arXiv:0808.0290v3 [quant-ph] 27 Dec 2008 De Broglie-Bohm Guidance Equations for Arbitrary Hamiltonians Ward Struyve Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada and Instituut voor Theoretische Fysica, K.U.Leuven Celestijnenlaan 200D, B-3001 Leuven, Belgium. 1 E–mail: Ward.Struyve@fys.kuleuven.be Antony Valentini Theoretical Physics Group, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, United Kingdom. E–mail: a.valentini@imperial.ac.uk Abstract In a pilot-wave theory, an individual closed system is described by a wavefunction ψ(q) and configuration q. The evolution of the wavefunction and configuration are respectively determined by the Schr¨ odinger and guidance equations. The guidance equation states that the velocity field for the configuration is given by the quantum current divided by the density |ψ(q)| 2 . We present the currents and associated guidance equations for any Hamiltonian given by a differential operator. These are derived directly from the Schr¨ odinger equation, and also as Noether currents arising from a global phase symmetry associated with the wavefunction in configuration space. 1 Introduction In the pilot-wave theory of de Broglie and Bohm [1–7], an individual closed system of nonrelativistic particles is described by a wavefunction ψ(x 1 ,..., x N ,t), which satisfies the nonrelativistic Schr¨odinger equation i∂ t ψ(x 1 ,..., x N ,t)=  − N  k=1  2 2m k ∇ 2 k + V (x 1 ,..., x N )  ψ(x 1 ,..., x N ,t), (1) 1 Present address. Currently Postdoctoral Fellow FWO. 1