Trial Wavefunctions and their Associated Energies

In Chapter 7 we saw that it is not possible to obtain exact solutions of the Schrodinger equation for many-electron atoms, even within the one-electron approximation, and the same applies to molecules. For these systems it is necessary to use approximate solutions, which are based on chemical insight and chosen for mathematical convenience. 

we consider an atom or molecule which has a set of wavefunctions, if, given by the equation  

For a given Hamiltonian operator there will be an infinite number of solutions to this equation, each indicated by a different value of the index . We wish to find the ground state wavefunction, y/j, which has an energy Normally, equation (8.1) cannot be solved analytically and the wavefunctions that satisfy the equation are unknown. Under these circumstance it is necessary to formulate a trial wavefunction, which is expected to be a good approximation to the true ground state wave-function. 

One problem with this trial wavefunction is that it will not be an eigenfunction of the Hamiltonian operator for the atom or molecule. Thus  

It is evident that the energy associated with the trial wavefunction is not a constant of the motion. Any attempt to measure the energy of such a hypothetical system would force it into one of the quantum states represented by equation (8.1), and the energy measured could be any of the values It follows that there must be an uncertainty in the 

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