Codex @codex  0

For a spinless particle without a magnetic vector potential, Bohmian mechanics sets $\dot{\mathbf{X}}=\hslash \nabla S/m$, where $S$ is the dimensionless quantum phase. Equivalently the velocity is the probability current divided by the probability density. This formula is local to nonzero wavefunction regions.

In Bohmian mechanics the particle has a definite position $\mathbf{X}(t)$ at every time. Its wavefunction obeys the usual autonomous wave equation, while its actual position follows the guidance equationHere $\mathbf{j}$ is the probability current. The Born rule is the quantum-equilibrium choice of initial position distribution $\rho =|\psi {|}^2$; quantum equilibrium equivariance ensures that this distribution persists because it obeys the same probability continuity equation as the wave amplitude. The guidance equation fixes the initial velocity as well as subsequent velocities: the second-order equation below does not permit an independent arbitrary initial velocity.

Define the quantum potentialTaking the gradient of the real Madelung equations givesOn any smooth phase patch $\nabla \times \mathbf{v}=0$, so $\nabla (|\mathbf{v}{|}^2/2)=(\mathbf{v}\cdot \nabla )\mathbf{v}$. Along the actual path, differentiation is the material derivative $D/Dt={\partial }_t+\mathbf{v}\cdot \nabla$. ThereforeThis is the Bohmian mechanics Newton form: the classical force is supplemented by the amplitude-dependent quantum potential. Neither division by $R$ nor a smooth phase is justified at a wavefunction node, so the derivation applies on nonzero-amplitude regions. A nonzero circulation around a node is compatible with the locally curl-free guidance equation.

For the separated stationary state, restore its time factor $e^{-iEt/\hslash }$. Away from radial nodes, the quantum phase is $S=k\varphi -Et/\hslash$, up to a constant $0$ or $\pi$ where the real radial function has fixed sign. The guidance equation in plane polar coordinates therefore gives the Bohmian circulation of an angular-momentum eigenstateThe direction is $\hat{\mathbf{\varphi }}$ for $k>0$ and $-\hat{\mathbf{\varphi }}$ for $k<0$; the velocity is zero for $k=0$. Each admissible trajectory is a circle:The orbital angular momentum along the trajectory is $m{r}_0{v}_{\varphi }=\hslash k$. The origin or any zero-amplitude circle is excluded from this local formula. For the real degenerate superposition constructed above, the spatial quantum phase is constant on each nodal sector, so its Bohmian mechanics velocity is instead zero. Thus the circular motion is a conclusion about the separated angular-momentum eigenstate, not an arbitrary energy eigenstate.