# The Superposition Principle in the Causal Interpretation of Quantum Mechanics

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The principle of quantum superposition states that the complex-valued wavefunction can be a linear combination of several quantum eigenstates simultaneously. The modulus (or absolute value) squared of the total wave function represents a probability density of finding a quantum particle only. In the causal interpretation of quantum theory, developed in 1952 by David Bohm, the concept of trajectories is introduced, in which the particles are guided by a real phase function (or quantum potential) from the total wavefunction (pilot wave). In the eikonal representation, the phase function for a superposed state from which the velocity is determined becomes very complicated. This Demonstration discusses a superposition of up to 15 states of the density distribution for a quantum particle in the box with infinitely high walls. As the superposed wave packet spreads in an infinite square well, the paths of the quantum particles approach in a way similar to a classical trajectory.

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Contributed by: Klaus von Bloh (January 2014)

After work by: Enrique Zeleny

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The guiding equation for the particle velocity is , which is calculated from the gradient of the phase of the wavefunction in the energy representation. In the eikonal form the total wavefunction becomes

.

In this case, the amplitude and the phase reduce to

and

,

from which the velocity is derived by differentiating the total phase function

with

and

,

where , and so on.

In the case that the phase function does not depend on the variable , the velocity becomes

with

and

,

which could be simplified to

.

Local singularities in the velocity do appear to exist for certain and values (), which makes the trajectories very unstable.

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References

[1] P. Holland, *The Quantum Theory of Motion*, Cambridge: Cambridge University Press, 1993.

[2] D. Bohm, *Quantum Theory*, New York: Prentice-Hall, 1951.

[3] I. Marzoli, F. Saif, I. Bialynicki-Birula, O. M. Friesch, A. E. Kaplan, and W. P. Schleich, "Quantum Carpets Made Simple." arXiv:quant-ph/9806033.