From Schrödinger’s equation to the quantum search algorithm

The quantum search algorithm is a technique for searching N possibilities in only O(√N) steps. Although the algorithm itself is widely known, not so well known is the series of steps that first led to it, these are quite different from any of the generally known forms of the algorithm. This paper describes these steps, which start by discretizing Schrödinger’s equation. This paper also provides a self contained introduction to quantum computing algorithms from a new perspective.     

A Flea on Schrödinger’s Cat

We propose a technical reformulation of the measurement problem of quantum mechanics, which is based on the postulate that the final state of a measurement is classical; this accords with experimental practice as well as with Bohr’s views. Unlike the usual formulation (in which the post-measurement state is a unit vector in Hilbert space), our version actually opens the possibility of admitting a purely technical solution within the confines of conventional quantum theory (as opposed to solutions that either modify this theory, or introduce unusual and controversial interpretative rules and/or ontologies). To that effect, we recall a remarkable phenomenon in the theory of Schrödinger operators (discovered in 1981 by Jona-Lasinio, Martinelli, and Scoppola), according to which the ground state of a symmetric double-well Hamiltonian (which is paradigmatically of Schrödinger’s Cat type) becomes exponentially sensitive to tiny perturbations of the potential as ?→0. We show that this instability emerges also from the textbook wkb approximation, extend it to time-dependent perturbations, and study the dynamical transition from the ground state of the double well to the perturbed ground state (in which the cat is typically either dead or alive, depending on the details of the perturbation). Numerical simulations show that adiabatically arising perturbations may (quite literally) cause the collapse of the wave-function in the classical limit. Thus, at least in the context of a simple mathematical model, we combine the technical and conceptual virtues of decoherence (which fails to solve the measurement problem but launches the key idea that perturbations may come from the environment) with those of dynamical collapse models à la grw (which do solve the measurement problem but are ad hoc), without sharing their drawbacks: single measurement outcomes are obtained (instead of merely diagonal reduced density matrices), and no modification of quantum mechanics is needed.     

Exact and approximate solutions to Schrödinger’s equation with decatic potentials

The one-dimensional Schrödinger’s equation is analysed with regard to the existence of exact solutions for decatic polynomial potentials. Under certain conditions on the potential’s parameters, we show that the decatic polynomial potential V (x) = ax ¹⁰ + bx ⁸ + cx ⁶ + dx ⁴ + ex ², a > 0 is exactly solvable. By examining the polynomial solutions of certain linear differential equations with polynomial coefficients, the necessary and sufficient conditions for corresponding energy-dependent polynomial solutions are given in detail. It is also shown that these polynomials satisfy a four-term recurrence relation, whose real roots are the exact energy eigenvalues. Further, it is shown that these polynomials generate the eigenfunction solutions of the corresponding Schrödinger equation. Further analysis for arbitrary values of the potential parameters using the asymptotic iteration method is also presented.     

New methods to solve the resonant nonlinear Schrödinger’s equation with time-dependent coefficients

In this study, the generalized \(\tan (\phi /2)\)-expansion method and He’s semi-inverse variational method (HSIVM) are applied to seek the exact solitary wave solutions for the resonant nonlinear Schrödinger equation with time-dependent coefficients. Using these methods, we investigate exact solutions for the nonlinear resonant Schrödinger equation with time-dependent coefficients two forms of nonlinearity, including power and dual-power law nonlinearity. Moreover, many new analytical exact solutions are obtained which are expressed by hyperbolic solutions, trigonometric solutions, and rational solutions. In addition, we obtained the bright soliton by HSIVM. These methods are powerful, efficient and those can be used as an alternative to establishing new solutions of different types of differential equations in mathematical physics and engineering.     

On Shtelen’s Solution of the Free Linear Schrödinger Equation

Abstract

The solution of the three-dimensional free Schrödinger equation due to W.M. Shtelen based on the invariance of this equation under the Lorentz Lie algebra so(1,3) of nonlocal transformations is considered. Various properties of this solution are examined, including its extension to n ≥ 3 spatial dimensions and its time decay; which is shown to be slower than that of the usual solution of this equation. These new solutions are then used to define certain mappings, F _n, on L ²(?ⁿ) and a number of their properties are studied; in particular, their global smoothing properties are considered. The differences between the behavior of F _n and that of analogous mappings constructed from usual solutions of the free Schrödinger equation are discussed.     

Location of Maximizers of Eigenfunctions of Fractional Schrödinger’s Equations

Eigenfunctions of the fractional Schrödinger operators in a domain D are considered, and a relation between the supremum of the potential and the distance of a maximizer of the eigenfunction from ? D is established. This, in particular, extends a recent result of Rachh and Steinerberger arXiv:1608.06604 (2017) to the fractional Schrödinger operators. We also propose a fractional version of the Barta’s inequality and also generalize a celebrated Lieb’s theorem for fractional Schrödinger operators. As applications of above results we obtain a Faber-Krahn inequality for non-local Schrödinger operators.     

Schrödinger’s Equation with Gauge Coupling Derived from a Continuity Equation

A quantization procedure without Hamiltonian is reported which starts from a statistical ensemble of particles of mass m and an associated continuity equation. The basic variables of this theory are a probability density ρ, and a scalar field S which defines a probability current j=ρ ? S/m. A first equation for ρ and S is given by the continuity equation. We further assume that this system may be described by a linear differential equation for a complex-valued state variable χ. Using these assumptions and the simplest possible Ansatz χ(ρ,S), for the relation between χ and ρ,S, Schrödinger’s equation for a particle of mass m in a mechanical potential V(q,t) is deduced. For simplicity the calculations are performed for a single spatial dimension (variable q). Using a second Ansatz χ(ρ,S,q,t), which allows for an explicit q,t-dependence of χ, one obtains a generalized Schrödinger equation with an unusual external influence described by a time-dependent Planck constant. All other modifications of Schrödinger’ equation obtained within this Ansatz may be eliminated by means of a gauge transformation. Thus, this second Ansatz may be considered as a generalized gauging procedure. Finally, making a third Ansatz, which allows for a non-unique external q,t-dependence of χ, one obtains Schrödinger’s equation with electrodynamic potentials A,φ in the familiar gauge coupling form. This derivation shows a deep connection between non-uniqueness, quantum mechanics and the form of the gauge coupling. A possible source of the non-uniqueness is pointed out.     

