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NOT a duplictae, see EDIT below
It is common knowledge that the Schrödinger equation can be solved exactly only for the simplest of systems - such the so-called toy models (particle in a box, etc), and the Hydrogen atoms; and not for relatively complex systems, such as the Helium atom and other multielectron systems.
I have been trying to get to the reason for this for a long time, and it seems that it has something to do with one or more of these -
- Correlation effects between the electrons as in the electrons trying to occupy positions opposite to each other with the nucleus between them.
- Some kind of quantum corellation due to entanglement.
- Appearance of inseparable 'cross-terms' in the expression for the Hamiltonian.
- As in 3 above, we can't figure out to model the fact that we don't know (from our theory) the total energy of the system (while we are making the model), and the kinetic and poten tial energies depend upon each other, so we can't actually find any of these.
Some references for the above-
However, complications can arise in the many-body problem. Since the potential energy depends on the spatial arrangement of the particles, the kinetic energy will also depend on the spatial configuration to conserve energy. The motion due to any one particle will vary due to the motion of all the other particles in the system ... For N interacting particles, i.e. particles which interact mutually and constitute a many-body situation, the potential energy function V is not simply a sum of the separate potentials (and certainly not a product, as this is dimensionally incorrect). The potential energy function can only be written as above: a function of all the spatial positions of each particle.
Unfortunately, the Coulomb repulsion terms make it impossible to find an exact solution to the Schrödinger equation for many-electron atoms and molecules even if there are only two electrons. The most basic approximations to the exact solutions involve writing a multi-electron wavefunction as a simple product of single-electron wavefunctions, and obtaining the energy of the atom in the state described by that wavefunction as the sum of the energies of the one-electron components.
Very often, only numerical solutions to the Schrödinger equation can be found for a given physical system and its associated potential energy. However, there exists a subset of physical systems for which the form of the eigenfunctions and their associated energies can be found.
The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of a large number of interacting particles ... In such a quantum system, the repeated interactions between particles create quantum correlations, or entanglement. As a consequence, the wave function of the system is a complicated object holding a large amount of information, which usually makes exact or analytical calculations impractical or even impossible.
I have been trying to figure out these things for years, but inspite of all the available information, I still have a question to which I cannot find a answer -
What is the fundamental reason behind our inability to solve the Schrödinger equation for multielectron atoms - Is it actually imposibble to solve it (any proof?) or are we just not good enough at maths? Supposing that we kept trying g ti solve it, can someone clarify whether a solution could be found in principle?
To clarify my question, existing answers to older questions already state that the problem has a very high computational complexity, so that finding a solution is extremly unlikely. My question is different. I already know from the linked resources (and others) that the equation is effectively unsolvable for us.
What I am asking for is a clarification whether we have any reason at all to believe in the existance of a solution in any form, and the reasons for the same ?
I know that there are other similar problems such as the 3 body problem in Classical mechancics and it would be great if the answer touches upon them too.
The reasons I listed are not exhaustive at any rate, they are just what came off the top of my head as some of the variations of "The math is too tough." that i have seen over the years.