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Working ScienceSupercomputers Help Solve a 50-Year-Old Quantum Physics Problemby Paul Preuss, Lawrence Berkeley National Laboratory For over half a century, theorists have tried and failed to provide a complete solution to the fundamental phenomenon of scattering (collisional breakup) in a quantum system of three charged particles. Such interactions are everywhere; ionization by electron impact, for example, is responsible for the glow of fluorescent lights and for the ion beams that engrave silicon chips. Now, scientists have used supercomputers to obtain a complete solution of the ionization of a hydrogen atom by collision with an electron, the simplest nontrivial example of the problem's last unsolved component.
Bill McCurdy, Associate Lab Director for Computing Sciences at Lawrence Berkeley National Laboratory, and Thomas Rescigno, staff physicist at Lawrence Livermore National Laboratory, and their collaborators Mark Baertschy, UC Davis, and William Isaacs, Berkeley Lab, used the SGI/Cray T3E at the National Energy Research Scientific Computing Center (NERSC) at LBNL and the IBM Blue Pacific computer at LLNL for their solution of the three-charged-body scattering problem.
This breakthrough, reported in detail in the December 24th issue of Science magazine, employs a mathematical transformation of the Schrödinger wave equation that makes it possible to treat the outgoing particles not as if their wave functions extend to infinityas they must be treated conventionallybut instead as if they simply vanish at large distances from the nucleus.
"Using this transformation we compute accurate solutions of the quantum-mechanical wave function of the outgoing particles, and from these solutions we extract all the dynamical information of the interaction," says McCurdy. Rescigno points out that "it wasn't until the late 1950s, using early computers, that accurate solutions were obtained even for the bound states of helium," an atom with two electrons closely orbiting the nucleus. "Scattering problems are a lot more difficult." As with all scattering problems, the electron-ionization of a hydrogen atom begins with a particle incoming at a certain velocity. This electron interacts with the atom, and two electrons fly out at an angle to each other, leaving the proton behind. The likelihood that a specific incoming state will result in an outgoing state with the particles at specific angles and energies is the "cross section" for that result. Cross sections of quantum-mechanical processes are derived from the system's wave function, solutions of the Schrödinger equation that yield probabilities of finding the entities involved in a certain state. In scattering problems, wave functions are not localized but extend over all space. Moreover, says McCurdy of the electromagnetic forces between charged particles, "Coulomb interactions are forever." These infinities make it impossible to define the final state of scattering exactly. "The form of the wave function where all three particles are widely separated is so intractable that no computer-aided numerical approach has been able to incorporate it explicitly." The method developed by McCurdy and Rescigno and their co-authors allows the calculation of a highly accurate wave function for the outgoing state that can be interrogated for details of the incoming state and interaction in the same way an experimenter would interrogate a physical system. They begin with a transformation of the Schrödinger equation called "exterior complex scaling," invented by Caltech's Barry Simon in 1979 to prove formal theorems in scattering theory. The transformation leaves the solution unchanged in regions which correspond to physical reality, producing the correct outgoing waveform based upon the angular separation and distances of two electrons far from the nucleus. 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. (Because electrons are identical, there is no way to distinguish between the initially bound and initially free electron). Comparison with real scattering experiments, such as those recently published by J. Röder et al, who scattered incoming 17.6 electron-volt electrons from hydrogen atoms and 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.
"Even if the specific methods have changed, quantum chemistry was founded when the helium atom with two bound electrons was solvedit showed that these problems were in principle solvable," McCurdy says. "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." This research is supported by the Office of Advanced Scientific Computing Research.
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