OPTIMAL WAVE FUNCTION FOR A HYDROGEN-LIKE ATOM

In several contexts we encounter the problem of the mean value of the Hamiltonian for a hydrogen-like atom (the a.u. are used throughout) 

OPTIMAL WAVE FUNCTION FOR A HYDROGEN-LIKE ATOM 

We may want to use the variational method for finding the ground-state wave function. In this method we minimize the mean value of the Hamiltonian with respect to parameters in the variational function We may treat c as such a parameter. Hence, minimizing e we force = 0, and therefore Copt = Z. Note that in this particular case  

The quantity — = = 2. For c = Copt = Zv/e have what is cahed the virial theorem 

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H. OPTIMAL WAVE FUNCTION FOR A HYDROGEN-LIKE ATOM... 

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