Solved Problems In Quantum Mechanics

Solved Problems In Quantum Mechanics-90
Now, collaborators at the Department of Energy's Lawrence Berkeley National Laboratory, Lawrence Livermore National Laboratory, and the University of California at Davis 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.They report their findings in the 24 December, 1999, issue of Science magazine.

Now, collaborators at the Department of Energy's Lawrence Berkeley National Laboratory, Lawrence Livermore National Laboratory, and the University of California at Davis 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.They report their findings in the 24 December, 1999, issue of Science magazine.(Because electrons are identical, there is no way to distinguish between the initially bound and initially free electron). Mc Curdy appears in Science magazine, 24 December 1999.

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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.

"An exact first-principles solution of the wave function for the hydrogen atom was vital to establishing the new quantum theory in the 1920s," says Rescigno.

"But even today, for systems with three or more charged particles, no analytic solutions exist"--that is, there are no explicit solutions to the Schrödinger equation for such systems.

Mathematically, they've come up with incredibly artful dodges, and some of them even seem to work." Earlier this year, however, in the Proceedings of the Royal Society, Colm T.

Whelan of Cambridge University and his colleagues published their conclusion that all such approximations perform inconsistently and that those few cases which appear to yield good agreement with experiment "are largely fortuitous." By contrast, the method developed by Mc Curdy 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.

Moreover, says Mc Curdy 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." But, Rescigno notes, "this obviously hasn't stopped people from working with plasmas and other ionization phenomena.

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