Quantum Motion of Two Particles in a 3D Trigonometric Pöschl-Teller Potential
Exact solutions of the nonrelativistic wave equations contain all the necessary information for the quantum system and have important applications in particle physics. This Demonstration discusses a solution of the Schrödinger equation in three-dimensional configuration space with the trigonometric Pöschl–Teller potential in the Bohm approach.
Quantum motion occurs in configuration-space particle trajectories associated with the de Broglie–Bohm causal interpretation of quantum mechanics. Previous Demonstrations (see Related Links) showed that the motion of a quantum particle could be obtained when the corresponding particle density (given by the modulus of the Schrödinger wavefunction) is not time dependent.
This Demonstration considers a three-dimensional anharmonic oscillator described by a trigonometric Pöschl–Teller potential in a degenerate stationary state with a constant phase shift. Two particles are placed on the margin of the potential randomly and separated by an initial distance .
A system with three degrees of freedom assigned by a superposition of three coherent stationary eigenfunctions that has commensurate energy eigenvalues and with a relative constant phase shift can exhibit chaotic motion in the associated de Broglie–Bohm picture, provided that the constant phase is not zero or an integer multiple of (see Bohm Trajectories for a Particle in an Infinite 3D Box). The origin of the motion lies in the relative phase of the total wavefunction, which has no analog in classical particle mechanics. The dynamical structure is quite complex. Some of the curves are closed and periodic, while others are quasiperiodic. In the region of nodal points of the wavefunction, the trajectories apparently become accelerated and chaotic. The parameters have to be chosen carefully, because of the singularities of the velocities and the large oscillations that can lead to very unstable trajectories. Further investigation to capture the full dynamics of the system is necessary.
The graphics show three-dimensional contour plots of the squared wavefunction (if enabled) and two initially neighboring trajectories. To modify the constant phase shift , change para in the initialization code and restart. Black points mark the initial positions of the two quantum particles and green points mark the actual positions. Blue points indicate the nodal point structure.
Associated Legendre polynomials arise as the solution of the Schrödinger equation:
with the integers , , , , and so on. A degenerate, unnormalized wavefunction with time period for for the three-dimensional case can be expressed by:
where , , are eigenfunctions, and are permuted eigenenergies of the corresponding stationary one-dimensional Schrödinger equation with . The eigenfunctions are defined by
where , , are associated Legendre polynomials. The parameter is a constant phase shift , and are the quantum numbers with and . The wavefunction is taken from .
For this Demonstration, the wavefuction is defined by:
Due to the permuted wavefunction structure and depending on the constant phase shift , most of the nodal points are positioned only in the , plane.
The velocity field is calculated from the gradient of the phase from the total wavefunction in the eikonal form (often called polar form) . The time-dependent phase function from the total wavefunction is:
The corresponding velocity field becomes time independent (autonomous) because of the gradient of the phase function.
The velocity in the direction becomes zero if , which is fulfilled for . The velocities in the other directions become zero for . The length of the curve depends on the constant phase shift .
In the program, if PlotPoints, AccuracyGoal, PrecisionGoal, and MaxIterations are increased, the results will be more accurate.
 G. Pöschl, E. Teller, "Bemerkungen zur Quantenmechanik des anharmonischen Oszillators," Zeitschrift für Physik,83 (3–4), 1933 pp. 143–151. doi:10.1007/BF01331132.
 M. Trott, The Mathematica GuideBook for Symbolics, New York: Springer-Verlag, 2006.