Bohm Trajectories for a Coupled Two-Dimensional Harmonic Oscillator
Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
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].
[more]
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.
Snapshots
Permanent Citation