Time evolution of the classical crossway system.

Ulrich Mutze 2022-12-21

The system under consideration is a system of two classical particles which move on the 'same streets' and have the same masses and feel the same forces as the quantum particles in the system treated in the previous animation. The relation between these two systems can be precisely expressed by saying that the quantum system originates from the classical system by quantization. Although 'quantization' can be done in different ways for systems with sufficiently complicated constitution ( expressed by a sufficiently complicated Lagrangian or Hamiltonian) , it is a uniquely defined procedure in our case. The only way to make the classical system behave in a way that is comparable with the quantum system is to give it suitable initial conditions.

If we would find an initial condition x1(0), v1(0), x2(0), v2(0), with v1(0) > 0 for which v1(t) < 0 for all sufficiently large times t then it would be proved that back-scattering as observed in the quantum system is not a pure quantum effect. Actually, the rational behind looking for a classical analogue of the crossway system was the the question whether back-scattering here is a pure quantum effect comparable to the tunnel effect or can also happen for classical trajectories.

The way we are choosing here is not to select and inspect individual trajectories but set the initial conditions as random variables. These are designed such that they translate as accurate as possible the statistical properties of the initial quantum state to a statistical ensemble of classical initial states. Let us see how this is done precisely:

For particle 1 the initial quantum state specifies a mean value for x10 of position, and a value for the standard deviation of position. For the velocity the initial quantum state gives specifies a mean value. For the corresponding classical statistical ensemble of x and v we specify the same mean values. For x also the same standard deviation. For the standard deviation σv we set from Heisenberg's uncertainty relation for for position and momentum σv = 1/(2 m1 σx).

For particle 2 the initial quantum state is set as the ground state of a harmonic oscillator with given particle mass m2 and standard deviation of position. How this translates to classical initial conditions for x and v ? We know the classical trajectories belonging to the ground state energy of the quantum oscillator. All such trajectories differ only by a constant phase difference. Therefore, a uniform random variable ranging from 0 to 2π determines position and velocity of a particle associated with this trajectory.

By letting a random generator create a series of instances of our random variables, we get a large number of trajectories. Recall the meaning of these trajectories. What our time evolution algorithm provides is a series of states each of characterized by a 5-tupel (t,x1,v1,x2,v2). For visualization of the process we ignore the velocities and are left with the 3-tupel (t,x1,x2) which characterizes what often is called a 'configuration' (in contrast to 'state'). Here x1 describes the position of particle 1 which moves along the x-axis and x2 describes the position of particle 2 which moves along the y-axis. If one considers the situation for two points in time which are close together we see a short path of particle 1 on the x-axis and a short path of particle 2 on the y-axis. If we have in mind not only one trajectory but lots of them, things become rather confusing. Therefore we make use of the graphical representation method that was already in use for the quantum system: represent (t,x1,x2) as (t,p) where p is a single point in the x,y-plane with x coordinate x1 and y coordinate x2, p=(x1,x2) for short. Then each trajectory becomes smooth line in the plane. When we present time as time in a video sequence the ignored velocity data get reintroduced by the speed impression of position change in the video stream.

The following video shows the fate of 8000 trajectories. In the first video frame each colored point on screen corresponds to one initial condition. The color of each point codes the conserved total energy of the 2-point system. The red color represents lowest energy, violet color highest. This is in accordance with the photon energy as a function of spectral color. Our palette also shows luminance modulation with dark lines in equidistant steps. This provide quantitative landmarks in the sequence of spectral colors analogous to the landmarks in the solar spectrum provided by the Fraunhofer lines. Since energy is a function of state and not of configuration it comes not as a surprise that in ensemble of initial configurations the colors are rather dispersed, so that no order or principle can be recognized. The following frames show how the initial states evolve in a way that the corresponding configurations form a continuous line.

The first part of the video shows 8000 steps of evolution with the same time step that was used in the quantum version. The quantized version. Time stepping is done by the Stoermer-Verlet method which for classical systems a bit more convenient to program than DALF and has the same stability and reversibility properties. Recall that the color of each curves indicates the constant total energy of the 2-particle system.

The next part of the video lets time run backwards (the DALF-integrator is strictly reversible) and the colors are set such that they indicate the position of particle 1 in the final state of the first part of the video. The final state of the second part equals the initial state of the whole run but with a coloring that shows the particles which will suffer back-scattering as red dotes and those which will show no excitation of the target (particle 2) as violet. One would have to consider much more trajectories if one would try to detect robust patterns in the distribution of colors. This is even more true if one would hunt for patters which correspond to the quite obvious patterns in the quantum wave function wave function of quantum crossway system and the system of classical trajectories.

The total computation time for creating the video frames was 440 s. 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.