Wolfram Demonstrations Project

In the nonrelativistic de Broglie–Bohm approach, the electron in a hydrogen-like atom is not moving in the stationary

state. More generally, this is true when the phase function does not depend on the spatial coordinates or when the wavefunction is a real-valued function. For the hydrogen-like atom in an eigenstate, the electron can be considered to be in motion if the magnetic quantum number

. The quantum particle orbits the

axis in a circle of constant radius

(for more details see [1]). In most other cases, any superposition state leads to a moving electron.

An appealing feature of the Bohmian description is that the particle has a well-defined trajectory in configuration space and we can impose initial positions for the trajectories, just as for a classical dynamical system. Such initial positions cannot by controlled by the experimenter, so there is an appearance of randomness in the pattern of detection. The wavefunction guides the particles in such a way that the particles are attracted to the regions in which the wave density is large (for more details see [1–3]). In regions with small wave density, the particles are accelerated as they pass through.

This Demonstration constructs a wavefunction in a degenerate stationary state in which the particle density, given by the square of the Schrödinger wavefunction, is time independent and the velocity term, given by the gradient of the total phase, is determined by an autonomous differential equation.

For this special case, an anisotropic Coulomb potential for a superposition of three eigenstates in parabolic coordinates with a constant phase shift factor

shows a complex dynamical structure that depends on the constant phase shift factor. Closed periodic trajectories with complex chaotic behavior would be associated with exponential divergence of initially neighboring trajectories.

The graphics show four possible orbits for the hydrogen atom, the time-independent wave density (if enabled), the velocity vector field (blue arrows), the initial starting points (shown as small red spheres) and the actual position (shown as small colored spheres). The initial distance between the two starting trajectories is determined by the factor

Contributed by: Klaus von Bloh

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The three-dimensional stationary Schrödinger equation with Coulomb potential

can be written in parabolic coordinates

with the wavefunction

, the Laplace operator

, the partial derivative

with respect to time and the Coulomb potential

. For simplicity, we set atomic units

In spherical polar coordinates, the unnormalized energy eigenfunctions for the hydrogen atom associated with the quantum state with the quantum numbers

is given by:

where

is a spherical harmonic. The radial function is

where

is a generalized Laguerre polynomial.

For a stationary case with

, the phase function

reduces to zero. Thus the Bohm momentum

of the ground state electron is

, and finally the electron is at rest [1]. For

, the electron is in motion.

If the wavefunction is expressed in parabolic coordinates (for more details see [4–6]), we find:

where

Here

and

is a Whittaker function.

The energy spectrum depends only on the integers

and

For the anisotropic Coulomb potential, the Schrödinger equation in Cartesian coordinates is:

with

and so on.

Due to computational limitations, the superposed total wavefunctions

are taken as a sum of just three eigenfunctions

, where the unnormalized wavefunction

for a particle is written as:

with

and (here,

Each part of the superposition of the three stationary state has the same energy:

The parabolic wavefunction

could be expressed by a linear combination of stationary states of spherical wavefunction

. For example,

Let

for convenience. After variable transformations from parabolic coordinates to Cartesian coordinates, we obtain the total phase function

The gradient of the phase

is obtained from the total wavefunction in polar form,

, with the quantum amplitude

. The velocity field

is calculated by the Bohm momentum

with mass

. From the Bohm momentum

it follows that

(here,

In the case of the superposition of the three stationary states with equal energy, the corresponding differential equation system for the velocity field becomes autonomous and obeys the time-independent part of the continuity equation

with the wave density

For

the wavefunction

in parabolic coordinates reduces to

with the total phase function

This could be simplified for

The trajectories reduce to circles with the velocities:

and

The velocity depends on the position of the particle only. For

and

, most of the orbits form "spirally circles" along the

axis, in which a spiral orbit is perpendicular on the circle orbit (for an example, see [7]).

In all other cases, the integration of the velocity vector

in respect to time leads to complex periodic or chaotic orbits (for examples, see [8, 9]). If

with

(

), it is strongly implied that chaotic motion occurs.

When PlotPoints, AccuracyGoal, PrecisionGoal and MaxSteps are increased (if enabled), the results will be more accurate.

References

[1] 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.

[2] Bohmian-Mechanics.net. (Dec 26, 2022) bohmian-mechanics.net.

[3] S. Goldstein, "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy (Summer 2017 Edition). (Dec 26, 2022) plato.stanford.edu/entries/qm-bohm.

[4] L. I. Schiff, Quantum Mechanics, 2nd ed., New York: McGraw-Hill, 1955.

[5] E. Merzbacher, Quantum Mechanics, New York: Wiley, 1961.

[6] J. F. Ogilvie, "The Hydrogen Atom According to Wave Mechanics: Part II. Paraboloidal Coordinates," Revista de Ciencia y Tecnologia, 32(2), 2016 pp. 25–39. arXiv:1612.05098.

[7] K. von Bloh. Bohm Trajectories for the Anisotropic Coulomb Potential with Constant Phase Shift: Part 3 [Video]. (Dec 28, 2022) www.youtube.com/watch?v=kIxbf256IlE.

[8] K. von Bloh. Bohm Trajectories for the Anisotropic Coulomb Potential in Parabolic Coordinates: Part 1 [Video]. (Dec 26, 2022) www.youtube.com/watch?v=mhkX1VArlT0.

[9] K. von Bloh. Bohm Trajectories for the Anisotropic Coulomb Potential with Constant Phase Shift: Part 2 [Video]. (Dec 26, 2022) www.youtube.com/watch?v=mmNFFvksa8Q.