STRUCTURE OF THE WAVE FUNCTION BY SHOT NOISE IN THE PARABOLIC POTENTIAL AND STATISTICAL DISTRIBUTION OF THE LIFETIME OF A PARTICLE WAVE PACKET
DOI:
https://doi.org/10.32782/mathematical-modelling/2024-7-1-16Keywords:
Schrödinger equation, parabolic potential, shot noise, stochastic buildup, wave function evolution, wave packet lifetime, lifetime probability density distributionAbstract
The properties of materials are associated with the excitation of nonlinear localized vibrations in the lattice, which affect the dynamics of particles in it. In this work, the problem of destroying the wave packet of a particle is posed and analyzed. We consider the quantum mechanical problem of the motion of a particle in a quadratic potential, which as a whole is subject to the stochastic process of shot noise and the dynamics of the wave function of the particle in it. Based on the found solutions to the nonstationary Schrödinger equation, the time evolution of the wave function is considered. The problem of destruction of the wave packet of a particle is formulated, which is realized when the condition of achieving the dispersion of the packet of a given size is met. A similar formulation arises in problems when the perturbation represents the trajectory of one-dimensional or two-dimensional process that models changes in the potential during the movement of a particle, in particular, when an electron moves along the crystal axis. In this case, the role of time in the problem is played the perturbation function that describes the forced vibrations of the crystal lattice. The time evolution of a particle in the ground state with an initial wave function in a potential that includes a square-integrable function – a stochastic shot noise process with zero mathematical expectation and dispersion – is considered. Based on the solution, which uses shot noise, particle density profiles are given in the form of profiles. The problem of the lifetime of a wave packet is considered, which is perturbed by the stochastic process of shot noise and is destroyed under the condition that its dispersion has reached a given size and the particle drops out of consideration (dies). Due to the stochastic nature of the disturbing process, the time interval before destruction will also be random. An analytical expression is obtained for the distribution density of the random variable – lifetime.
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