Schrodinger problem, three-body

The quantum treatment of this molecule appears to be a straightforward extension of that for the hydrogen atom (see Section 5.1 for the solutions). One more proton has been added, and symmetry of the system has changed from spherical to cylindrical. But this apparently small change has converted the system into the so-called three-body problem, for which there is no general solution, even in classical physics. Fortunately, there is an excellent approximation that treats the nuclear and electronic motions almost as if they were independent. This method allows us to solve the Schrodinger equation exactly for the electron in Hj and provides an approximate solution for the protons as if they were almost independent. 

The term "many-electron atoms includes all atoms and atomic ions with more than one extra-nuclear electron. Helium is the simplest many-electron atom. Even in this case the Schrodinger equation cannot be solved analytically. The helium atom, with two electrons and one nucleus, is an example of the Three-body Problem, the equation of motion of which remains unsolved also in Classical Mechanics. The difficulty is that the motion of every particle, in a many-body problem, is coupled to the motion of all the other particles. 

Harmonium may be defined as a quantum three-body problem described by the Schrodinger equation with harmonic interactions between particles 1—3 and 2 — 3 and the Coulombic interaction between particles 1—2. The... 

The trimer states, which in most cases can be called Efimov trimers, are interesting objects. Their existence can be seen from the Born-Oppenheimer picture for two heavy atoms and one light atom in the gerade state. Within the Born-Oppenheimer approach the three-body problem reduces to the calculation of the relative motion of the heavy atoms in the effective potential created by the light atom. For the light atom in the gerade state, this potential is + (/ ), found in the previous subsection. The Schrodinger equation for the wavefunction of the relative motion of the heavy atoms, Xv(R), reads... 

For pedagogical reasons, we shall start this review with very elementary reminders of the two-body problem, followed by some mathematical developments on the two-body Schrodinger equation, which will be useful for their three-body analogues discussed in chapters 8,9 and 10. The three-body problem... 

To date, there is no known analytic solution to the second-order differential Schrodinger equation for the helium atom. This does not mean that there is no solution, or that wavefunctions do not exist. It simply means that we know of no mathematical function that satisfies the differential equation. In fact, for atoms and molecules that have more than one electron, the lack of separability leads directly to the fact that there are no known analytical solutions to any atom larger than hydrogen. Again, this does not mean that the wavefunctions do not exist. It simply means that we must use other methods to understand the behavior of the electrons in such systems. (It has been proven mathematically that there is no analytic solution to the so-called three-body problem, as the He atom can be described. Therefore, we must approach multielectron systems differently.)... 

We have already noted that the difficulties of the many-body problem were overcome by Kohn and Scham, who showed that the task of finding the right electron density can be expressed in a way that involves solving of a set of equations in which each equation only involves a sin e electron. The main difference is that the Kohn-Sham equations are missing the summations that appear inside the Hamiltonian (8.4) of the full Schrodinger equation. This is because the solutions of the Kohn-Sham equations are single-electron eigenfimctions that depend only on three spatial variables and spin. 

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