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Usually we have a Hamiltonian H and we construct a state |ψ> using some trial states. Then we minimize E = <ψ|H|ψ> and get an upper bound for the ground state energy. In many cases the state |ψ> is then used to calculate other quantum mechanical observables O. E.g. in the non-relativistic quark model one is interested in the magnetic moment of the nucleon and things like that.

The interesting fact is that in order to do that one must assume that we have not only

*weak convergence*of <ψ|H|ψ> to the true ground state energy E° but instead

*strong convergence*of |ψ> to the true ground state |ψ°> .

And this is my question: under which conditions does this strong convergence follow from the variational principle?