相关论文: Arnold's potentials and quantum catastrophes
The well known phenomenon of avoided level crossing (ALC) can be perceived as a quantum analogue of the Thom's catastrophes in classical dynamical systems. One-dimensional Schr\"{o}dinger equation is chosen for illustration. In constructive…
We use path-integrals to derive a general expression for the semiclassical approximation to the partition function of a one-dimensional quantum-mechanical system. Our expression depends solely on ordinary integrals which involve the…
The existence of quantum tunneling opens the possibility of a sudden spatial relocalization of a system after a minor modification of its parameters. Such a quantum analogue of the Thom's classical catastrophe would manifest itself,…
We use path-integrals to derive a general expression for the semiclassical approximation to the partition function of a one-dimensional quantum-mechanical system. Our expression depends solely on ordinary integrals which involve the…
In usual dimensional counting, momentum has dimension one. But a function f(x), when differentiated n times, does not always behave like one with its power smaller by n. This inevitable uncertainty may be essential in general theory of…
Non-separable $D-$dimensional partial differential Schr\"{o}dinger equations are considered at $D=2$ and $D=3$, with the even-parity local potentials $V(x,y,\ldots)$ which are polynomials of degree four (cusp catastrophe resembling case)…
Singularity of the potential function makes quantum tunneling problem mathematically underdetermined. To circumvent the difficulties it introduced in physics, a potential singularity cutoff is often used, followed by a reverse limit…
In this paper we start from the Schr\"odinger equation to revisit some classical quantum mechanics from the perspective of phase transition process. Here the relativistic effect of particles moving at high speed can be regarded as the phase…
Classical transport equations with probabilistic initial conditions can be viewed as quantum systems. In a discrete version they are probabilistic automata. The time-local probabilistic information is encoded in a classical wave function.…
By integrating curvatures multiplied non-trivial densities, we introduce an integral expression of the Arnold strangeness that is a celebrated plane curve invariant. The key is a partition function by Shumakovitch to reformulate Arnold…
It was proposed recently that the Schr\"odinger wave function can be reconstructed exactly from a discrete superposition of classical action branches weighted by associated classical densities, without semiclassical approximations. We…
In this paper we consider the discrete one-dimensional Schroedinger operator with quasi-periodic potential v_n = \lambda v (x + n \omega). We assume that the frequency \omega satisfies a strong Diophantine condition and that the function v…
The Copenhagen interpretation of quantum mechanics assumes the existence of the classical deterministic Newtonian world. We argue that in fact the Newton determinism in classical world does not hold and in classical mechanics there is…
Classical mechanics is a singular theory in that real-energy classical particles can never enter classically forbidden regions. However, if one regulates classical mechanics by allowing the energy E of a particle to be complex, the particle…
The classical limit $\hbar$->0 of quantum mechanics is known to be delicate, in particular there seems to be no simple derivation of the classical Hamilton equation, starting from the Schr\"odinger equation. In this paper I elaborate on an…
The parametric ladder climbing (successive Landau-Zener-type transitions) and the quantum saturation of the threshold for the classical parametric autoresonance due to the zero point fluctuations at low temperatures are discussed. The…
We investigate the effect of classical singularities in the quantum properties of non-random Hamiltonians. We present explicit results for the case of a kicked rotator with a non-analytical potential though extensions to higher…
We consider the Schr\"odinger operator with a periodic potential on quasi-1D models of armchair single-wall nanotubes. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite…
The Schrodinger equation for stationary states in a central potential is studied in an arbitrary number of spatial dimensions, say q. After transformation into an equivalent equation, where the coefficient of the first derivative vanishes,…
We present a detailed theory of spectacular semiclassical catastrophes happening during the time evolution of a kicked quantum rotor (Phys.Rev. Lett. {\bf 87}, 163601 (2001)). Both two- and three-dimensional rotational systems are analyzed.…