Interpretation of the Quantum Measurement: Example of a Linearly Coupled Harmonic Oscillator

We argue that it may be possible to consistently explain the quantum measurement by assuming that the wave function is in one-to-one correspondence with objective physical reality and has no probabilistic interpretation. In the context of such approach we consider the model of a harmonic oscillator linearly coupled to a heat bath and treat the oscillator as the system being measured. Three classes of initial pure states for the bath are considered. Exact expressions for the average values and variances of the oscillator coordinate and momentum as functions of time are considered for each class of pure states. It is shown that these quantities exhibit different asymptotic behavior for different classes of initial states of the bath. In particular, if each mode of the bath is initially in a coherent state, then for an arbitrary initial state of the oscillator the variances of the oscillator coordinate and momentum asymptotically approach the same values as for a coherent state of the free oscillator, while the averages of coordinate and momentum show a Brownian-like behavior. We argue that such behavior shows several features of the quantum measurement and supports our interpretation of the wave function.     

Regularization of some linear ill-posed problems with discretized random noisy data

For linear statistical ill-posed problems in Hilbert spaces we introduce an adaptive procedure to recover the unknown solution from indirect discrete and noisy data. This procedure is shown to be order optimal for a large class of problems. Smoothness of the solution is measured in terms of general source conditions. The concept of operator monotone functions turns out to be an important tool for the analysis.

     

Numerical differentiation from a viewpoint of regularization theory

In this paper, we discuss the classical ill-posed problem of numerical differentiation, assuming that the smoothness of the function to be differentiated is unknown. Using recent results on adaptive regularization of general ill-posed problems, we propose new rules for the choice of the stepsize in the finite-difference methods, and for the regularization parameter choice in numerical differentiation regularized by the iterated Tikhonov method. These methods are shown to be effective for the differentiation of noisy functions, and the order-optimal convergence results for them are proved.

     

Quantum transport in chains with noisy off-diagonal couplings

We present a model for conductivity and energy diffusion in a linear chain described by a quadratic Hamiltonian with Gaussian noise. We show that when the correlation matrix is diagonal, the noise-averaged Liouville-von Neumann equation governing the time evolution of the system reduces to the [Lindblad, Commun. Math. Phys. 48, 119 (1976)] equation with Hermitian Lindblad operators. We show that the noise-averaged density matrix for the system expectation values of the energy density and the number density satisfies discrete versions of the heat and diffusion equations. Transport coefficients are given in terms of model Hamiltonian parameters. We discuss conditions on the Hamiltonian under which the noise-averaged expectation value of the total energy remains constant. For chains placed between two heat reservoirs, the gradient of the energy density along the chain is linear.     

Sulfonyl Nitrene and Amidyl Radical: Structure and Reactivity

Photocatalytic generation of nitrenes and radicals can be used to tune or even control their reactivity. Photocatalytic activation of sulfonyl azides leads to the elimination of N₂ and the resulting reactive species initiate C−H activations and amide formation reactions. Here, we present reactive radicals that are generated from sulfonyl azides: sulfonyl nitrene radical anion, sulfonyl nitrene and sulfonyl amidyl radical, and test their gas phase reactivity in C−H activation reactions. The sulfonyl nitrene radical anion is the least reactive and its reactivity is governed by the proton coupled electron transfer mechanism. In contrast, sulfonyl nitrene and sulfonyl amidyl radicals react via hydrogen atom transfer pathways. These reactivities and detailed characterization of the radicals with vibrational spectroscopy and with DFT calculations provide information necessary for taking control over the reactivity of these intermediates.     

On the optimization of projection-iterative methods for the approximate solution of ill-posed problems

We consider a new version of the projection-iterative method for the solution of operator equations of the first kind. We show that it is more economical in the sense of amount of used discrete information.     

Self–regularization by projection for noisy pseudodifferential equations of negative order

It is well known, that pseudodifferential equations of negative order considered in Sobolev spaces with small smoothness indices are ill–posed. On the other hand, it is known that efficient discretization schemes with properly chosen discretization parameters allow to obtain a regularization effect for such equations. The main accomplishment of the present paper is the principle for the adaptive choice of the discretization parameters directly from noisy discrete data. We argue that the combination of this principle with wavelet–based matrix compression techniques leads to algorithms which are order–optimal in the sense of complexity.     

Optimal discretization of Ill-posed problems

We present a survey of results on the optimal discretization of ill-posed problems obtained in the Institute of Mathematics of the Ukrainian National Academy of Sciences.