In Bohmian mechanics, do electrons move inside an atom?

Question

Look at http://www.bohmian-mechanics.net/whatisbm_pictures_hydrogen.html. It is mentioned that in the rest states of a bound electron, the position of the electron is stationary, since the wavefunction's phase is not changing. In the Bohmian picture where the electron DOES have an exact position, it should be supposed to stay at a certain point, since it has no Bohmian trajectory.

However, we KNOW that the electron has a kinetic energy of ~13.6eV, which make it have a velocity of ~2E6 m/s. So isn't this a contradiction in the Bohmian picture of the atom?

Answer 1

So isn't this a contradiction in the Bohmian picture of the atom?

In the Bohmian view, what would be kinetic energy can be 'stored' in the guiding wave. Here's an extensive excerpt from Reflections on the deBroglie–Bohm Quantum Potential

Consider the simple case of a cubical box with no classical potential inside and infinite potential outside. In other words, the particle cannot escape while the box remains intact. If an initially free (spinless) particle is trapped in such a box of side length $L$, then its wavefunction would take on the stationary waveform:

$$\psi = (2L)^{3/2}|\sin(n_1\pi\,x/L)\sin(n_2\pi\,y/L)\sin(n_3\pi\,z/L)|e^{iE_nt/\hbar} = Re^{iS/\hbar}$$

with total energy $E=(n^2_1 + n^2_2 + n^2_3)(\pi^2\hbar^2/2mL^2)$; where $n_1,\,n_2,\,n_3$ are positive integers. In Orthodox Quantum Theory, the particle is assumed to have kinetic energy only and to be bouncing back and forth between the walls of the box. In deBroglie–Bohm Theory, the value of the quantum potential is given by: $Q = -(\hbar^2/2m)(\nabla^2R)/R = (n^2_1 + n^2_2 + n^2_3)(\pi^2\hbar^2/2mL^2)$: This is the same magnitude as the particle’s kinetic energy is assumed to have in Orthodox Quantum Theory. However, since $S = -Et,\,\nabla S = 0$; i.e., the particle has zero momentum and therefore zero kinetic energy. All the energy of the system is potential with the kinetic energy of the quantum particle having become stored in the wave field (Bohm1952, 184; Riggs1999, 3072). What’s more, this energy will be returned to the particle if the wave field’s stationary state is disturbed, e.g., if any side of the box is removed. Surprisingly, this explanation was originally suggested by David Bohm when he wrote:

... the kinetic energy of the particle will come from the $\psi$ field, which is able to store up even macroscopic orders of energy when its wave-length is small (1953, 14, italics mine).

Answer 2

This is indeed one of the many disadvantages of Bohmian mechanics: in the ground state the electron is predicted to hover still in midair forever. Besides being exceptionally strange for a theory that sells itself on being classically intuitive, it flies in the face of experimental measurements of the electron's velocity, momentum, or kinetic energy.

The Bohmian reply to this problem is that all such experimental measurements are wrong. In other words, any attempts to measure properties of the Bohmian particle (besides its position) are doomed to accidentally instead measure properties of the pilot wave. The electron really is hovering still in midair, but we just can't observe it doing that. Several tricks like this allow Bohmian mechanics to evade contradiction with the experimentally verified uncertainty principle. Whether this is a deal breaker is up to you.

On the plus side, for highly excited states, the Bohmian trajectories are somewhat intuitive. For example, for atomic states, they give concrete realization to the orbits postulated in the Bohr model, so it's not all bad.

Answer 3

In Bohmian mechanics, kinetic energy splits into a contribution from the actual motion of the particle and a contribution from the quantum potential. In the situation you reference, all of the standard quantum mechanical kinetic energy is stored in the quantum potential.