Bohm Trajectories for a Coupled Two-Dimensional Harmonic Oscillator

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Exact analytic solutions of the stationary Schrödinger equation enable the exploration of detailed properties of a quantum system. We note also that the integrability of a classical dynamical system need not necessarily imply its quantum integrability [1].


This Demonstration considers two-dimensional Bohm trajectories in a harmonic potential perturbed by a coupling term . Analytic expressions of the Schrödinger equation for this potential are known for a limited number of eigenstates. There exist solutions for the ground state and the first and second excited states [1–3]. Obviously, as , the coupled harmonic oscillator behaves asymptotically as two independent one-dimensional oscillators. Here the one-dimensional harmonic oscillator is referred to as a single quantum particle. The motion of two particles is represented in a two-dimensional configuration space by the coordinates along a trajectory. A superposition state leads to an entangled or nonlocal behavior in a two-dimensional configuration space. In general, an entangled wavefunction cannot be factorized into a product of two single-particle wavefunctions.

The motion ranges from motionless, periodic to quasi-periodic to fully chaotic, depending on the parameter and the constant phase shift . In the de Broglie–Bohm (or causal) interpretation of quantum mechanics [4–6], the particle position and momentum are well defined, and the motion can be described by continuous evolution according to the time-dependent Schrödinger equation.

The velocity vector of a superposition state , as expressed in the guiding equation, for one particle will depend upon the positions of the other, whenever the total wavefunction is not a product of single-particle wavefunctions (nonfactorizability) [4]. The projection of the trajectory onto two-dimensional configuration space leads to a decomposition of two spatially divided motions in one-dimensional real space. This is quantum entanglement in the de Broglie–Bohm interpretation. In our case, the one-dimensional quantum particles in a superposition state behave manifestly nonlocally, because of the coupling factor . The coupling factor determines the shapes of the orbits.

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 coupled harmonic potential (if enabled) is shown with blue/black contour lines.


Contributed by: Klaus von Bloh (May 8)
Open content licensed under CC BY-NC-SA


The two-dimensional stationary Schrödinger equation with potential , a harmonic potential with a coupling term, can be written:

with the potential


with reduced mass , Planck's constant , the constants and the Laplacian operator in Cartesian coordinates. For simplicity, set , and equal to 1.

With the variable transformation , , we get two different quantum harmonic oscillators with frequencies and .

Each single part of the superposition state can be expressed by an unnormalized product state


the are Hermite polynomials.

The solution of the Schrödinger equation will return a set of eigenvalues and coupled eigenfunctions in the original coordinates .

With the energy eigenvalue , the unnormalized, time-dependent wavefunction gives [2]:


with and . For the energy becomes complex valued.

This special equation , obeys the Schrödinger equation only for or 1.

An unnormalized wavefunction for two one-dimensional particles, from which the trajectories are calculated, can be defined by a superposition state :

with ; ; and

or in detail:




For the frequencies are equal, with .

Some examples:

For the quantum particles are at rest.

For and and with , the velocity field becomes autonomous and obeys the time-independent part of the continuity equation

with ; the trajectories reduce to circles with the velocities and :

and .

For , and and with , the velocity field becomes time dependent, but the orbits reduce to fractions or multiples of a circle:

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 . You can change the coupling constant potential , the constant phase shift and the superposition factor parameters in the program.

Special thanks to Vikram Athalye from Cummins College of Engineering for Women, Pune, India for his support.


[1] R. S. Kaushal, "Quantum Mechanics of Noncentral Harmonic and Anharmonic Potentials in Two-Dimensions," Annals of Physics, 206(1), 1991 pp. 90–105. doi:10.1016/0003-4916(91)90222-T.

[2] I. H. Naeim, J. Batle and S. Abdalla, "Solving the Two-Dimensional Schrödinger Equation Using Basis Truncation: A Hands-on Review and a Controversial Case," Pramana, 89(5), 2017 70. doi:10.1007/s12043-017-1467-z.

[3] R. M. Singh, F. Chand and S. C. Mishra, "The Solution of the Schrödinger Equation for Coupled Quadratic and Quartic Potentials in Two Dimensions," Pramana, 72(4), 2009 pp. 647–654. doi:10.1007/s12043-009-0058-z.

[4] P. R. Holland, The Quantum Theory of Motion: An Account of the de Broglie–Bohm Causal Interpretation of Quantum Mechanics, New York: Cambridge University Press, 1993.

[5] (Jan 31, 2023)

[6] S. Goldstein, "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy (Summer 2017 Edition). (Jan 31, 2023)


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