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I have seen many ground state wave functions.

From where are they derived? How can one calculate them? Where can one find a list of all ground state wavefunctions discovered?

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I'm not sure on what level you're asking this, so ignore the bits of this answer that don't make sense.

Wavefunctions, ground state and otherwise, aren't designed. For some system, e.g. a hydrogen atom, you write down the Schrodinger equation and solve it. The solutions are the wavefunctions. All you need to know to write down the Schrodinger equation is the potential energy of your system e.g. for the hydrogen atom this is the electrostatic attraction between the electron and proton.

However life is rarely this simple. For example the Schrodinger equation is only an approximation and doesn't take relativistic effects into account. For that you need the Dirac equation and the extra complexity of this makes exact wavefunctions impossible to write down. Instead we start with the solutions of the Schrodinger equation and include the relativistic effects as perturbations.

Even sticking to the Schrodinger equation, all but simple systems are impossible to solve exactly and we have to use approximate methods and wheel out our computers. Even the H$_2$ molecule doesn't have a simple exact solution.

There is no list of all ground state wavefunctions discovered. Any such list would be enormously long. Did you have any particular examples in mind?

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in web, there is paper called orthogonal polynomial from hermitian matrices, every polynomial have a ground state wave function, i am not sure from B and D to ground state or from ground state to B and D – M-Askman Feb 14 '12 at 11:43
if you read, do you know where do small r come from in the part determination of B and D and eta, there are many possible solutions, does maple can derive these? – M-Askman Feb 14 '12 at 11:58
-1: sorry, but everything here is wrong. Ground state wavefunctions are almost never found by doing anything remotely like solving the Schrodinger equation, except for in textbooks. The Feynman description of liquid He4 begins with an ansatz for the ground state wavefunction, which is found by general principles of positivity and doing variational approximation, as is the He ground state. Similar ansatzes, even without the variation justification are common. Further, the Dirac equation is solved exactly for the H atom, and the analysis is no more difficult than the Schrodinger equation. – Ron Maimon Feb 14 '12 at 12:52
Do you mean ? If so, I'd have to sit down and work through the paper before I could comment, and time pressures make this unlikely. In any case, you obviously know more about this area than i do! – John Rennie Feb 14 '12 at 12:53
Stupid me, I forgot Feynman, Karabali, and Nair's ansatz for the ground state of the 2+1 dimensional gauge theory! This is one of the most remarkable applications of "guess the ground state". There is also the exact ground state of noninteracting QED, which is either Schwinger or folklore (or both!) These the two high-energy examples. You might, by a stretch, include Nambu and Higgs stuff, but I think if the ground state is just a classical VEV, it's not too hard to guess, so it wouldn't be a good example. – Ron Maimon Feb 14 '12 at 13:05

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