Spatial confinement of quantum systems has been used to model atoms under high pressure, quantum dots, and Bose–Einstein condensates. Spherical boxes are commonly employed, yet they are energy-minimizing, and other box geometries will be more relevant for some applications. We study generic single-particle quantum systems comprising a central physical potential confined within a three-dimensional infinite potential well corresponding to an impenetrable box having prescribed geometric properties; we refer to this as geometric confinement. There is a marked distortion of the wavefunction and energy spectrum, relative to the standard unconfined system, when a confining box is made small enough. We employ asymptotic analysis in combination with numerical simulations to better understand this behaviour, determining lengthscales on which each of the physical potential and confining box respectively dominate the structure of quantum states, with Padé approximants employed to connect asymptotic scalings between these two regimes. This combination of methods permits us to determine, in a fairly general manner, how and to what extent box size and shape modify a given single-particle quantum system held under geometric confinement. We provide a variety of examples of single-particle central potentials and confining geometries to illustrate our methods and motivate the theoretical findings.
Spatial confinement of quantum systems has been used to model atoms under high pressure, quantum dots, and Bose–Einstein condensates. Spherical boxes are commonly employed, yet they are energy-minimizing, and other box geometries will be more relevant for some applications. We study generic single-particle quantum systems comprising a central physical potential confined within a three-dimensional infinite potential well corresponding to an impenetrable box having prescribed geometric properties; we refer to this as geometric confinement. There is a marked distortion of the wavefunction and energy spectrum, relative to the standard unconfined system, when a confining box is made small enough. We employ asymptotic analysis in combination with numerical simulations to better understand this behaviour, determining lengthscales on which each of the physical potential and confining box respectively dominate the structure of quantum states, with Padé approximants employed to connect asymptotic scalings between these two regimes. This combination of methods permits us to determine, in a fairly general manner, how and to what extent box size and shape modify a given single-particle quantum system held under geometric confinement. We provide a variety of examples of single-particle central potentials and confining geometries to illustrate our methods and motivate the theoretical findings.
Bose–Einstein condensates (BECs) comprise a state of matter in which separate atoms or subatomic particles coalesce into a single macroscopic wave function when cooled to near absolute zero. BECs are confined within traps when carrying out experiments, and in the present paper, we model BECs confined within three-dimensional infinite potential wells corresponding to impenetrable boxes with prescribed geometric properties. To do this, we study the Gross–Pitaevskii equation (GPE) governing the macroscopic wave function within generic three-dimensional finite boxes, first describing analytical and numerical approaches for solving the ground state macroscopic wave function, and then employing these methods to understand how the geometry of a confining box influences properties of the BEC wave function. We show that the chemical potential of the BEC is sensitive to the size and shape of the box, with the chemical potential lowest for a spherical box and comparatively large for boxes with more extreme aspect ratios. Compared to commonly employed harmonic traps, geometrically confined BECs exhibit larger variations of the chemical potential with the self-interaction parameter, yet geometrically confined attractive BECs may be more robust against collapse. In particular, we show that the collapse threshold for an attractive BEC can be modified by the specific manner of geometric confinement.
A succinct history of the Confined Atom problem is presented. The hydrogen atom confined to the centre of an impenetrable sphere counts amongst the exactly soluble problems of physics, alongside much more noted exact solutions such as Black Body Radiation and the free Hydrogen atom in absence of any radiation field. It shares with them the disadvantage of being an idealisation, while at the same time encapsulating in a simple way particular aspects of physical reality. The problem was first formulated by Sommerfeld and Welker [1]—henceforth cited as SW—in connection with the behaviour of atoms at very high pressures, and the solution was published on the occasion of Pauli's 60th birthday celebration. At the time, it seemed that there was not much other connection with physical reality beyond a few simple aspects connected to the properties of atoms in solids, for which more appropriate models were soon developed. Thus, confined atoms attracted little attention until the advent of the metallofullerene, which provided the first example of a confined atom with properties quite closely related to those originally considered by SW. Since then, the problem has received much more attention, and many more new features of quantum confinement, quantum compression, the quantum Faraday cage, electronic reorganisation, cavity resonances, etc have been described, which are relevant to real systems. Also, a number of other situations have been uncovered experimentally to which quantum confinement is relevant. Thus, studies of the confined atom are now more numerous, and have been extended both in terms of the models used and the systems to which they can be applied. Connections to thermodynamics are explored through the properties of a confined two‐level atom adapted from Einstein's celebrated model, and issues of dynamical screening of electromagnetic radiation by the confining shell are discussed in connection with the Faraday cage produced by a confining conducting shell. The conclusions are shown to be relevant to a proposed 'quantum computer'. The description of the actual geometry of C60, as opposed to a purely spherical approximation, leads to some qualification of the computed results.
