Question

The Hamiltonian of the helium atom, under the assumption that the mass of the nucleus is much greater than that of the electrons and ignoring the spin, is of the form:

Where are the position and momentum of the electron and $Z=2$ is the atomic number of helium. Note that the first four terms are simply the sum of two Hamiltonians corresponding to a hydrogen atom for each electron; while the last term represents the interaction between both electrons.

i) Investigate the experimental energy of the base state of the helium atom.

ii) Making use of the variational method, and using as a test function the product of two base states of the hydrogen atom (one for each electron):

Calculate an approximation to the energy of the base state of the helium atom, considering Z as a variable and minimizing it.

1T2 Vi,p;) 4πε0 73-Z (ri + r2) егр πα do

Answer 1

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Using hydrogenic wave function we use the expectation values for kinetic and potential energy operators and also for the expectation value for the intrection energy between the electrons.