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 two neutrons, depending on the isotope, held together by the strong force. Unlike for the hydrogen atom, 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. Historically, the first attempt to obtain the helium spectrum from quantum mechanics was done by Albrecht Unsöld in 1927. Egil Hylleraas obtained an accurate approximation in 1929. Its success was considered to be one of the earliest signs of validity of Schrödinger's wave mechanics.
Introduction
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
H ( r 1 , r 2 ) = ∑ i = 1 , 2 ( − ℏ 2 2 μ ∇ r i 2 − Z e 2 4 π ε 0 r i ) − ℏ 2 M ∇ r 1 ⋅ ∇ r 2 + e 2 4 π ε 0 r 12 {\displaystyle H(\mathbf {r} _{1},\,\mathbf {r} _{2})=\sum _{i=1,2}\left(-{\frac {\hbar ^{2}}{2\mu }}\nabla _{r_{i}}^{2}-{\frac {Ze^{2}}{4\pi \varepsilon _{0}r_{i}}}\right)-{\frac {\hbar ^{2}}{M}}\nabla _{r_{1}}\cdot \nabla _{r_{2}}+{\frac {e^{2}}{4\pi \varepsilon _{0}r_{12}}}}
where μ = m M m + M {\displaystyle \mu ={\frac {mM}{m+M}}} is the reduced mass of an electron with respect to the nucleus, r 1 {\displaystyle \mathbf {r} _{1}} and r 2 {\displaystyle \mathbf {r} _{2}} are the electron-nucleus distance vectors and r 12 = | r 1 − r 2 | {\displaystyle r_{12}=|\mathbf {r} _{1}-\mathbf {r} _{2}|} . It operates in a 6-dimensional configuration space ( r 1 , r 2 ) {\displaystyle (\mathbf {r} _{1},\,\mathbf {r} _{2})} . The nuclear charge, Z {\displaystyle Z} is 2 for helium. In the approximation of an infinitely heavy nucleus, M = ∞ {\displaystyle M=\infty } we have μ = m {\displaystyle \mu =m} and the mass polarization term ℏ 2 M ∇ r 1 ⋅ ∇ r 2 {\textstyle {\frac {\hbar ^{2}}{M}}\nabla _{r_{1}}\cdot \nabla _{r_{2}}} disappears, so that in operator language, the Hamiltonian simplifies to:
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