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Semiconductor Physics Quantum Electronics & Optoelectronics

Integral equations for the wave function of particle systems

Костянтин Вікторович Авдонін

Published Sep 16, 2019• 1 citation
Publisher

Abstract

Constructions of integral equations for the wave function of particle systems in bound states have been proposed in this work. We obtain the kernel of the Fredholm-type integral equation for an odd number of particles in explicit form. Besides, an integral equation that can be attributed to the class of Volterra integral equations was built. Their equivalence to known equations in limiting cases has been shown.

DOI
10.15407/spqeo22.03.319

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Constructions of integral equations for the wave function of particle systems in bound states have been proposed in this work. We obtain the kernel of the Fredholm-type integral equation for an odd number of particles in explicit form. Besides, an integral equation that can be attributed to the class of Volterra integral equations was built. Their equivalence to known equations in limiting cases has been shown.
In this work, exact solutions are obtained for two-dimensional partial integral equations of the Logunov-Tavkhelidze type for a system of two particles of equal mass in the case of a «delta-circle» potential and a superposition of two such potentials, defined in the relativistic configuration representation. It is shown that for a single «delta-circle» potential, depending on its parameters, the system can have either one bound state or none at all. For a superposition of two «delta-circle» potentials, cases with one or two bound states, or their absence, are possible. Analysis of the wave functions revealed differences in their properties across different representations: in the relativistic configuration representation, the number of zeros of the partial wave function equals the state number of the particle system, whereas in the momentum representation, the number of zeros exceeds that in the configuration representation. The nonrelativistic limit of the obtained solutions is found, which agrees with the solutions of the two-dimensional Schrödinger equation for similar potentials in the momentum and coordinate representations.
We investigate finite-time field-induced ionization from a one-dimensional attractive delta-function well in a uniform dc electric field of arbitrary strength. Our aim is to determine the physical bound-state survival amplitude ab(t)a_b(t)ab​(t) at any observation time without constructing the complete time-dependent propagator or wavefunction. In the gauge-equivalent Kramers--Henneberger representation, the field appears as motion of the contact point, and the resulting dynamics reduces to a closed Volterra equation. Exact endpoint-phase factorization organizes the full chronological rescattering history into a relative-time convolution hierarchy and an exact resolvent. By introducing the accumulated bound-state amplitude and reorganizing its two-time domain in relative and complementary center times, the final contact contribution becomes an explicit boundary integral. All spatial integrations are carried out analytically. The field-driven contact-free term is obtained in closed form using the Faddeeva function, while the direct and repeated-rescattering terms are expressed as explicitly evaluable time integrals. The result applies at arbitrary dc-field strength and finite time, with no weak-field expansion, rescattering truncation, or asymptotic-time approximation. It is verified through the exact field-free limit and an independent numerical solution of the original physical Volterra equation. The formulation gives an exact nonperturbative solution of a fundamental ionization model and reveals otherwise hidden analytical structure in driven quantum dynamics.
We study the time-evolution of a quantum particle subjected to time-dependent zero-range forces in two dimensions. After establishing a conceivable ansatz for the solution to the Schrödinger equation, we prove that the wave packet time-evolution is completely specified by the solutions of a system of Volterra-type equations – the charge equations – involving the coefficients of the singular part of the wave function, thus extending to the two-dimensional case known results in one and three dimensions.