Weyl’s Type Estimates on the Eigenvalues of Critical Schrödinger Operators

We consider on a bounded domain \(\Omega \subset {\mathbb{R}}^N\) , the Schrödinger operator ? Δ ? V supplemented with Dirichlet boundary solutions. The potential V is either the critical inverse square potential V(x) = (N ? 2)²/4|x|^?2 or the critical borderline potential V(x) =  (1/4)dist(x, ?Ω)^?2. We present explicit asymptotic estimates on the eigenvalues of the critical Schrödinger operator in each case, based on recent results on improved Hardy–Sobolev type inequalities.     

Schrödinger’s Cat Meets Einstein’s Twins: A Superposition of Different Clock Times

The phenomenon of quantum superposition, which allows a physical system to exist in different states ‘simultaneously’, is one of the most bizarre notions in physics. Here we illustrate an even more bizarre example of it: a superposed state of a physical system consisting of both an ‘older’ version and a ‘younger’ version of that system. This can be accomplished by exploiting the special relativistic effect of time dilation featuring in Einstein’s famous twin paradox.     

Energy minimization related to the nonlinear Schrödinger equation

In this the window of the Sobolev gradient technique to the problem of minimizing a Schrödinger functional associated with a nonlinear Schrödinger equation. We show that gradients act in a suitably chosen Sobolev space (Sobolev gradients) can be used in finite-difference and finite-element settings in a computationally efficient way to find minimum energy states of Schrödinger functionals.     

High-energy optical Schrödinger solitons

The conditions for the existence of a Lax pair were determined and exact analytic solutions to the nonlinear evolution equations of the Schrödinger type with complex and nonuniform potentials were found. In particular, these solutions provide a basis for the soliton management concept in applied problems and solve the problems of optimal energy accumulation by a Schrödinger soliton in an active medium and soliton amplification in optical fiber communication lines and soliton lasers.     

The surfboard Schrödinger equations

We study the large time behavior of solutions of time dependent Schrödinger equationsiu/t=–(1/2)u+t V(x/t)u with bounded potentialV(x). We show that (1) if>–1, all solutions are asymptotically free ast, (2) if–1 a solution becomes asymptotically free if and only if it has the momentum support outside of suppV for large time, (3) if –1 <0 all solutions are still asymptotically modified free ast and that (4) if 0 <2, for each local minimumx ₀ ofV(x), there exist solutions which are asymptotically Gaussians centered atx=tx ₀ and spreading slowly ast.     

Topological Solitons of the Nonlinear Schrödinger’s Equation with Fourth Order Dispersion

This paper obtains the topological 1-soliton solution of the nonlinear Schrödinger’s equation, in a non-Kerr law media, with fourth order dispersion. An exact 1-soliton solution is obtained. The types of nonlinearity that are studied in this paper are Kerr law and power law.     

Almost periodic Schrödinger operators

We studyH=–d ²/dx ²+V(x) withV(x) limit periodic, e.g.V(x)=a _n cos(x/2 ⁿ ) with a _n <. We prove that for a genericV (and for generica _n in the explicit example), (H) is a Cantor ( nowhere dense, perfect) set. For a dense set, the spectrum is both Cantor and purely absolutely continuous and therefore purely recurrent absolutely continuous.Research partially supported by NSF Grant MCS78-01885On leave from Department of Physics, Princeton UniversityOn leave from Departments of Mathematics and Physics, Princeton University; during 1980–81 Sherman Fairchild Visiting Scholar at Caltech     

Quantum Circuits and Schr?dinger’s Cat States

Quantum systems that are confined to circuit geometries are called quantum circuits. Macroscopic superconducting circuits are quantum circuits which can be modelled using a Quantisation by Parts scheme based on the macroscopic wave function approach of Feynman. This paper studies the circuit composed of an input wire and an output plate. We find that in order to achieve a consistent theory of supercurrent flow we have to generalize the quantisation by parts scheme to quantise in a path space. The generalized theory predicts a current flow down the wire into the plane. In addition to a current flowing radially outwards in the plane, the theory allows a circulating current round the origin. Strikingly, the circulating current can flow clockwise or anti-clockwise in such a way as to generate a magnetic moment of magnitude half of a Bohr magneton for an orbiting electron in an atom and a magnetic flux half that of the magnetic flux quantum of a superconducting ring. There is also the possibility of a macroscopic superposition of the two states of opposing circulating currents resembling a Schr?dinger’s cat situation. Furthermore, we outline a setup involving an external magnetic field that may allow experimental tests of the theory.     

Asymptotic Properties of Solutions to 3-Particle Schrödinger Equations

We construct a generalized Fourier transformation ℱ(λ) associated with the 3-body Schr?dinger operator H=−Δ+Σ ^a V ^a (x ^a ) and characterize all solutions of (H−λ)u= 0 in the Agmon–H?rmander space ℬ^* as the image of ℱ(λ)^*. These stationary solutions admit asymptotic expansions in ℬ^* in terms of spherical waves associated with scattering channels. Received: 20 September 2000 / Accepted: 20 May 2001