Quantum mechanical virial-like theorem for confined quantum systems

Mukherjee, Neetik; Roy, Amlan K.

doi:10.1103/physreva.99.022123

Cited by 16 publications

(18 citation statements)

References 52 publications

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“…Recently a Virial-like equation has been derived by the authors for confined quantum systems [183]. It has the form,…”

Section: A Virial-like Theoremmentioning

confidence: 99%

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Information analysis in free and confined harmonic oscillator

Mukherjee¹,

Roy²

2021

Preprint

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In this chapter we shall discuss the recent progresses of information theoretic tools in the context of free and confined harmonic oscillator (CHO). Confined quantum systems have provided appreciable interest in areas of physics, chemistry, biology, etc., since its inception. A particle under extreme pressure environment unfolds many fascinating, notable physical and chemical changes. The desired effect is achieved by reducing the spatial boundary from infinity to a finite region. Similarly, in the last decade, information measures were investigated extensively in diverse quantum problems, in both free and constrained situations. The most prominent amongst these are: Fisher information (I), Shannon entropy (S), Rényi entropy (R), Tsallis entropy (T ), Onicescu energy (E) and several complexities. Arguably, these are the most effective measures of uncertainty, as they do not make any reference to some specific points of a respective Hilbert space. These have been invoked to explain several physico-chemical properties of a system under investigation. Kullback-Leibler divergence or relative entropy describes how a given probability distribution shifts from a reference distribution function. This characterizes a measure of discrimination between two states. In other words, it extracts the change of information in going from one state to another.The one-dimensional confined harmonic oscillator (1DCHO), defined by v), can be classified into two forms, (a) symmetrically confined harmonic oscillator (SCHO) (when d m = 0), (b) asymmetrically confined harmonic oscillator (ACHO) (corresponding to d m = 0). Further, in latter case, confinement can be accomplished two different ways: (i) by changing the box boundary, keeping box length and d m fixed at zero; (ii) another route is to adjust d m by keeping box length and boundary fixed. SCHO can be treated as an intermediate between a particle-ina-box (PIB) and a 1DQHO. Though the Schrödinger equation for SCHO can be solved exactly, for ACHO it cannot be. We have employed two different methods for the latter, viz., (i) an imaginary time propagation (ITP), leading to minimization of an expectation value (ii) a variation-induced exact diagonalization procedure that utilizes SCHO eigenfunctions as basis. It is found that, at very low x c region, I, S, R, T, E remain invariant with change confinement length, x c . At moderate x c region S x , R x , T x progress and I x , E x decrease with rise in x c . Additionally, special attention has been paid to study relative information in SCHO and ACHO.Analogously, a 3DCHO (radically confined within an impenetrableFree and confined harmonic oscillator • • • spherical well) can act as a bridge between particle in a spherical box (PISB) and isotropic 3DQHO. The time-independent Schrödinger equation for D-dimensional harmonic oscillator in both free and confined condition can be solved exactly, within a Dirichlet boundary condition. Here a detailed exploration of information measures has been carried out in r and p spaces, for a 3DCHO. Some ex...

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“…Recently a Virial-like equation has been derived by the authors for confined quantum systems [183]. It has the form,…”

Section: A Virial-like Theoremmentioning

confidence: 99%

“…Consequently, the difference between mean square and average values of T , V tends to rise. For more details, please consult [183].…”

Section: Symmetrically Confined Harmonic Oscillator (Scho)mentioning

confidence: 99%

Information analysis in free and confined harmonic oscillator

Mukherjee¹,

Roy²

2021

Preprint

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“…With these values, we obtain the ground state energy = −2.9033 u.a and the nonlinear parameter that represent the screening factor is K = 3.78. The corresponding wave function can be written using equations (6) and 15…”

Section: Variational Calculationsmentioning

confidence: 99%

“…The number of parameters increases with the system size up to several hundreds and the optimization requires efficient strategies [2]. Several kinds of trial functions are used to deal with two-electron atomic or ion systems [3][4][5][6]. A comparative study of two-electron systems with different potentials is done by authors of Ref.…”

