A theory of active sonar (or radar) pulse wave form design, for optimal target detection in cluttered environments, is presented. The received target signal is maximized via a cost function L that incorporates both the signal-to-noise ratio and a generalization of the Heisenberg uncertainty principle, which is used to balance bandwidth (or range resolution) against signal gain. The optimal pulse wave form is the ground state solution to a one-dimensional Schrödinger-type equation in frequency space, with an effective potential energy that tends to concentrate pulse energy in frequency bands where the target reflectivity dominates the clutter reflectivity.