Causal Interpretation of the Free Quantum Particle

A solution of the one-dimensional free Schrödinger equation leads to a time-dependent Gaussian wave packet (unnormalized, ) with initial width and group velocity in space. According to the orthodox interpretation of quantum theory, the probability of finding the particle within a spatial interval decreases with time as the wave packet disperses. In the causal interpretation of quantum theory of David Bohm, the wavefunction is a real wave in configuration space. With this assumption it could be shown that a quantum particle possesses well-defined positions and velocities at all times. According to the causal interpretation there is a quantum force proportional to a quantum potential that guides the quantum particle inside the wave packet.
The trajectories run along the maximum plateau of the squared wavefunction. The width of the wave packet increases with time from its initial value. In contrast to classical mechanics, particles are accelerated at the front and decelerated at the rear of the packet, because increases in magnitude as the amplitude of the wavefunction decreases. In the center of the packet the motion of the particle is uniform, which is the predicted motion for the classical case. Inside the wave the starting points for the particles are distributed according to the density of the wave at . The equation of the motion for the particles is given analytically.
On the right side, the graphic shows the squared wavefunction and the trajectories. On the left side, you can see the position of the particles, the squared wavefunction (blue), the quantum potential (red), and the velocity (green). The quantum potential and the velocity are scaled down.


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F. J. Belinfante, A Survey of Hidden-Variables Theories, Oxford: Pergamon Press, 1973.
P. R. Holland, The Quantum Theory of Motion, Cambridge: Cambridge University Press, 1993.
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