertainty Measures of Multidimensional Quantum Systems with Central Potentials.

The spreading of the stationary states of the multidimensional single-particle systems with a central potential is quantified by means of Heisenberg-like measures (radial and logarithmic expectation values) and entropy-like quantities (Fisher, Shannon, Rényi) of position and momentum probability densities. Since the potential is assumed to be analytically unknown, these dispersion and information-theoretical measures are given by means of inequality-type relations which are explicitly shown to depend on dimensionality and state’s angular hyperquantum numbers. The spherical-symmetry and spin effects on these spreading properties are obtained by use of various integral inequalities (Daubechies–Thakkar, Lieb–Thirring, Redheffer–Weyl, ...) and a variational approach based on the extremization of entropy-like measures. Emphasis is placed on the uncertainty relations, upon which the essential reason of the probabilistic theory of quantum systems relies.

Multidimensional hydrogenic states: position and momentum expectation values

The position and momentum probability densities of a multidimensional quantum system are fully characterized by means of the radial expectation values ⟨r α ⟩ and , respectively. These quantities, which describe and/or are closely related to various fundamental properties of realistic systems, have not been calculated in an analytical and effective manner up until now except for a number of three-dimensional hydrogenic states. In this work we explicitly show these expectation values for all discrete stationary D-dimensional hydrogenic states in terms of the dimensionality D, the strength of the Coulomb potential (i.e. the nuclear charge) and the D state’s hyperquantum numbers. Emphasis is placed on the momentum expectation values (mostly unknown, specially the ones with odd order) which are obtained in a closed compact form. Applications are made to circular, S-wave, high-energy (Rydberg) and high-dimensional (pseudo-classical) states of three- and multidimensional hydrogenic atoms. This has been possible because of the analytical algebraic and asymptotical properties of the special functions (orthogonal polynomials, hyperspherical harmonics) which control the states’ wavefunctions. Finally, some Heisenberg-like uncertainty inequalities satisfied by these dispersion quantities are also given and discussed.

Rydberg multidimensional states: Rényi and Shannon entropies in momentum space

In this work the momentum spreading of a multidimensional hydrogenic system in highly excited (Rydberg) states is quantified by means of the Rényi and Shannon entropies of its momentum probability density. These quantities, which rest at the core of numerous fields from atomic and molecular physics to quantum technologies, are determined by means of a methodology based on the strong degree-asymptotics of a modified -norm of the Gegenbauer polynomials which control the wavefunctions of these states in momentum space. The leading term of these entropic quantities is found from first principles, i.e. by means of the Coulomb potential parameters (space dimensionality, the nuclear charge) and the states’s hyperquantum numbers, in a rigorous simple manner. It is shown that they fulfill a logarithmic growth scaling law with the principal hyperquantum number n which characterizes the Rydberg state.

One-dimensional harmonic oscillator problem and its hidden SU(1,1) symmetry in the presence of a minimal length

The one-dimensional harmonic oscillator in the presence of a minimal length possesses a hidden SU(1,1) symmetry. This symmetry makes it possible to construct the deformed canonical coherent states. These states turn out to be ground states of the deformed dipolar coupling. Indeed, the coupling to the external field may be considered as a dipolar coupling of a charged dipole to an electric field in one dimension. The spectrum is constructed algebraically. The wave functions in the momentum space are then reported. We have found that the momentum probability density of order n possesses n+1 vertices and these vertices of excited states disappear from n−1 when the deformation parameter increases. So, the particle in these states loses its point nature and can even explore space. The Heisenberg uncertainty relation involving a non-zero minimal length induces then a finite lower bound to spatial localisation.

Quantum information entropies of multiple quantum well systems in fractional Schrödinger equations

AbstractIn this work, we study the position and momentum information entropies of multiple quantum well systems in fractional Schrödinger equations, which, to the best of our knowledge, have not so far been studied. Through a confining potential, their shape and number of wells (NOW) can be controlled by using a few tuning parameters; we present some interesting quantum effects that only appear in the fractional Schrödinger equation systems. One of the parameters denoted by the Ld can affect the position and momentum probability densities if the system is fractional (1 &lt; α &lt; 2). We find that the position (momentum) probability density tends to be more severely localized (delocalized) in more fractional systems (ie, in smaller values of α). Affecting the Ld on the position and momentum probability densities is a quantum effect that only appears in the fractional Schrödinger equations. Finally, we show that the Beckner Bialynicki‐Birula‐Mycieslki (BBM) inequality in the fractional Schrödinger equation is still satisfied by changing the confining potential amplitude Vconf, the NOW, the fractional parameter α, and the confining potential parameter Ld.

