Studying the quantum crossway system together with Bohmian particles.

Ulrich Mutze 2023-02-18

Let us recall the definition of the crossway system in a way that emphasizes the structural aspects that became evident by comparing the quantum system with its classical analog. An important concept is the configuration space (space of configurations). This space allows the state space (space of dynamical states) to be realized as a space of functions on the configuration space. For quantum systems these functions are complex-valued and for classical systems they take values in the real vector space of tangent vectors. In quantum mechanics the flavor of the theory that does not work in general complex Hilbert spaces but with complex-valued functions on configuration space is well known as wave mechanics.

The great advantage of wave mechanics is that here the passage from a one-particle system to a many-particle system is simply achieved by forming the Cartesian product of configuration spaces, whereas in the Hilbert space frame work we have to construct the many-particle state space as a tensor product by means of multi-linear algebra. If, as is the case in the present study, we are interested in computational models, working with configuration spaces is particularly convenient since replacing continua by finite sets of points is a well understood technique in computational physics. In the situations shown in the two videos, the configuration space can be viewed in too ways:

In the original version we have two mass-points moving on different 'streets' which cross at a right angle at the center of the graphical window. One of the points (called particle 2 throughout) feels a harmonic oscillator potential which has its equilibrium position just at the crossing point. The other point (particle 1) moves freely with regard to its street. Both particles feel an interaction potential which is a function of their mutual distance.

In the version discovered later, there is only one mass-point (the particle). It moves in a Euclidian plane that has the two 'streets' of the original picture as axes of a coordinate system. It feels a potential which depends on its distance from the origin. This potential, as a function of distance, is the same as the interaction potential of the two points in the first version. Further it feels a potential that depends on the distance from coordinate axis 1 which as a function is the same which acted on particle 2 in the original version. The project classicalcrossway has more on the relation between these two views.

A further interesting aspect of configurations in quantum theory is the basic insight of the Bohmian view on quantum mechanics that the wave function associated with the motion of a particle also defines an equation of motion for particle configurations (i.e. positions). So the statement 'this particle now is at position x' implies 'this particle is on its way along a trajectory which ascribes to the time 'now' the position x. The trajectory can be computed from the time-dependent wave function of the particle and from the mentioned information concerning x.

The position of a particle (such as a proton) could therefore considered a beable according to the well-known definition by John Bell: "The beables of the theory are those elements which might correspond to elements of reality, to things which exist." This is different from the Copenhagen view which considers positions of particles only then as meaningful if the position came out as a documented event in an experiment. One should notice that the Copenhagen attitude, when followed strictly, makes it very difficult (if not impossible) to describe the admirable biochemical reaction chains like photosynthesis, citric acid cycle, and aerobic respiration. The stability and persistence of the particles asks for notions that don't become inapplicable if no laboratory like setup is around. Speaking about actually existing particles (as considered in Bohmian mechanics) with a microscopically sharply defined position seems to be the most natural way to go.

Recall that the equation of motion of Bohmian particles depend on a wave function which is a solution of the time dependent Schroedinger equation. With the number of interacting particles the computational workload for evolving the wave function is easily seen to grow exponentially so that any underlying informational machinery will break down for mildly complex physical situations. This does not mean that nature will suffer from a irreparable break-down. It has, however, to implement a recovery strategy for local break-downs which will happen on a regular basis. In this (my personal) view of things these break-downs do not break causality: they are real events that happen inevitably since wave functions start out from meaningful local situations (such as collisions of two particles, both initially in their ground state), by 'coming into interaction' with more and more particles become more and more 'computational demanding' and finally asks for more 'computational resources' than are available. See Computational power needed to propagate a non-relativistic n-particle wave function.

The recovery strategy will be based on the positions and kinds (proton, electron, ...) of the particles involved. In the ontological view of Bohm these will remain available although the wave function ceased to exit. It is instructive to mentally look into a electronic camera (e.g. the one in your smart phone). The configuration space of the multi-pixel semiconductor system of the image sensor offers so many degrees of freedom to an approaching photon that a complete treatment of all of them is impossible for any computer-like system (even one of the size of the universe). So we end up in a recovery state which provides one additional electron in one of the pixel-cells and with the photon no longer in existence.

The evolution of the wave function is typically much larger a computational burden than moving Bohmian particles. In the case of the following two videos, removing 9999 of the 10000 particles reduced the execution time from 1537 s to 1467 s which is only 4.3 %. It is quite obvious that adding sufficiently Bohmian particles to the wave function (as is the case in the first video) increases our chance to get some kind of understanding for the complex dynamics of the wave function. This is particularly so for the last third of the video where a reflected wave interferes with an outgoing wave and the flow of particles lets us easily see where one of the two directions 'has more followers'. Having viewed the first video several times I actually felt some understanding for the process. After I had made (for different reasons) the video with a single Bohmian particle I was quite surprised to see a behavior that seemed to conflict with by 'insights' in the mass behavior. Bohr's complementarity comes to mind.

Unlike the classical trajectories, the Bohmian trajectories present no uncertainties how the initial conditions should be set. These are determined by the initial condition of the wave function and the guiding condition. Unlike the classical trajectories the Bohmian trajectories don't show conservation of energy. What is shown as energy of particles in the third (final) non-animated diagram in the two videos does not take into account the Bohmian 'quantum potential'. There seems to be no Bohmian claim that energy along a Bohmian trajectory becomes less variable if contribution of the quantum potential are taken into account. Notice that energy of wave function which looks perfectly constant in the same diagram, actually shows a very small variability that is shown in the second non-animated diagram. This is an effect of space dicretization and also depends on the integrator. This is discussed in The direct midpoint method as a quantum mechanical integrator and Ulrich Mutze "On the Stability Limit of Leapfrog Methods", Wolfram Demonstrations Project Published: September 7 2011 For all integrators of the leap-frog family there seems to exist an explicit formula connecting the relative error of the norm and the relative error of the energy. Comparing the two diagrams for these error quantities in the videos suggests such an connection.

The interested reader is invited to inspect the files cpmdata...txt, cpmcerr...txt, ...cpp, ...ini in the folder of the video to see how the many parameters of a 'video production' get their value and what the values of some dependent parameters are. For instance we read in cpmdata_t1_b.txt the entry 'nOsc: total number of oscillations during run = 1.74877' that the oscillator particle performed nearly two full oscillations in this simulation. For further video production I plan to store the corresponding version of these four files together with the video to be always able to reproduce results. Notice that many of the source files which started as part of a particular project tend to migrate into the C+- library /cpm/cpm0 - /cpm/cpm4. For instance most of the functionality of the project bohmiancrossway and other crossway-related projects now comes from the files /cpm/cpm3/include/cpmcrossway.h and /cpm/cpm3/source/cpmcrossway.cpp and thus are not documented in the four documentation files mentioned above. Therefore I also plan to hold older versions of /cpm/cpm0 - /cpm/cpm4 on this website.

These computing tools were used: CPU 4.464 GHz; RAM 16 GB; OS Ubuntu; SW C++20, OpenGL, FreeGlut, Eigen3, Code::Blocks, ffmpeg (for video creation from ppm-files), C+- (self-made C++ library); Compiler GNU.