# Connection between theory of superconductivity and Bohmian mechanics

I stumbled upon an interesting chapter of Feynman Lectures about Superconductivity and was surprised to see that the equations of motion of the superconducting electron fluid (21.38 and 21.39) hold a term known as Quantum Potential from Bohmian mechanics (the last term in the first equation): $$\begin{equation} \label{Eq:III:21:38} \left.m\frac{dv}{dt}\right|_{\text{comoving}}\kern{-0.5ex}= q(E\!+\!v\!\times\!B)\,+\,\nabla{\frac{\hbar^2}{2m^2} \!\biggl(\!\frac{1}{\sqrt{\rho}}\nabla^2\!\!\sqrt{\rho}\!\biggr)}. \end{equation}$$ $$\nabla \times v=-\frac{q}{m}\,B.$$ Seeing this I was wondering if Bohmian Mechanics might have been misinterpreted in the past as a theory about a single electron? Maybe it is in fact a theory describing the behaviour of many charged massive bosons (like cooper pairs of electrons in the case of superconductivity), where the wave function becomes a classical observable (as Feynman explains quite nicely)?

• Two things being described by the same equation doesn't mean they're the same thing. There simply aren't that many simple-looking rotationally invariant PDEs. – knzhou Jul 11 '18 at 9:52
• It does mean that you can import your intuition from Bohmian mechanics, if you want, but that doesn't mean this is Bohmian mechanics. The context is quite different, as here the superconducting field behaves like a classical field, while the whole point of Bohmian mechanics is to explain quantum behavior. Really the only link is that fields are involved, which is about as descriptive as saying two programs both have for loops. – knzhou Jul 11 '18 at 9:54

It happens quite often in physics that equations from rather different areas have a similar form (A professor I knew, for example, used to say: "Statistical physics is exactly like quantum mechanics without the $i$".) So no, Bohmian mechanics has not been "misinterpreted" just because there is some similarity to other equations in physics. You may want to read up more about Bohmian mechanics for example here to understand its merits describing non-relativistic quantum particles: https://arxiv.org/abs/quant-ph/0408113
From the Schrödinger equation, the following continuity equation follows. Here $\rho = |\psi|^2$. $$\frac{\partial \rho(x)}{\partial t} = - \nabla \cdot j(x)$$ with the so-called quantum current $j(x) = 2 \Im (\psi^*(x) \nabla \psi(x))$. This type of equation can be found also in the dynamics of fluids, where then $\rho$ is the density of the fluid and $j$ its current. For fluids, this can be expressed as $j = \rho v$, with velocity field $v$. If you take the analogy further, and also read the quantum continuity equation as a similar thing, you get for the velocity: $$v(x) = \frac{j(x)}{\rho(x)} = 2 \Im \frac{\psi^*(x) \nabla \psi(x) }{|\psi|^2(x)}.$$ Actually, this is just the guiding law in Bohmian mechanics, which gives the change of the positions $v(Q(t)) = \frac{d}{dt} Q(t)$ of the particles. So you see that a strong analogy holds between the movement of particles in a fluid and the movement of quantum particles guided by a wave function in Bohmian mechanics. This is also the reasion that experiments like here: https://www.youtube.com/watch?v=WIyTZDHuarQ show the same behaviour as quantum systems to some extent.