Causal Interpretation of the Quantum Harmonic Oscillator

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The harmonic oscillator is an important model in quantum theory that could be described by the Schrödinger equation: , (
) with
. In this Demonstration a causal interpretation of this model is applied. A stable (nondispersive) wave packet can be constructed by a superposition of stationary eigenfunctions of the harmonic oscillator. The solution is a wave packet in the (
,
) space where the center of the packet oscillates harmonically between
with frequency
. From the wavefunction in the eikonal representation
, the gradient of the phase function
and therefore the equation for the motion could be calculated analytically. The motion is given by
, where
are the initial starting points. The trajectories of the particles oscillate with the amplitude
and frequency
and they never cross. In practice, it is impossible to predict or control the quantum trajectories with complete precision. The effective potential is the sum of quantum potential (QP) and potential
that leads to the time-dependent quantum force:
.
Contributed by: Klaus von Bloh (March 2011)
Open content licensed under CC BY-NC-SA
Snapshots
Details
P. Holland, The Quantum Theory of Motion, Cambridge, England: Cambridge University Press, 1993.
D. Bohm, Quantum Theory, New York: Prentice–Hall, 1951.
Permanent Citation
"Causal Interpretation of the Quantum Harmonic Oscillator"
http://demonstrations.wolfram.com/CausalInterpretationOfTheQuantumHarmonicOscillator/
Wolfram Demonstrations Project
Published: March 7 2011