Intertwining relations of non-stationary Schrödinger operators

Intertwining of exactly solvable generalized Schrodinger equations

The Darboux transformation operator technique in differential and integral forms is applied to the generalized Schrodinger equation with a position-dependent effective mass and with linearly energy-dependent potentials. Intertwining operators are obtained in an explicit form and used for constructing generalized Darboux transformations of an arbitrary order. A relation between supersymmetry and the generalized Darboux transformation is considered. The method is applied to generation of isospectral potentials with additional or removal bound states or construction of new partner potentials with…

Straightforward Derivation of the Schr{ö}dinger Equation from Classical Mechanics and the Planck Postulate

According to the widely accepted notion, the Schr{ö}dinger equation (SE) is not derivable in principle. Contrary to this belief, we present here a straightforward derivation of SE. It is based on only two fundamentals of mechanics: the classical Hamilton-Jacobi equation(HJE) and the Planck postulate about the discrete transfer of energy at micro-scales. Our approach is drastically different from the other published derivations of SE which either employ an ad hoc underlying assumption about the probabilistic or the statistical nature of the micro-scale phenomena, or rely on the prior knowledge …

Implementation of traveling odd Schr$ö$dinger cat states in circuit-QED

We propose a realistic scheme of generating a traveling odd Schr$ö$dinger cat state and a generalized entangled coherent state in circuit quantum electrodynamics (circuit-QED). A squeezed vacuum state is used as initial resource of nonclassical states, which can be created through a Josephson traveling-wave parametric amplifier, and travels through a transmission line. Because a single-photon subtraction from the squeezed vacuum gives with very high fidelity an odd Schr$ö$dinger cat state, we consider a specific circuit-QED setup consisting of the Josephson amplifier creating the traveling res…

The Schr$\ddot{o}$dinger-Poisson equations as the large-N limit of the Newtonian N-body system: applications to the large scale dark matter dynamics

In this paper it is argued how the dynamics of the classical Newtonian N-body system can be described in terms of the Schr$\ddot{o}$dinger-Poisson equations in the large $N$ limit. This result is based on the stochastic quantization introduced by Nelson, and on the Calogero conjecture. According to the Calogero conjecture, the emerging effective Planck constant is computed in terms of the parameters of the N-body system as $\hbar \sim M^{5/3} G^{1/2} (N/<ρ>)^{1/6}$, where is $G$ the gravitational constant, $N$ and $M$ are the number and the mass of the bodies, and $<ρ>$ is their average densit…

A Nonperturbative Perspective on Inner Product Quantization: Highly Accurate Solutions to the Schr{ö}dinger Equation

We devise a new and highly accurate quantization procedure for the inner product representation, both in configuration and momentum space. Utilizing the representation $Ψ(ξ) = \sum_{i}a_i[E]ξ^i R_β(ξ)$, for an appropriate reference function, $R_β(ξ)$, we demonstrate that the (convergent) zeroes of the coefficient functions, $a_i[E] = 0$, approximate the exact bound/resonance state energies with increasing accuracy as $i \to \infty$. The validity of the approach is shown to be based on an extension of the Hill determinant quantization procedure. Our method has been applied, with remarkable succ…

分钟级自旋5/2原子的薛定谔猫态

Minutes-scale Schr{ö}dinger-cat state of spin-5/2 atoms

本研究展示了光阱中自旋5/2镱原子的长寿命薛定谔猫态，相干时间达1400秒，并利用拉姆齐干涉法实现海森堡极限计量，为量子精密测量提供新方案。