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Concept
Helium atom
Summary
A helium atom is an atom of the chemical element helium. Helium is composed of two electrons bound by the electromagnetic force to a nucleus containing two protons along with either one or two neutrons, depending on the isotope, held together by the strong force. Unlike for hydrogen, a closed-form solution to the Schrödinger equation for the helium atom has not been found. However, various approximations, such as the Hartree–Fock method, can be used to estimate the ground state energy and wavefunction of the atom.
The quantum mechanical description of the helium atom is of special interest, because it is the simplest multi-electron system and can be used to understand the concept of quantum entanglement. The Hamiltonian of helium, considered as a three-body system of two electrons and a nucleus and after separating out the centre-of-mass motion, can be written as
where is the reduced mass of an electron with respect to the nucleus, and are the electron-nucleus distance vectors and . The nuclear charge, is 2 for helium. In the approximation of an infinitely heavy nucleus, we have and the mass polarization term disappears, so that in atomic units the Hamiltonian simplifies to
It is important to note, that it operates not in normal space, but in a 6-dimensional configuration space . In this approximation (Pauli approximation) the wave function is a second order spinor with 4 components , where the indices describe the spin projection of both electrons (z-direction up or down) in some coordinate system. It has to obey the usual normalization condition . This general spinor can be written as 2×2 matrix and consequently also as linear combination of any given basis of four orthogonal (in the vector-space of 2×2 matrices) constant matrices with scalar function coefficients as . A convenient basis consists of one anti-symmetric matrix (with total spin , corresponding to a singlet state)
and three symmetric matrices (with total spin , corresponding to a triplet state)
It is easy to show, that the singlet state is invariant under all rotations (a scalar entity), while the triplet can be mapped to an ordinary space vector , with the three components
Since all spin interaction terms between the four components of in the above (scalar) Hamiltonian are neglected (e.
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