This Demonstration considers twodimensional Bohm trajectories in a central field represented by an isotropic harmonic oscillator augmented by an inverse quadratic potential plus a constant . This is called a pseudoharmonictype potential, with the form . Exact solutions of the Schrödinger equation for this potential are known. An analogous potential in three dimensions can represent the interaction of some diatomic molecules [1]. Obviously, the pseudoharmonic oscillator behaves asymptotically as a harmonic oscillator, but has a singularity at For , there is a small region where the potential exhibits a repulsive inversesquaretype character. Possible trajectories can then exhibit a rich dynamical structure, depending on the parameters , of the potential and the initial starting points. The motion ranges from periodic to quasiperiodic to fully chaotic. In the de Broglie–Bohm (or causal) interpretation of quantum mechanics [2, 3], the particle position and momentum are well defined, and the motion can be described by continuous evolution according to the timedependent Schrödinger equation. In contrast to [4], the conditions for chaotic behavior can occur in a system with two degrees of freedom and for a superposition of two stationary states. See, for example, , (in the variable in the program), , , , and the initial position , two trajectories with initial distance . The graphic shows the trajectories (white/blue), the velocity vector field (red), the absolute value of the wavefunction and the initial and final points of the trajectories. The initial point is shown as a white square, which you can drag. The final point is shown as a small white or blue dot. The pseudoharmonictype potential (if enabled) is shown with blue/black contour lines.
The twodimensional stationary Schrödinger equation with potential , a function only of the distance from the origin, can be written: , reduced mass , Planck's constant , the constants and the Laplacian operator in plane polar coordinates. For simplicity, set and equal to 1. The solution of the Schrödinger equation for the quantum system with the pseudoharmonictype potential gives to the eigenfunction (for detailed information, see [5]) in plane polar coordinates: with , the integer and with the eigenenergy : . In Cartesian coordinates ( , ), the solution reads: . An unnormalized wavefunction for a particle, from which the trajectories are calculated, can be defined by a superposition state : or in Cartesian coordinates: For , the velocity field becomes autonomous and obeys the timeindependent part of the continuity equation with ; the trajectories reduce to circles with the velocities and : and . When PlotPoints, AccuracyGoal, PrecisionGoal and MaxSteps are increased (if enabled), the results will be more accurate. The initial distance between the two starting trajectories is determined by the factor . In the program, you can change the parameters of the potential. [1] K. J. Oyewumi and K. D. Sen,"Exact Solutions of the Schrödinger Equation for the Pseudoharmonic Potential: An Application to Some Diatomic Molecules," Journal of Mathematical Chemistry, 50(5), 2012 pp. 1039–1059. doi:10.1007/s1091001199674. [4] R. H. Parmenter and R. W. Valentine, "Deterministic Chaos and the Causal Interpretation of Quantum Mechanics," Physics Letters A, 201(1), 1995 pp. 1–8. doi:10.1016/03759601(95)00190E. [5] S. M. Ikhdair and R. Sever, "Exact Solutions of the Radial Schrödinger Equation for Some Physical Potentials," Central European Journal of Physics, 5(4), 2007 pp. 516–527. doi:10.2478/s1153400700229.
