Finally, we note that a more rigorous derivation of the above quantum trajectory expressions is also possible, based on a Lagrangian/action extremization procedure. In this approach, the Lagrangian is equal to the usual classical one, with the addition of a quantum contribution-i.e., L = T -V -L Q , where T is the kinetic energy. It is important to note that a gauge freedom exists in the definition of L Q . Thus in general, L Q need not be Qalthough this particular choice is allowed, and has the advantage that the resultant action, obtained by integrating the Lagrangian over time, or equivalently