for dealing with the node problem, based on a bipolar decomposition of the wavefunction. The idea is to partition the wavefunction into two (or in principle, more) component functions, i.e. $\Psi = \Psi_1
+ \Psi_2$. One then applies QTM propagation separately to $\Psi_1$ and $\Psi_2$, which can be linearly superposed to generate $\Psi$ itself at any desired later time. In essence, this works because the  equation itself is linear, but the equivalent Bohmian mechanical, or quantum Hamilton’s equations of motion (QHEM) are not.[@poirier04bohmI] In principle, therefore, one may improve the numerical performance of QTM calculations simply by judiciously dividing up the initial wavepacket into pieces.

Although bipolar decompositions have been around for quite some time,[@floyd94; @brown02] their use as a tool for circumventing the node problem for QTM calculations is quite recent. Two promising new exact methods that seek to accomplish this are the so-called “counter-propagating wave” method (CPWM),[@poirier04bohmI] and the “covering function” method (CFM).[@babyuk04] In the CPWM, the bipolar decomposition is chosen to correspond to the semiclassical WKB approximation,[@poirier04bohmI] for which all of the hydrodynamic field functions are smooth and classical-like, and the component wavefunctions are node-free. Interference is achieved naturally, via the superposition of left- and right-traveling (i.e. 