Many-particle systems pose commonly known computational challenges in quantum theory. The obstacles arise from the difficulty in finding sets of eigenvalues and eigenvectors of the underlying Hamiltonian while enforcing fermion or boson statistics, not to mention the prohibitive increase in the computational cost with the system’s size. The first obvious step in this direction is to elaborate the theory for Fermi or Bose gases without inter-particle interactions. The traditional approach to the work is with the ideal gases confined in a cubic container with impenetrable walls (in arbitrary dimensions). This approach allows one to find the particle’s spectra and compute all thermodynamic quantities of the confined gas. In the present work, we consider the gas confined in a spherical container (in other words, an infinitely deep spherical potential well in D dimensions), solving the corresponding Schrödinger equation using zero boundary conditions. We address the case of a finite number of particles N, either bosons or fermions, in the spherical potential box, as well as the thermodynamic limit, corresponding to N → ∞. Owing to Weyl’s relations, in the latter limit, the results do not depend on the shape of the box and thus approach the commonly known ones valid in the infinite space. Owing to the underlying SO(D) symmetry, we are dealing with particles carrying a well-defined angular momentum that, together with sorted energy eigenvalues, imparts a shell structure to the system.
Quantum confinement is studied by numerically solving time-dependent (TD) Schrödinger equation (SE). An imaginary-time evolution technique is employed in conjunction with the minimization of an expectation value, to reach the global minimum. Excited states are obtained by imposing the orthogonality constraint with all lower states. Applications are made on three important model quantum systems, namely, harmonic, repulsive and quartic oscillators; enclosed inside an impenetrable box. The resulting diffusion equation is solved using finite-difference method. Both symmetric and asymmetric confinement are considered for attractive potential; for others only symmetrical confinement. Accurate eigenvalue, eigenfunction and position expectation values are obtained, which show excellent agreement with existing literature results. Variation of energies with respect to box length is followed for small, intermediate and large sizes. In essence, a simple accurate and reliable method is proposed for confinement in quantum systems.
Abstract Size-invariant shape transformation is a geometric technique that allows for a clear separation between quantum size and shape effects by modifying the shape of the confinement domain without altering its size. The impact of shape on the behavior of confined systems is significantly different from that of size, making it an emerging area of research. The recent realization of flat-bottomed optical box traps has further contributed to the study of quantum gases in complex confinement geometries. Here, we propose shape-induced Bose–Einstein condensation at a fixed size, temperature, and density. We investigate the impact of pure quantum shape effects on a non-interacting Bose gas confined within nested square domains, where the shape parameter is defined and controlled by the rotation angle between the inner and outer squares. Our findings reveal that specific heat exhibits an additional low-temperature peak at certain shapes. This work opens new avenues for controlling quantum systems through geometric manipulation and provides insights into the thermodynamic properties of Bose gases under shape-induced quantum effects.
Quantum entanglement, a phenomenon where two or more particles become linked regardless of distance, has captivated physicists and philosophers alike. This research explores the potential of utilizing geometric principles to understand and manipulate entanglement, proposing a novel model to represent quantum entanglement as a geometric structure. We aim to investigate how the inherent geometry of entangled states can be leveraged to optimize quantum computation and communication protocols. The core claim is that establishing a geometric model of quantum entanglement will unlock new avenues for enhancing the performance of these technologies. This paper delves into the foundational concepts and proposes a mechanism for translating entanglement into a geometric framework, ultimately seeking to reveal hidden properties and potential optimizations within the space of entangled quantum states.