Why is the Bohm quantum potential considered a potential?

Question

In Bohmian mechanics, the term $$\begin{equation} Q = -\frac{\hbar^2}{2m}\frac{\nabla^2 R}{R} \end{equation}$$

is regarded as the quantum potential term. However this is merely a term from the real part of the quantum kinetic energy equation. $$\begin{equation} T = -\frac{\hbar^2}{2m}\frac{\nabla^2 \Psi}{\Psi} \end{equation}$$

I completely fail to see why this should be regarded as anything except kinetic energy. Can someone explain?

(Yes, I know about dividing by complex quantities. When you plug in the polar form of $\Psi$ the exponentials cancel and you're left with just $R$ so dealing with the $\Psi*$ is unnecessary.)

Answer 1

In the de Broglie-Bohm model, we write $\Psi(x) = R(x) e^{iS(x)/\hbar}$ and identify $p(x)\equiv \big(\nabla S\big)(x)$ as the momentum of a particle at the point $x$. After some standard manipulation (see my answer here for more details), we find that

$$ -\frac{\hbar^2}{2m}\nabla^2 \Psi + V\Psi = E\Psi$$ $$\implies E = \frac{(\nabla S)^2}{2m} - \frac{\hbar^2 \nabla^2 R}{2mR} + V $$

Making the identification $p\equiv \nabla S$ means that the first term becomes $p^2/2m$, which we interpret as kinetic energy; the rest is interpreted as potential energy, which includes a classical contribution $V$ and a "quantum" contribution $-\hbar^2\nabla^2 R/2mR$.

There's not much more depth to it than that. In the standard formulation of quantum mechanics, the quantum potential term would be considered part of the "kinetic energy" term, but in de Broglie-Bohm the interpretation of the various terms is slightly different.

Answer 2

The notion that this is some kind of potential is historic. It was Bohm himself who introduced it. In recent times many researchers working on Bohmian Mechanics have mostly abandoned that notion except sometimes as a pedagogical analogy which can only be taken so far. Just like, for example, in special relativity the interpretation that mass increases for objects moving near the speed of light is no longer considered useful.

The origin for the notion of a quantum potential was that the Hamilton-Jacobi equations are very similar to the equation of motion for $S$, the phase of the wave function when it is written $\psi = R e^{iS/\hbar}$. The Schrodinger equation gives:

$$\frac{\partial S }{\partial t} = -\left(\frac{|\nabla S|^2}{2m} + V-\frac{\hbar^2}{2m}\frac{\nabla^2 R}{R}\right)$$

...which is just the Hamiltonian Jacobi equation but with an extra term, which you likely notice is equal to your $Q$. This could be interpreted as a modification to the potential. But as I already mentioned, this interpretation has shown its limits and is therefore not to be taken too literally. The notion of a wave function acting as a guide to point particles is sufficient.