Bohm Trajectories for an Isotropic Harmonic Oscillator with Added Inverse Quadratic Potential This Demonstration considers two-dimensional 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 pseudoharmonic-type 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 inverse-square-type 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 quasi-periodic 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 time-dependent 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 pseudoharmonic-type potential (if enabled) is shown with blue/black contour lines.
Details The two-dimensional stationary Schrödinger equation with potential only of the distance from the origin, can be written:
with the potential 1
, a function
, 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 pseudoharmonic-type 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 time-independent part of the continuity equation
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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.
Figure 1: Trajectories in configuration space for different initial conditions (see the Mathematica notebook)
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References [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/s10910-011-99674. [2] Bohmian-Mechanics.net. (Oct 8, 2018) www.mathematik.unimuenchen.de/~bohmmech/BohmHome/index.html. [3] S. Goldstein, "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy (Summer 2017 Edition). (Oct 2, 2018) plato.stanford.edu/entries/qm-bohm. [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:10.1016/0375-9601(95)00190-E. [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/s11534-007-0022-9. RELATED LINKS Bohm, David Joseph (1917–1992) (ScienceWorld) Schrödinger Equation (Wolfram MathWorld) Spherical Bessel Function of the First Kind (Wolfram MathWorld) Separation of Variables (Wolfram MathWorld) Polar Coordinates (Wolfram MathWorld) Laplacian (Wolfram MathWorld) Quasiperiodic Motion (Wolfram MathWorld) Particle in an Infinite Circular Well (Wolfram Demonstrations Project) Chaos (Wolfram MathWorld) Nodal Points in Bohmian Mechanics (Wolfram Demonstrations Project)
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Influence of a Moving Nodal Point on the "Causal Trajectories" in a Quantum Harmonic Oscillator Potential (Wolfram Demonstrations Project) Bohm Trajectories for a Particle in a Two-Dimensional Circular Billiard (Wolfram Demonstrations Project) Bohm Trajectories for the Two-Dimensional Coulomb Potential (Wolfram Demonstrations Project) PERMANENT CITATION Klaus von Bloh "Bohm Trajectories for an Isotropic Harmonic Oscillator with Added Inverse Quadratic Potential" http://demonstrations.wolfram.com/BohmTrajectoriesForAnIsotropicHarmonicOscillat orWithAddedInv/ Wolfram Demonstrations Project Published: October 19, 2018 Betreff: Your submission to the Wolfram Demonstrations Project Absender: Wolfram Demonstrations Project Empfänger:
[email protected] Datum: 7. November 2018 13:30
Dear Klaus von Bloh, We are happy to inform you that your submission Bohm Trajectories for an Isotropic Harmonic Oscillator with Added Inverse Quadratic Potential to the Wolfram Demonstrations Project has been accepted for publication. Your Demonstration will now be available to all visitors to the Wolfram Demonstrations Project site. The permanent URL for your Demonstration is: http://demonstrations.wolfram.com/BohmTrajectoriesForAnIsotropicHarmonicOscillatorWithA ddedInv/ It will be available within the next 24 hours. We encourage you to cite this Demonstration in other publications, and to send a link to the Demonstration to anyone you feel is appropriate.
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