Periodic Quantum Motion of Two Particles in a 3D Harmonic Oscillator Potential
Bohmian quantum mechanics allows for both waves and particles, whereby particles are guided by the phase of the total wavefunction. The velocity of the particles becomes autonomous only with a periodic trajectory structure, when the corresponding wavefunction is in a degenerate stationary state of two eigenfunctions differing by a constant phase shift. This Demonstration studies the dynamic structure of the superposition of two three-dimensional eigenstates of the harmonic oscillator, which leads to periodic motion in configuration space. Two particles are placed on the margin of the harmonic potential randomly and separated by an initial distance .
The parameters have to be chosen carefully because singularities in the velocities or large oscillations can lead to very unstable trajectories. The motion is determined by the relative phase of the total wavefunction, which has no analog in classical particle mechanics. Changing the constant phase shift does not influence the structure of the trajectory; it only changes the length of the path.
The graphics show three-dimensional contour plots of the squared wavefunction (if enabled) and two initially neighboring trajectories. Red 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 Hermite polynomials arise as the solution of the Schrödinger equation: , with , , and so on. A degenerate, unnormalized, complex-valued wavefunction for the three-dimensional case can be expressed by:
where , , and are eigenfunctions, and are permuted eigenvalues of the corresponding stationary one-dimensional Schrödinger equation with . The eigenfunctions are defined by:
where , , and are Hermite polynomials. The parameter is a constant phase shift, and the energies depend on the quantum numbers, with , with .
For this Demonstration, the wavefunction is defined by:
In this case, the square of the Schrödinger wavefunction , where is its complex conjugate, is time independent:
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.
In the program, if PlotPoints, AccuracyGoal, PrecisionGoal, MaxSteps, and MaxIterations are enabled and increased, the results will be more accurate.