the same partial differential equation. The main difference is that in ordinary quantum mechanics, the Schrödinger equation is connected to reality by the Born postulate, which states that the probability density of the particle's position is given by Pilot wave theory considers the guidance equation to be the fundamental law, and sees the Born rule as a derived concept. The second equation is a modified Hamilton–Jacobi equation for the action : where is the quantum potential defined by If we choose to neglect , our equation is reduced to the Hamilton–Jacobi equation of a classical point particle. So, the quantum potential is responsible for all the mysterious effects of quantum mechanics. One can also combine the modified Hamilton–Jacobi equation with the guidance equation to derive a quasi-Newtonian equation of motion where the hydrodynamic time derivative is defined as Mathematical formulation for multiple particles The Schrödinger equation for the many-body wave function is given by The complex wave function can be represented as: The pilot wave guides the motion of the particles. The guidance equation for the jth particle is: The velocity of the jth particle explicitly depends on the positions of the other particles. This means that the theory is nonlocal. Empty wave function Lucien Hardy and John Stewart Bell have emphasized that in the de Broglie–Bohm picture of quantum mechanics there can exist empty waves, represented by wave functions propagating in space and time but not carrying energy or momentum, and not associated with a particle. The same concept was called ghost waves (or "Gespensterfelder", ghost fields) by Albert Einstein. The empty wave function notion has been discussed controversially. In contrast, the many-worlds interpretation of quantum mechanics does not call for empty wave functions. See also Hydrodynamic quantum analogues Free-fall atomic model Quantum potential Notes References External links "Pilot-wave hydrodynamics" Bush, J.W.M, 2014, Annu. Rev. Fluid Mech., 49, 269–292. "Quantum mechanics writ large", Bush, J.W.M, 2010. "Pilot waves, Bohmian metaphysics, and the foundations of quantum mechanics" , lecture course on pilot wave theory by Mike Towler, Cambridge University (2009). "Hydrodynamic quantum analogues" Research on hydrodynamic quantum analogues and hydrodynamic pilot-wave theory, by John Bush (MIT) and coworkers. More complete HTML encyclopedic page about the subject. Klaus von Bloh’s Bohmian mechanics demonstrations in: Wolfram Demonstrations Project Hidden variable theory Interpretations of quantum mechanics Quantum measurement