Section: Introductionmentioning

confidence: 99%

Atomic Form Factor Calculations of S-states of Helium

Diallo¹,

Faye²,

Gomis³

et al. 2019

AJMP

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Variational calculations of the helium atom states are performed using highly compact 26-parameter correlated Hylleraas-type wave functions. These correlated wave functions used here yield an accurate expectation energy values for helium ground and two first excited states. A correlated wave function consists of a generalized exponential expansion in order to take care of the correlation effects due to N-corps interactions. The parameters introduced in our model are determined numerically by minimization of the total atomic energy of each electronic configuration. We have calculated all integrals analytically before dealing with numerical evaluation. The 1S 2 1 1 S and 1S2S 2 1, 3 S states energies, charge distributions and scattering atomic form factors are reported. The present work shows high degree of accuracy even with relative number terms in the trial Hylleraas wave functions definition. The results presented here, indicate that the highly compact twenty-six variational parameters model will have the quantitative and qualitative applicability for the study of electronic correlation. The correlated wave functions are used to calculate the atomic form factor for the diffusion of electrons by the helium atom. The atomic form factor is evaluated as the Fourier transform of the electron density distribution of an atom or ion, which is calculated from theoretical correlated wave functions for free atoms. Finally, suggestions are made as to the way the atomic form factor of the helium atom may be approximated by a sum of Gaussians for efficiency use.

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“…This has provoked a fast development of a density functional theory of independent particles moving in multidimensional central potentials with various analytical forms (see, eg, refs. ).…”

Section: Introductionmentioning

confidence: 97%

Two‐dimensional confined hydrogen: An entropy and complexity approach

Estañón

Aquino

Puertas‐Centeno

et al. 2020

Int J of Quantum Chemistry

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The position and momentum spreading of the electron distribution of the two-dimensional confined hydrogenic atom, which is a basic prototype of the general multidimensional confined quantum systems, is numerically studied in terms of the confinement radius for the 1s, 2s, 2p, and 3d quantum states by means of the main entropy and complexity information-theoretical measures. First, the Shannon entropy and the Fisher information, as well as the associated uncertainty relations, are computed and discussed. Then, the Fisher-Shannon, lopezruiz-mancini-alvet, and LMC-Rényi complexity measures are examined and mutually compared. We have found that these entropy and complexity quantities reflect the rich properties of the electron confinement extent in the two conjugated spaces. K E Y W O R D SFisher information of the two-dimensional confined hydrogen atom, Fisher-Shannon complexity measure of the two-dimensional confined hydrogen atom, LMC-Rényi complexity measure of the two-dimensional confined hydrogen atom, Shannon entropy of the two-dimensional confined hydrogen atom, two-dimensional confined hydrogen atom | INTRODUCTIONThe fundamental and practical relevance of the spherically confined quantum systems has been manifested from the early days of quantum physics [1,2] up until now. [3][4][5][6][7] They have been used as prototypes to explain numerous phenomena and systems not only in the three-dimensional world but also for nonrelativistic and relativistic D-dimensional (D ≥ 2) chemistry and physics. [8][9][10][11][12][13][14] For example, pioneered by Gerhard Ertl, the 2007 Nobel Laureate in Chemistry, and his collaborators, surface chemistry has been extensively studied and has become a key branch in chemistry. [11][12][13] In physics of materials, two-dimensional systems have been used to gain insight into the properties of semiconductors (see, eg, refs. [15, 16]), and they start to play a fundamental role in bringing strong and new forms of control to the atomic-scale limit over the dynamics of matter in the extreme confinement of electromagnetic energy by phonon polaritonics. [10] Moreover, the properties of the real fluids can be studied by means of crystalline fluids (ie, with a symmetry) with nonstandard dimensions (see, eg, ref. [17]).The idea of two-and multidimensional spherical confinement of atoms has been used not only to simulate the effect of high pressure on the static dipole polarizability in hydrogen [18] but also to model a great deal of nanotechnological objects such as quantum dots; quantum wells and quantum wires; [19,20] atoms and molecules embedded in nanocavities, for example, in fullerenes, zeolites cages, and helium droplets; [4,6,7,[21][22][23] dilute bosonic and fermionic systems in magnetic traps of extremely low temperatures; [24][25][26] and a variety of quantum-information elements. [27,28] This has provoked a fast development of a density functional theory of independent particles moving in multidimensional central potentials with various analytical forms (see, eg, refs. [5-7, 29-...

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Quantum mechanical virial-like theorem for confined quantum systems

Cited by 16 publications

References 52 publications

Information analysis in free and confined harmonic oscillator

Information analysis in free and confined harmonic oscillator

Atomic Form Factor Calculations of S-states of Helium

Two‐dimensional confined hydrogen: An entropy and complexity approach

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