Quantum information measures of infinite spherical well

In this work, we study the Shannon information entropies [Formula: see text] and [Formula: see text] of an infinite spherical well. The Shannon entropy [Formula: see text] is calculated numerically in terms of the analytical result of the wave function in momentum space. Some typical features of the position and momentum probability densities [Formula: see text] and [Formula: see text] as well as the information entropy densities [Formula: see text] and [Formula: see text] are demonstrated. We find that the position entropy [Formula: see text] increases with the radius a of the spherical well for given quantum numbers l, m and n. It is interesting to note that the position entropy [Formula: see text] decreases with the quantum numbers l and n for a fixed radius a and quantum number m. The position entropy [Formula: see text] is almost independent of the quantum numbers l, m and n. The momentum entropy [Formula: see text] first increases and then decreases with respect to the radius a. We also note that the [Formula: see text] increases with the radius a and finally arrives at a constant. In addition, the Bialynicki–Birula–Mycielski (BBM) inequality is verified and also hold for this confined system.

Gaussian diffusion interrupted by Lévy walk

We investigate a stochastic process that links two processes in one model: Gaussian diffusion and one-sided Levy walks leading to anomalous diffusion. The motivation for our study is an observation of ‘heavy-tailed’ momentum probability density in a standard map and its modifications when accelerated modes emerge in the phase space. In this stochastic model, we derive a fractional differential equation for its probability density function. Numerical and analytical solutions for certain values of parameters are presented.

Scaling Limit Analysis of Borromean Halos

The analysis of the core recoil momentum distribution of neutron-rich isotopes of light exotic nuclei is performed within a model of the halo nuclei described by a core and two neutrons dominated by the $s-$wave channel. We adopt the renormalized three-body model with a zero-range force, that accounts for the universal Efimov physics. This model is applicable to nuclei with large two-neutron halos compared to the core size, and a neutron-core scattering length larger than the interaction range. The halo wave function in momentum space is obtained by using as inputs the two-neutron separation energy and the energies of the singlet neutron-neutron and neutron-core virtual states. Within our model, we obtain the momentum probability densities for the Borromean exotic nuclei Lithium-11 ($^{11}$Li), Berylium-14 ($^{14}$Be) and Carbon-22 ($^{22}$C). A fair reproduction of the experimental data was obtained in the case of the core recoil momentum distribution of $^{11}$Li and $^{14}$Be, without free parameters. By extending the model to $^{22}$C, the combined analysis of the core momentum distribution and matter radius suggest (i) a $^{21}$C virtual state well below 1 MeV; (ii) an overestimation of the extracted matter $^{22}$C radius; and (iii) a two-neutron separation energy between 100 and 400 keV.

Momentum distributions in light halo nuclei and structure constraints

The core recoil momentum distribution of neutron-rich isotopes of light exotic nuclei is studied within a three-body model, where the nuclei are described by a core and two neutrons, with interactions dominated by the s-wave channel. In our framework, the two-body subsystems should have large scattering lengths in comparison with the interaction range allowing to use a three-body model with a zero-range force. The ground-state halo wave functions in momentum space are obtained by using as inputs the two-neutron separation energy and the energies of the singlet neutron-neutron and neutron-core virtual states. Within our model, we obtain the momentum probability densities for the Borromean exotic nuclei 11 Li and 22 C. In the case of the core recoil momentum distribution of 11Li, a fair reproduction of the experimental data was obtained, without free parameters, considering only the two-body low-energies. By analysing the obtained core momentum distribution in face of recent experimental data, we verify that such data are constraining the 22 C two-neutron separation energy to a value between 100 and 400 keV.

Probability Densities of Superconductors

The energy, current density and momentum probability densities of superconductors were studied from London, Ginzburg-Landau and BSC theories by treating cooper pair as a particle moving in a magnetic field through analytical and numerical techniques. The London and GL solution were exactly the same at the classical limit for NbN. Considering a Cooper pair as a complete classical particle, the momentum probability density was derived by using the Maxwell velocity distribution and the quantum mechanical momentum probability density was derived by using the radial wave function of the cooper pairs for Zn. The quantum mechanical and classical momentum probability densities overlap at zero momentum.

