Classical and quantum motion in an inverse square potential (original) (raw)
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Journal of Mathematical Physics, 2013
The classical 2D dynamics of a particle moving under an inverse square potential, − k/r 2 , is analysed. We show that such problem is an example of a geometric system since its negative energy orbits are equivalent to free motion on a certain hypersurface. We then solve in momentum space, the corresponding unrenormalized quantum problem showing that there is no discrete energy spectrum and, particularly, no ground state. C 2013 AIP Publishing LLC. [http://dx.
Falling of a quantum particle in an inverse square attractive potential
The European Physical Journal D
Evolution of a particle in an inverse square potential is studied. We derive an equation of motion for r 2 and solve it exactly. It gives us a possibility to identify the conditions under which a falling of a quantum particle into an attractive centre is possible. We get the time of falling of a particle from an initial state into the centre. An example of a quasi-stationary state which evolves with r 2 being constant in time is given. We demonstrate the existence of quantum limit of falling, namely, a particle does not fall into the attractive centre, when coupling constant is smaller then some critical value. Our results are compared with experimental measurements of neutral atoms falling in the electric field of a charged wire. Moreover, we propose modifications of the experiment, which allow to observe quantum limit of falling.
Renormalization of the Inverse Square Potential
Physical Review Letters, 2000
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.
Quantum Mechanics of Singular Inverse Square Potentials Under Usual Boundary Conditions
viXra, 2017
The quantum mechanics of inverse square potentials in one dimension is usually studied through renormalization, self-adjoint extension and WKB approximation. This paper shows that such potentials may be investigated within the framework of the position-dependent mass quantum mechanics formalism under the usual boundary conditions. As a result, exact discrete bound state solutions are expressed in terms of associated Laguerre polynomials with negative energy spectrum using the Nikiforov-Uvarov method for the repulsive inverse square potential.
Towards an inverse scattering theory for two-dimensional nondecaying potentials
Theoretical and Mathematical Physics, 1998
The inverse scattering method is considered for the nonstationary Schr6dinger equation with the potential U(Xl, x2) nondecaying in a finite number of directions in the x plane. The general resolvent approach, which is particularly convenient for this problem, is tested using a potential that is the B~cklund transformation of an arbitrary decaying potential and that describes a soliton superimposed on an arbitrary background. In this example, the resolvent, Jost solutions, and spectral data are explicitly constructed, and their properties are analyzed. The characterization equations satisfied by the spectral data are derived, and the unique solution of the inverse problem is obtained. The asymptotic potential behavior at large distances is also studied in detail. The obtained resolvent is used in a dressing procedure to show that with more general nondecaying potentials, the Jost solutions may have an additional cut in the spectral-parameter complex domain. The necessary and sufficient condition for the absence of this additional cut is formulated.
Localization and Critical Diffusion of Quantum Dipoles in Two Dimensions
Physical Review Letters, 2011
We discuss quantum propagation of dipole excitations in two dimensions. This problem differs from the conventional Anderson localization due to existence of long range hops. We found that the critical wavefunctions of the dipoles always exist which manifest themselves by a scale independent diffusion constant. If the system is T-invariant the states are critical for all values of the parameters. Otherwise, there can be a "metal-insulator" transition between this "ordinary" diffusion and the Levy-flights (the diffusion constant logarithmically increasing with the scale). These results follow from the two-loop analysis of the modified non-linear supermatrix σ-model. 71.55.Jv Anderson showed that a quenched disorder can localize a quantum particle, i.e. completely suppress its diffusion. Later , it was realized that in two dimensions (2D) localization occurs for an arbitrary weak disorder. This conclusion was reached by studying the scaling behavior of the dimensionless Thouless conductance g(L) as a function of the linear size of the system L (the observable electrical conductance of the system of the e charged particles is given by g × e 2 / , and we will set Planck constant = 1 hereinafter). Localization implies that g L → 0 as L → ∞. This is always true when the time reflection symmetry (T-invariance) is broken (so called unitary ensemble, GUE). For T-invariant systems it is still correct if the orbital and spin degrees of freedom are decoupled or when the particles have an integer spin (orthogonal ensemble, GOE). For the particles with halfinteger spin, the theory [4] predicts that the spin-orbital coupling causes antilocalization g(L → ∞) → ∞ if disorder is weak, while for a stronger disorder g(L → ∞) → 0 (metal-insulator transition for symplectic ensemble) .
Renormalization of the singular attractive 1 / r 4 potential
We study the radial Schrödinger equation for a particle of mass m in the field of a singular attractive g 2 / r 4 potential with particular emphasis on the bound-states problem. Using the regularization method of Beane et al. ͓Phys. Rev. A 64, 042103 ͑2001͔͒, we solve analytically the corresponding "renormalization-group flow" equation. We find in agreement with previous studies that its solution exhibits a limit cycle behavior and has infinitely many branches. We show that a continuous choice for the solution corresponds to a given fixed number of bound states and to low-energy phase shifts that vary continuously with energy. We study in detail the connection between this regularization method and a conventional method modifying the short-range part of the potential with an infinitely repulsive hard core. We show that both methods yield bound-states results in close agreement even though the regularization method of Beane et al. does not include explicitly any new scale in the problem. We further illustrate the use of the regularization method in the computation of electron bound states in the field of neutral polarizable molecules without dipole moment. We find the binding energy of s-wave polarization bound electrons in the field of C 60 molecules to be 17 meV for a scattering length corresponding to a hard-core radius of the size of the molecule radius ͑ϳ3.37 Å͒. This result can be further compared with recent two-parameter fits using the Lennard-Jones potential yielding binding energies ranging from 3 to 25 meV.