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 .[more]
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) . 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.[less]
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 :
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 .
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 :
For , and and with , the velocity field becomes time dependent, but the orbits reduce to fractions or multiples of a circle:
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.
 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.
 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.
 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.
 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.
 Bohemian-Mechanics.net. (Jan 31, 2023) www.mathematik.uni-muenchen.de/~bohmmech/index.php.
 S. Goldstein, "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy (Summer 2017 Edition). (Jan 31, 2023)plato.stanford.edu/entries/qm-bohm.