Mathematics and Computation: Quantum Pinball  
What makes scattering problems involving charged particles even more difficult is that "Coulomb interactions are forever," says Bill McCurdy, Berkeley Lab�s Associate Laboratory Director for Computing Sciences. As far as the mathematics of the Schr�dinger equation is concerned, the electromagnetic force between charged particles acts over infinitely long distances. These infinities make it essentially impossible to find an exact formula for the final state of scattering. McCurdy says, "The form of the wave function where all three particles are widely separated is so intractable that no computeraided numerical approach has been able to incorporate it explicitly." McCurdy and his longtime collaborator Thomas Rescigno, a staff physicist at Livermore Lab, working with doctoral candidate Mark Baertschy of UC Davis and postdoctoral fellow William Isaacs at Berkeley Lab, tackled the threechargedbody scattering problem using the Cray T3E supercomputer at the National Energy Research Scientific Computing Center (NERSC), located at Berkeley Lab, and the IBM Blue Pacific computer at Livermore Lab.
The method developed by McCurdy and Rescigno and their colleagues, on the other hand, allows the calculation of a highly accurate wave function for the outgoing state that can be interrogated for details about the collision dynamics in the same way an experimenter would interrogate a physical system. They begin with a mathematical transformation of the Schr�dinger equation called "exterior complex scaling," invented by Caltech�s Barry Simon in 1979, which makes it possible to treat the outgoing particles not as if their wave functions extend to infinity—as they must be treated conventionally— but instead as if they simply vanish at large distances from the nucleus. "Using this transformation we can compute accurate solutions of the quantummechanical wave function of the outgoing particles," McCurdy says, because in regions that correspond to physical reality, the transformation gives the correct outgoing waveform. "From these solutions we extract all the dynamical information of the interaction."����� Once the wave function has been calculated, it must be analyzed by computing the "quantum mechanical flux," a means of finding the distribution of probability densities that dates from the 1920s. This computationally intensive process can yield the probability of producing electrons at specific energies and directions from the ionized atom.
Comparison with real scattering experiments, such as those recently published by J. R�der et al, who scattered electrons incoming at precise energies from hydrogen atoms, then carefully measured the angles and energies of the outgoing electrons, prove the accuracy of the new method. The experimental data points match the graph of the cross sections calculated by Rescigno, Baertschy, Isaacs, and McCurdy with astonishing exactitude. "Quantum chemistry was founded when the helium atom with two bound electrons was solved in the 1950s," McCurdy says. Even if specific methods have changed, "those calculations showed that the molecular problems of quantum chemistry were in principle solvable. What we have done is analogous. The details of our method probably won�t survive, but we�ve taken a big step toward treating ionizing collisions of electrons with more complicated atoms and molecules." While noting the irony, the researchers find it encouraging that the same supercomputing power and sophisticated software tools built to investigate the complexity of increasingly larger systems could be used—indeed, were essential— "to answer a basic physics question for one of the simplest systems imaginable in physics and chemistry."� — Paul Preuss 

