The Causal Interpretation of a Particle in a Two-Dimensional Square Box

time steps

50

quantum potential

yes

no

th

m

eigenstate

1

2

3

4

5

th

n

eigenstate

1

2

3

4

5

constant phase factor

0

1

2

π

2

3π

2

The de Broglie–Bohm interpretation of quantum theory contradicts the opinion that in the case of a macro system, the motion of the quantum system should approach the motion following from classical mechanics. Without measuring the momentum of the particle, there are some cases where the unobserved quantum particle is, in contradiction to the classical counterpart, at rest. In the case of motion, a quantum particle possesses highly nonclassical but well-defined trajectories. Furthermore, the motion of a quantum particle could be obtained when the corresponding particle density (given by the modulus of the Schrödinger wavefunction) is not time dependent. To study this effect, we consider a two-dimensional square box with infinite potential walls in a degenerated stationary state with a constant phase shift. A free particle is contained between impenetrable and perfectly reflecting walls, separated by a distance

L

. In this case, the energy eigenvalues and eigenfunctions for the two-dimensional box can be derived from those of the infinite square-well solutions of the one-dimensional Schrödinger equation. This quantum system can exhibit motion in the associated de Broglie–Bohm theory. The origin of the motion lies in the relative phase of the total wavefunction, which has no classical analogue in particle mechanics.

The graphic shows the squared wavefunction, the particles, the trajectories (yellow), and the velocity field (red) for various constant phase factors. If the "quantum potential" button is active, you see the trajectories and the velocity field with the associated quantum potential.