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In this paper we used the Fredholm method in Schroedingers integral equation in the investigation of the scattering effect near the center of it between a stationary quantum wave function and an electrostatic potential. Two potentials are studied one Coulombian and the other Podolsky. The result shows the importance of the proposal of Podolsky to regularize the effect near the scattering center in the quantum wave function. Being that the coulombian potential presents with strong variation in the amplitude of the wave after the scattering. In the case of Podolskys potential, this is corrected by adopting a constant that removes this strong variation.
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We construct a quantum algorithm that creates the Laughlin state for an arbitrary number of particles $n$ in the case of filling fraction one. This quantum circuit is efficient since it only uses $n(n-1)/2$ local qudit gates and its depth scales as $2n-3$. We further prove the optimality of the circuit using permutation theory arguments and we compute exactly how entanglement develops along the action of each gate. Finally, we discuss its experimental feasibility decomposing the qudits and the gates in terms of qubits and two qubit-gates as well as the generalization to arbitrary filling fraction.
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We address the impossibility of achieving exact time reversal in a system with many degrees of freedom. This is a particular example of the difficult task of aiming an initial classical state so as to become a specific final state. We also comment on the classical-to-quantum transition in any non-separable closed system of $n geq 2$ degrees of freedom. Even if the system is initially in a well defined WKB, semi-classical state, quantum evolution and, in particular, multiple reflections at classical turning points make it completely quantum mechanical with each particle smeared almost uniformly over all the configuration space. The argument, which is presented in the context of $n$ hard discs, is quite general. Finally, we briefly address more complex quantum systems with many degrees of freedom and ask when can they provide an appropriate environment to the above simpler systems so that quantum spreading is avoided by continuously leaving imprints in the environment. We also discuss the possible connections with the pointer systems that are needed in the quantum-to-classical collapse transitions.
The bound state Bethe-Salpeter amplitude was expressed by Nakanishi using a two-dimensional integral representation, in terms of a smooth weight function $g$, which carries the detailed dynamical information. A similar, but one-dimensional, integral representation can be obtained for the Light-Front wave function in terms of the same weight function $g$. By using the generalized Stieltjes transform, we first obtain $g$ in terms of the Light-Front wave function in the complex plane of its arguments. Next, a new integral equation for the Nakanishi weight function $g$ is derived for a bound state case. It has the standard form $g= N g$, where $N$ is a two-dimensional integral operator. We give the prescription for obtaining the kernel $ N$ starting with the kernel $K$ of the Bethe-Salpeter equation. The derivation is valid for any kernel given by an irreducible Feynman amplitude.
The behaviour of quantum systems in non-inertial frames is revisited from the point of view of affine coherent state (ACS) quantization. We restrict our approach to the one-particle dynamics confined in a rotating plane about a fixed axis. This plane is considered as punctured due to the existence of the rotation center, which is viewed as a singularity. The corresponding phase space is the affine group of the plane and the ACS quantization enables us to quantize the system by respecting the affine symmetry of the true phase space. Our formulation predicts the appearance of an additional quantum centrifugal term, besides the usual angular momentum one, which prevents the particle to reach the singular rotation center. Moreover it helps us to understand why two different non-inertial Schr{o}dinger equations are obtained in previous works. The validity of our equation can be confirmed experimentally by observing the harmonic oscillator bound states and the critical angular velocity for their existence.
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