For the models of $N$-body identical harmonic oscillators interacting through potentials of homogeneous degree -2, the unitary operator that transforms a system of time-dependent parameters into that of unit spring constant and unit mass of different timescale is found. If the interactions can be written in terms of the differences between positions of two particles, it is also shown that the Schrödinger equation is invariant under a unitary transformation. These unitary relations can be used not only in finding coherent states from the given stationary states in a system, but also in finding exact wave functions of the Hamiltonian systems of time-dependent parameters from those of time-independent Hamiltonian systems. Both operators are invariant under the exchange of any pair of particles. The transformations are explicitly applied for some of the Calogero-Sutherland models to find exact coherent states.
We use the invariant-based inverse engineering subject to the quasiadiabatic condition to produce robust and high fidelity coherent superposition of quantum states. The inverse engineering provides shortcuts to the desired quantum-state evolution while the quasiadiabaticity provides robustness with respect to errors. We derive simple pulses with low areas which are robust with respect to pulse area and detuning.
Using the {\it nonlinear coherent states method}, a formalism for the construction of the coherent states associated to {\it "inverse bosonic operators"} and their dual family has been proposed. Generalizing the approach, the "inverse of $f$-deformed ladder operators" corresponding to the nonlinear coherent states in the context of quantum optics and the associated coherent states have been introduced. Finally, after applying the proposal to a few known physical systems, particular nonclassical features as sub-Poissonian statistics and the squeezing of the quadratures of the radiation field co…
The Schrödinger-like equation written in terms of the displacement operator is solved analytically for a inverse square plus Coulomb-like potential. Starting from the new Hamiltonian, the effects of the spatially dependent mass on the bound states and normalized wave functions of the "usual" inverse square plus Coulomb interaction are discussed.
Coherent gate errors are a concern in many proposed quantum computing architectures. These errors can be effectively handled through composite pulse sequences for single-qubit gates, however, such techniques are less feasible for entangling operations. In this work, we benchmark our coherent errors by comparing the actual performance of composite single-qubit gates to the predicted performance based on characterization of individual single-qubit rotations. We then propose a compilation technique, which we refer to as hidden inverses, that creates circuits robust to these coherent errors. We pr…
This article is an illustration of the construction of coherent and generalized intelligent states which has been recently proposed by us for an arbitrary quantum system $[ 1] $. We treat the quantum system submitted to the infinite square well potential and the nonlinear oscillators. By means of the analytical representation of the coherent states à la Gazeau-Klauder and those à la Klauder-Perelomov, we derive the generalized intelligent states in analytical ways.
The Levinson theorem for two-dimensional scattering is generalized for potentials with inverse square singularities. By this theorem, the number of bound states in a given m-th partial wave is related to the phase shift and the singularity strength of the potential. For the m-wave phase shift the asymptotic behaviour is calculated for short wavelengths.
We present the exact solution of the 1D Dirac equation for the inverse-square-root potential $1/\sqrt{x}$ for several configurations of vector, pseudo-scalar and scalar fields. Each fundamental solution of the problem can be written as an irreducible linear combination of two Hermite functions of a scaled and shifted argument. We derive the exact equations for bound-state energy eigenvalues and construct accurate approximations for the energy spectrum.
We introduce an exactly integrable singular potential for which the solution of the one-dimensional stationary Schrödinger equation is written through irreducible linear combinations of the Gauss hypergeometric functions. The potential, which belongs to a general Heun family, is a short-range one that behaves as the inverse square root in the vicinity of the origin and vanishes exponentially at the infinity. We derive the exact spectrum equation for the energy and discuss the bound states supported by the potential.
The unitary operator which transforms a harmonic oscillator system of time-dependent frequency into that of a simple harmonic oscillator of different time-scale is found, with and without an inverse-square potential. It is shown that for both cases, this operator can be used in finding complete sets of wave functions of a generalized harmonic oscillator system from the well-known sets of the simple harmonic oscillator. Exact invariants of the time-dependent systems can also be obtained from the constant Hamiltonians of unit mass and frequency by making use of this unitary transformation. The g…
Using the Tridiagonal Representation Approach, we obtain solutions (energy spectrum and corresponding wavefunctions) for a new five-parameter potential box with inverse square singularity at the boundaries.
The quantum-mechanical D-dimensional inverse square potential is analyzed using field-theoretic renormalization techniques. A solution is presented for both the bound-state and scattering sectors of the theory using cutoff and dimensional regularization. In the renormalized version of the theory, there is a strong-coupling regime where quantum-mechanical breaking of scale symmetry takes place through dimensional transmutation, with the creation of a single bound state and of an energy-dependent s-wave scattering matrix element.
We use the Tridiagonal Representation Approach (TRA) to obtain exact scattering and bound states solutions of the Schrödinger equation for short-range inverse-square singular hyperbolic potentials. The solutions are series of square integrable functions written in terms of the Jacobi polynomial with the Wilson polynomial as expansion coefficients. The series is finite for the discrete bound states and infinite, but bounded, for the continuum scattering states.
We solve analytically the Schrödinger equation for the N-dimensional inverse square potential in quantum mechanics with a minimal length in terms of Heun's functions. We apply our results to the problem of a dipole in a cosmic string background. We find that a bound state exists only if the angle between the dipole moment and the string is larger than π/4. We compare our results with recent conflicting conclusions in the literature. The minimal length may be interpreted as a radius of the cosmic string.
This paper is a continuation of the previous one [Journal xx, xxxxx (2022)]. Here, we reformulate the same J-matrix theory by regularizing the inverse square singular potential. The objective is to restore rapid convergence of the calculation in the theory and recover the conventional tridiagonal representation. Partial success is achieved.
