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 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.
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.
The Hamiltonian of the helium atom, under the assumption that the mass of the nucleus is much greater than that of the e...
10.13. (a) Consider the helium atom to be a fixed point nucleus (charge 2e) with two spin-half fermion elec trons. What is the degeneracy of its ground state? (That is, how many independent states of the whole atom have the ground state energy?) (b) Suppose instead that the electron was a spin-half boson. (It is an experimental fact that all spin-half particles are fermions, but there is nothing to stop us imagining a spin-half boson.) What then would be the...
hope you can answer all the questions and don't copy the others answer i'll rate it Ex.7 Consider an atom with Z electrons. Suppose that we want to model this atom without considering the Pauli exclusion principle. It is then reasonable to assume that all electrons are in the same single particle state. In this single particle state, the average distance between the neucleus and the electron is R. Furthermore, by using the uncertain principle, p = Ap = ħ/R,...
A system consists of two particles of mass mi and m2 interacting with an interaction potential V(r) that depends only on the relative distancer- Iri-r2l between the particles, where r- (ri,/i,21) and r2 22,ひ2,22 are the coordinates of the two particles in three dimensions (3D) (a) /3 pointsl Show that for such an interaction potential, the Hamiltonian of the system H- am▽ri _ 2m2 ▽22 + V(r) can be, put in the form 2M where ▽ and ▽ are the...