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Studies in Microeconomics

Description of the Bluffing Phenomenon in the Untrusted Seller–Buyer Relationship via the Wigner Function of the q-Deformed Quantum Harmonic Oscillator Model

E. I. Jafarov

Published Aug 17, 2024• 0 citations
Publisher

Abstract

The q -deformed quantum harmonic oscillator model is applied to the description of the untrusted seller–buyer relationship. We assumed that the demand law by the buyer is described through the square modulus of the wavefunctions of the q -oscillator model in the position representation, whereas the supply law by the seller is described through the square modulus of the wavefunctions in the momentum representation. Thus, their untrusted relationship is described through the q -oscillator Wigner function of the stationary states in terms of the Al Salam-Chihara polynomials. As an example, we study the first excited state and show that the finite-difference differentiation parameter h or the deformation parameter q generated through it can be considered as a parameter defining the level of the bluffing phenomenon in this untrusted relationship. JEL Classifications: C02, D01, D11, D46, D58, D81

DOI
10.1177/23210222241262171

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The q -deformed quantum harmonic oscillator model is applied to the description of the untrusted seller–buyer relationship. We assumed that the demand law by the buyer is described through the square modulus of the wavefunctions of the q -oscillator model in the position representation, whereas the supply law by the seller is described through the square modulus of the wavefunctions in the momentum representation. Thus, their untrusted relationship is described through the q -oscillator Wigner function of the stationary states in terms of the Al Salam-Chihara polynomials. As an example, we study the first excited state and show that the finite-difference differentiation parameter h or the deformation parameter q generated through it can be considered as a parameter defining the level of the bluffing phenomenon in this untrusted relationship. JEL Classifications: C02, D01, D11, D46, D58, D81
The phase space representation for a q -deformed model of the quantum harmonic oscillator is constructed. We have found explicit expressions for both the Wigner and Husimi distribution functions for the stationary states of the q -oscillator model under consideration. The Wigner function is expressed as a basic hypergeometric series, related to the Al-Salam-Chihara polynomials. It is shown that, in the limit case h → 0 ( q → 1), both the Wigner and Husimi distribution functions reduce correctly to their well-known non-relativistic analogues. Surprisingly, examination of both distribution functions in the q -deformed model shows that, when q ≪ 1, their behaviour in the phase space is similar to the ground state of the ordinary quantum oscillator, but with a displacement towards negative values of the momentum. We have also computed the mean values of the position and momentum using the Wigner function. Unlike the ordinary case, the mean value of the momentum is not zero and it depends on q and n . The ground-state-like behaviour of the distribution functions for excited states in the q -deformed model opens quite new perspectives for further experimental measurements of quantum systems in the phase space.
We construct a new model of the quantum oscillator, whose energy spectrum is equally-spaced and lower-bound, whereas the spectra of position and of momentum are a denumerable non-degenerate set of points in [− 1, 1] that depends on the deformation parameter q ∊ (0, 1). We provide its explicit wavefunctions, both in position and momentum representations, in terms of the discrete q -Hermite polynomials. We build a Hilbert space with a unique measure, where an analogue of the fractional Fourier transform is defined in order to govern the time evolution of this discrete oscillator. In the limit when q → 1 − , one recovers the ordinary quantum harmonic oscillator.
The boson expansion method is applied to find the spectrum of a q-deformed harmonic oscillator. We use two different boson expansions, each of them including a deformation parameter, defined in terms of exponential and logarithmic functionals. The resulting Hamiltonians are bilinear forms of the transformed operators. Physical effects resulting from the deformation of the generators of the algebra are studied by comparing known finite-range potentials and the effective potentials obtained for each of the considered Hamiltonians.
The finite q -oscillator is a model that obeys the dynamics of the harmonic oscillator, with the operators of position, momentum and Hamiltonian being functions of elements of the q -algebra su q (2). The spectrum of position in this discrete system, in a fixed representation j , consists of 2 j + 1 'sensor'-points , and similarly for the momentum observable. The spectrum of energies is finite and equally spaced, so the system supports coherent states. The wavefunctions involve dual q -Kravchuk polynomials, which are solutions to a finite-difference Schrödinger equation. Time evolution (times a phase) defines the fractional Fourier– q -Kravchuk transform. In the classical limit as q → 1 we recover the finite oscillator Lie algebra, the N = 2 j → ∞ limit returns the Macfarlane–Biedenharn q -oscillator and both limits contract the generators to the standard quantum-mechanical harmonic oscillator.
We define a Wigner distribution function for a one-dimensional finite quantum system, in which the position and momentum operators have a finite (multiplicity-free) spectrum. The distribution function is thus defined on discrete phase-space, i.e. on a finite discrete square grid. These discrete Wigner functions possess a number of properties similar to the Wigner function for a continuous quantum system such as the quantum harmonic oscillator. As an example, we consider the so-called oscillator model in dimension 2 j + 1, which is known to tend to the canonical oscillator when j tends to infinity. In particular, we compare plots of our discrete Wigner functions for the oscillator with the well known plots of Wigner functions for the canonical quantum oscillator. This comparison supports our approach to discrete Wigner functions.
We propose a phase-space representation concept in terms of the Wigner function for a quantum harmonic oscillator model that exhibits the semiconfinement effect through its mass varying with the position. The new method is used to compute the Wigner distribution function exactly for such a semiconfinement quantum system. This method suppresses the divergence of the integrand in the definition of the quantum distribution function and leads to the computation of its analytical expressions for the stationary states of the semiconfined oscillator model. For this quantum system, both the presence and absence of the applied external homogenous field are studied. Obtained exact expressions of the Wigner distribution function are expressed through the Bessel function of the first kind and Laguerre polynomials. Furthermore, some of the special cases and limits are discussed in detail.