Motion of a quantum particle in 1-dimensional space under the influence of an external potential. Preliminary!!

Ulrich Mutze 2022-10-01

We study quantum dynamics of a single non-relativistic particle on a discrete 1-dimensional space under the influence of an external potential. The initial state of the particle is given by a wave function which is localized in an interval the size of which is 20 Compton wavelengths of the particle and the discretization is chosen such that the distance between neighboring particle positions is a tenth of a Compton wave length. As a consequence the particle state is fully described by 200 complex numbers subject to an obvious normalization condition. We obtain these numbers from a complex valued function of a real variable, actually a product of a real Gaussian and a complex monochromatic wave chosen such that the wave function approximately vanishes at the endpoints of its discrete domain. For technical reasons we multiply this preliminary wave function with a smooth windowing function which brings these numbers down to zero exactly. The wave-like factor 'boosts' the wave function to a constant velocity, to the right in our case. An algorithm for bringing to live this velocity as motion needs space to move into. The situation is particularly simple in our 1-dimensional case. Most evolution algorithms, as the one employed here, need one additional point position to the right and one to the left, for each integration step. Thus the list of points involved in the motion of the wavefunction grows. For reasons of computational economy we don't add one point after the other but a bunch of 50 points at once. Only if the wave function as expanded to the end of the new space and has grown to a value well above numerical noise (1.e-7 in our case) a new bunch of points will be added. In this way the available space is titely connected with the wave function and its particle. This conceivable relation between size of available space and motion of the particle is well expressed by referring to this space as the particle's biotope. I'll adhere to this wording in what follows.

Actually, the present note is an exercise in implementing the concept of expanding space ( following the expansion of wave functions) starting from the most simple non-trivial situation. To go a bit further, consider two identical particles, two electrons, say with two separated biotopes and let the particles move towards each other so that their biotopes will intersect in the near future. As long as this intersection is void, the troubling question concerning symmetrizing or anti-symmetrizing with wave functions of distant particles (the proverbial 'electron behind the moon') cannot be formulated and, thus, has not to be answered. An instructive simulation of this situation, for two particles with discretized biotopes on a common 1-dimensional position space should be within the reach of my methods and tools.

Back to our single particle system:

Reflection of a particle from a potential barrier

Acceleration of a particle encountering a downhill potential ramp

The two videos show situations which differ only by the sign of the potential. The dynamic state of the particle is initially given as a Gaussian placed in a part of space where the potential is zero. The initial velocity is directed to the right, were the particle will encounter a ramp-like change of potential, increasing in the first case and decreasing in the second one. After an x-interval of smooth change the potential becomes constant. For the initial state of the dynamic system, there is only a very limited part of space provided as a data structure, actually as a list of equi-distant particle positions which thus form a linear finite lattice. It is convenient to refer to the space in which a particle is allowed to move as the particles (and its wavefunctions) biotope. When the wave function reached the rim of biotope, the time-evolution algorithm enlarges the biotope by attaching new pieces carrying zero-values of the wave function and constantly continued potential. From this ever growing biotope the videos show only a cutout on which the the wave function shows most of its relevant behavior. The most obvious observation is that the phase velocity changes sign near the turning points of the potential. In the case of barrier the energy level meets the potential curve. In the WKB-approximation this point plays a special role. Surprisingly the dynamics of the wave function seems not take notice of a particularity there.

Subsequent to the time-evolution there are shown two diagrams which show the spurious deviations from energy conservation and norm conservation which are characteristic features of the DALF integration algorithm An asynchronous leapfrog method II (Nov 2013, 14 MB, http://www.arxiv.org/abs/1311.6602) of error measures over the time span covered by the present simulation. These errors can be shown scale with the square of the time step and to develop no trend (neither decreasing nor increasing). There is a critical time step, which for the present simulation is by a factor 4 larger than the employed time step, above which both errors grow exponentially, leading very soon to explosive growth. There are a cheap ways to compute the critical time step (not making use of matrix diagonalization, of course). In the legend of the diagrams x = 83000 is the number of the integration step, y the value of the error measure and n = 3601 the number of space points in the observed part of the biotope at the end of the simulation. The total biotope then had 37900 space points, more than 10 times more than the visual biotope. The computation time was 709 s with creating the movie file and 300 s without creating (and filling) a movie file, and 219 s even without creating screen images. These computing tools were used: CPU 4.464 GHz; RAM 16 GB; OS Ubuntu; SW C++20, OpenGL, FreeGlut, Code::Blocks, Mathematica (for video creation), C+- (selfmade C++ library); Compiler GNU.