The J-matrix method was developed to handle regular short-range scattering potentials. Its accuracy, stability, and convergence properties compare favorably with other successful scattering methods. Recently, we extended the method to the treatment of potentials with 1/r singularity. In this work, we do the same for 1/r^2 singular potentials.
We examine a simple tabletop experimental setup for probing the inverse-square law of gravity and detecting eventual deviations therefrom. The nature of the setup allows indeed to effectively reach for shorter distances compared to what is allowed by other methods. Furthermore, we show that the same setup could also in principle be used to probe the interaction between gravitomagnetism and the intrinsic angular spin of quantum particles. Moreover, we show that the setup allows to have a gravitationally induced harmonic oscillator, introducing thus the possibility of studying in a novel way the…
We study higher-order corrections to the Grimus-Stockinger theorem dealing with the large-distance asymptotic behavior of the wave-packet modified neutrino propagator within the framework of field-theoretical description of the neutrino oscillation phenomenon. We discuss the possibility that these corrections are responsible for breakdown of the classical inverse-square law (ISL) at the macroscopic distances. In particular the ISL violation can be an explanation of the well-known reactor antineutrino anomaly.
The problem of a particle of mass m in the field of the inverse square potential is studied in quantum mechanics with a generalized uncertainty principle, characterized by the existence of a minimal length. Using the coordinate representation, for a specific form of the generalized uncertainty relation, we solve the deformed Schrödinger equation analytically in terms of confluent Heun functions. We explicitly show the regularizing effect of the minimal length on the singularity of the potential. We discuss the problem of bound states in detail and we derive an expression for the energy spectru…
We study the problem of the attractive inverse square potential in quantum mechanics with a generalized uncertainty relation. Using the momentum representation, we show that this potential is regular in this framework. We solve analytically the s-wave bound states equation in terms of Heun's functions. We discuss in detail the bound states spectrum for a specific form of the generalized uncertainty relation. The minimal length may be interpreted as characterizing the dimension of the system.
Engineered micro- and nanomechanical resonators with ultra-low dissipation constitute the ideal systems for applications ranging from high-precision sensing such as magnetic resonance force microscopy, to quantum transduction between disparate quantum systems. Traditionally, the improvement of the resonator's performance - often quantified by its Qf product (where Q is quality factor and f is frequency) - through nanomechanical engineering such as dissipation dilution and strain engineering, has been driven by human intuition and insight. Such an approach is inefficient and leaves aside a plet…
A distorted-wave version of the renormalisation group is applied to scattering by an inverse-square potential and to three-body systems. In attractive three-body systems, the short-distance wave function satisfies a Schroedinger equation with an attractive inverse-square potential, as shown by Efimov. The resulting oscillatory behaviour controls the renormalisation of the three-body interactions, with the renormalisation-group flow tending to a limit cycle as the cut-off is lowered. The approach used here leads to single-valued potentials with discontinuities as the bound states are cut off. T…
A Green's function approach is presented for the D-dimensional inverse square potential in quantum mechanics. This approach is implemented by the introduction of hyperspherical coordinates and the use of a real-space regulator in the regularized version of the model. The application of Sturm-Liouville theory yields a closed expression for the radial energy Green's function. Finally, the equivalence with a recent path-integral treatment of the same problem is explicitly shown.
We study the quantum mechanics of a system with inverse square potential in noncommutative space. Both the coordinates and momentums are considered to be noncommutative, which breaks the original so(2,1) symmetry. The energy levels and eigenfunctions are obtained. The generators of the so(2,1) algebra are also studied in noncommutative phase space and the commutators are calculated, which shows that the so(2,1) algebra obtained in noncommutative space is not closed. However the commutative limit Θ,\barΘ\to 0 for the algebra smoothly goes to the standard so(2,1) algebra.
We study the radial Schrödinger equation for a particle of mass $m$ in the field of the inverse-square potential $α/r^{2}$ in the medium-weak-coupling region, i.e., with $-1/4\leq2mα/\hbar^{2}\leq3/4$. By using the renormalization method of Beane \textit{et} \textit{al.,}with two regularization potentials, a spherical square well and a spherical $δ$ shell, we illustrate that the procedure of renormalization is independent of the choice of the regularization counterterm. We show that, in the aforementioned range of the coupling constant $α$, there exists at most one bound state, in complete agr…
The Self-Adjoint Extension in the Schrodinger equation for potentials behaved as an attractive inverse square at the origin is critically reviewed. Original results are also presented. It is shown that the additional non-regular solutions must be retained for definite interval of parameters, which requires a necessity of performing a Self-Adjoint Extension (SAE) procedure of radial Hamiltonian.The Pragmatic approach is used and some of its consequences are considered for wide class of transitive potentials. Our consideration is based on the established earlier by us a boundary condition for th…
We investigate the entanglement content of the ground state of a system characterized by effective elementary degrees of freedom with fractional statistics. To this end, we explicitly construct the ground state for a chain of $N$ spins with inverse square interaction (the Haldane-Shastry model) in the presence of an external uniform magnetic field. For such a system at zero temperature, we evaluate the entanglement in the ground state both at finite size and in the thermodynamic limit. We relate the behavior of the quantum correlations with the spinon condensation phenomenon occurring at the s…
Quantum anomalies in the inverse square potential are well known and widely investigated. Most prominent is the unbounded increase in oscillations of the particle's state as it approaches the origin when the attractive coupling parameter is greater than the critical value of 1/4. Due to this unphysical divergence in oscillations, we are proposing that the interaction gets screened at short distances making the coupling parameter acquire an effective (renormalized) value that falls within the weak range 0 to 1/4. This prevents the oscillations form growing without limit giving a lower bound to …
