Studying a classical analog to the quantum crossway system.

Ulrich Mutze 2023-02-09

The primary motivation for studying a classical analog for the crossway quantum system was the desire to understand the mechanism of back-scattering. In our case the classical analog is defined straightforwardly: it has to have the same functional form of the Hamiltonian. Since in our case all forces on the particles are defined by potentials, which are given as functions of particle positions this means simply that the same functions for the potentials (and the same masses) are used in the equations of motion for two Newtonian particles as are used in the two particle Schrödinger equation. The classical equivalent to the wave function of the quantum system is a large number of mutually independent (non-interacting) classical trajectories. Since both systems are deterministic, we have to match the statistical ensemble of trajectories to the wave function only for the initial state. To this task there is a canonical solution which a well-known element of the Bohmian formulation of quantum mechanics: The particle momentum is the imaginary part of the logarithmic derivative of the wave function. For our assumption concerning the initial wave function this means that particle 1 has a velocity which is independent of position and that particle 2 has velocity 0. If we consider the quantum mechanical (i.e. Bohmian) trajectory instead of the classical one we have to take into consideration Bohm's quantum potential which is a function of particle positions (no velocities!). For a particle in a stationary state the forces derived from the quantum potential have a great influence. As a result the classical and the quantum trajectory behave quite differently: The quantum trajectory shows a particle at rest (until another particle approaches and exerts forces on it) whereas the classical trajectory shows an oscillatory motion around the deepest point of the potential unless its initial position coincides with this deepest point. The main characteristic of a quantum groundstate, that a particle in ground state never delivers energy to a fellow particle only may absorb energy from there, cannot be realized with classical particles. So, we observe in the early phase of time evolution a pulsating behavior of the particle positions that is not seen in the quantum simulation. Initial position distribution and momentum distribution of particle 2 (the harmonic oscillator particle) is chosen to imitate the ground state of the harmonic oscillator in quantum groundstate. The qroundstate of the classical harmonic oscillator (staying at rest at the place of least potential energy) would obviously not a good choice. So we need to steel from quantum theory the initial state of our oscillator particle. That the classical particles are stangers in the quantum world becomes obvious when particle 1 and 2 collide. It may well happen that particle 2 looses energy to particle 2, whereas a quantum particle in ground state could only take energy from particle 1.

These computing tools were used: CPU 4.464 GHz; RAM 16 GB; OS Ubuntu; SW C++20, OpenGL, FreeGlut, Code::Blocks, ffmpeg (for video creation from ppm-files), C+- (self-made C++ library); Compiler GNU.