Probability Densities of Superconductors

The energy, current density and momentum probability densities of superconductors were studied from London, Ginzburg-Landau and BSC theories by treating cooper pair as a particle moving in a magnetic field through analytical and numerical techniques. The London and GL solution were exactly the same at the classical limit for NbN. Considering a Cooper pair as a complete classical particle, the momentum probability density was derived by using the Maxwell velocity distribution and the quantum mechanical momentum probability density was derived by using the radial wave function of the cooper pairs for Zn. The quantum mechanical and classical momentum probability densities overlap at zero momentum.

Exact quantum correlations of conjugate variables from joint quadrature measurements

We demonstrate that for two canonically conjugate operators $\hat{q},\hat {p} $,the global correlation $\langle \hat{q} \hat {p} + \hat{p} \hat {q} \rangle -2 \langle \hat{q}\rangle \langle \hat {p}\rangle$, and the local correlations $\langle \hat{q} \rangle (p) - \langle \hat{q}\rangle$ and $\langle \hat{p} \rangle (q)-\langle \hat {p}\rangle$ can be measured exactly by Von Neumann-Arthurs-Kelly joint quadrature measurements . These correlations provide a sensitive experimental test of quantum phase space probabilities quite distinct from the probability densities of $ q,p $. E.g. for EPR states, and entangled generalized coherent states, phase space probabilities which reproduce the correct position and momentum probability densities have to be modified to reproduce these correlations as well.

Entropy and complexity analysis of hydrogenic Rydberg atoms

The internal disorder of hydrogenic Rydberg atoms as contained in their position and momentum probability densities is examined by means of the following information-theoretic spreading quantities: the radial and logarithmic expectation values, the Shannon entropy, and the Fisher information. As well, the complexity measures of Crámer-Rao, Fisher-Shannon, and López Ruiz-Mancini-Calvet types are investigated in both reciprocal spaces. The leading term of these quantities is rigorously calculated by use of the asymptotic properties of the concomitant entropic functionals of the Laguerre and Gegenbauer orthogonal polynomials which control the wavefunctions of the Rydberg states in both position and momentum spaces. The associated generalized Heisenberg-like, logarithmic and entropic uncertainty relations are also given. Finally, application to linear (l = 0), circular (l = n − 1), and quasicircular (l = n − 2) states is explicitly done.

Momentum Probabilities for a Single Quantum Particle in Three-Dimensional Regular ‘Infinite’ Wells: One Way of Promoting Understanding of Probability Densities

Students often wrestle unsuccessfully with the task of correctly calculating momentum probability densities and have difficulty in understanding their interpretation. In the case of a particle inside an ‘infinite’ potential well, its momentum can take values that are not just those corresponding to the particle’s quantised energies but will be found to have values that occur with probabilities given by applying the Born Statistical Postulate. Momentum probability densities for several ‘infinite’ potential wells are plotted and some representative numerical values calculated and discussed. Students will benefit from these solutions as they will aid the understanding of momentum space descriptions of quantum systems, momentum probability densities and their visualisation.

Morse Potential in the Momentum Representation

The momentum representation of the Morse potential is presented analytically by hypergeometric function. The properties with respect to the momentum p and potential parameter β are studied. Note that |Ψ(p)| is a nodeless function and the mutual orthogonality of functions is ensured by the phase functions arg[Ψ(p)]. It is interesting to see that the |Ψ(p)| is symmetric with respect to the axis p = 0 and the number of wave crest of |Ψ(p)| is equal to n + 1. We also study the variation of |Ψ(p)| with respect to β. The amplitude of |Ψ(p)| first increases with the quantum number n and then deceases. Finally, we notice that the discontinuity in phase occurs at some points of the momentum p and the position and momentum probability densities are symmetric with respect to their arguments.

Quantum carpets woven by cold atoms in a laser field

Propagation of wave packets of cold two-level atoms in a standing-wave laser field can be interpreted in the dressed-state basis as motion in two optical potentials. The three distinct regimes of the wavepacket motion are specified by the ratio of the squared atom–laser detuning to the normalized Doppler shift. We calculate the momentum and position probability densities, which form patterns with minima and maxima of probability both in the momentum and the position spaces known as quantum carpets. At small and large detunings, the atomic motion is substantially adiabatic, and the quantum carpets have a simple form. At intermediate detunings, the wave packet moves nonadiabatically, splitting at each node of the standing wave, which causes a proliferation or branching of atomic trajectories with a single atom. Nonadiabatic transitions produce beautiful quantum carpets with a rich structure.