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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