# 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].

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Contributed by: Klaus von Bloh (May 8)

Open content licensed under CC BY-NC-SA

## Details

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:

with

and

.

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.

References

[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] Bohemian-Mechanics.net. (Jan 31, 2023) www.mathematik.uni-muenchen.de/~bohmmech/index.php.

[6] S. Goldstein, "Bohmian Mechanics." *The Stanford Encyclopedia of Philosophy (Summer 2017 Edition)*. (Jan 31, 2023)plato.stanford.edu/entries/qm-bohm.

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