Related papers: Wave function representation of probability distri…
In quantum mechanics, wave functions and density matrices represent our knowledge about a quantum system and give probabilities for the outcomes of measurements. If the combined dynamics and measurements on a system lead to a density matrix…
Given a quantum state in the finite-dimensional Hilbert space $ \C^n $, the range of possible values of a quantum observable is usually identified with the discrete spectrum of eigenvalues of a corresponding Hermitian matrix. Here any such…
We report an infinite number of orthonormal wave functions bases for the quantum problem of a free particle in presence of an applied external magnetic field. Each set of orthonormal wave functions (basis) is labeled by an integer $p$,…
Properties of certain $q$-orthogonal polynomials are connected to the $q$-oscillator algebra. The Wall and $q$-Laguerre polynomials are shown to arise as matrix elements of $q$-exponentials of the generators in a representation of this…
Series representations consisting of spherical harmonics are obtained for characteristic exponents and probability density functions of multivariate stable distributions under various conditions. A esult potentially applicable in a…
We study the decrease of fluctuations of diagonal matrix elements of observables and of Husimi densities of quantum mechanical wave functions around their mean value upon approaching the semi-classical regime ($\hbar \rightarrow 0$). The…
Orthogonal polynomials of two real variables can often be represented in complex variables. We explore the connection between the two types of representations and study the structural relations of complex orthogonal polynomials. The complex…
We provide probabilistic interpretation of resonant states. This we do by showing that the integral of the modulus square of resonance wave functions (i.e., the conventional norm) over a properly expanding spatial domain is independent of…
We express the probabilistic character associated to the wave function by treating it as a stochastic variable. This is accomplished by means of a stochastic equation for the wave function whose noise changes the phase of the wave function…
The meaning of the wave function of the Universe was actively discussed in 1980s. In most works on quantum cosmology it is accepted that the wave function is a probability amplitude for the Universe to have some space geometry, or to be…
We introduce the probability distributions describing quantum observables in conventional quantum mechanics and clarify their relations to the tomographic probability distributions describing quantum states. We derive the evolution equation…
$W$-representation realizes partition functions by an action of a cut-and-join-like operator on the vacuum state with a zero-mode background. We provide explicit formulas of this kind for $\beta$- and $q,t$-deformations of the simplest…
Polynomials are common algebraic structures, which are often used to approximate functions including probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of…
The problem of quantum harmonic oscillator with "regular+random" square frequency, subjected to "regular+random external force, is considered in framework of representation of the wave function by complex-valued random process. Average…
The notion of standard positive probability distribution function (tomogram) which describes the quantum state of universe alternatively to wave function or to density matrix is introduced. Connection of the tomographic probability…
Asymptotic representations for large values of the hyperradius are constructed for the scattering wave function of a system of $ N $ particles considered as a generalized function of angular variable coordinates. The coefficients of the…
We obtain a positive probability distribution or Q-function for an arbitrary fermionic many-body system. This is different to previous Q-function proposals, which were either restricted to a subspace of the overall Hilbert space, or used…
We determine the inner product on the Hilbert space of wavefunctions of the universe by imposing the Hermiticity of the quantum Hamiltonian in the context of the minisuperspace model. The corresponding quantum probability density reproduces…
In contrast to wave functions in nonrelativistic quantum mechanics interpreted as probability amplitudes, wave functions in relativistic quantum mechanics have generalized meanings such as charge-density amplitudes, energy-density…
It is usually believed that a picture of Quantum Mechanics in terms of true probabilities cannot be given due to the uncertainty relations. Here we discuss a tomographic approach to quantum states that leads to a probability representation…