Helium atom

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. σ 0 0 = 1 2 ( 0 1 − 1 0 ) = 1 2 ( ↑↓ − ↓↑ ) {displaystyle {oldsymbol {sigma }}_{0}^{0}={frac {1}{sqrt {2}}}{egin{pmatrix}0&1\-1&0end{pmatrix}}={frac {1}{sqrt {2}}}(uparrow downarrow -downarrow uparrow )} σ 0 1 = 1 2 ( 0 1 1 0 ) = 1 2 ( ↑↓ + ↓↑ ) ; {displaystyle {oldsymbol {sigma }}_{0}^{1}={frac {1}{sqrt {2}}}{egin{pmatrix}0&1\1&0end{pmatrix}}={frac {1}{sqrt {2}}}(uparrow downarrow +downarrow uparrow );;} σ 1 1 = ( 1 0 0 0 ) = ↑↑ ; {displaystyle {oldsymbol {sigma }}_{1}^{1}={egin{pmatrix}1&0\0&0end{pmatrix}}=;uparrow uparrow ;;} σ − 1 1 = ( 0 0 0 1 ) = ↓↓ . {displaystyle {oldsymbol {sigma }}_{-1}^{1}={egin{pmatrix}0&0\0&1end{pmatrix}}=;downarrow downarrow ;.} σ x = 1 2 ( 1 0 0 − 1 ) {displaystyle sigma _{x}={frac {1}{sqrt {2}}}{egin{pmatrix}1&0\0&-1end{pmatrix}}} , σ y = i 2 ( 1 0 0 1 ) {displaystyle sigma _{y}={frac {i}{sqrt {2}}}{egin{pmatrix}1&0\0&1end{pmatrix}}} and σ z = 1 2 ( 0 1 1 0 ) {displaystyle sigma _{z}={frac {1}{sqrt {2}}}{egin{pmatrix}0&1\1&0end{pmatrix}}} . ϕ x = 1 2 ( φ a ( r → 1 ) φ b ( r → 2 ) ± φ a ( r → 2 ) φ b ( r → 1 ) ) {displaystyle phi _{x}={frac {1}{sqrt {2}}}(varphi _{a}({vec {r}}_{1})varphi _{b}({vec {r}}_{2})pm varphi _{a}({vec {r}}_{2})varphi _{b}({vec {r}}_{1}))} 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 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 {vec {r}}_{1}} and r → 2 {displaystyle {vec {r}}_{2}} are the electron-nucleus distance vectors and r 12 = | r 1 → − r 2 → | {displaystyle r_{12}=|{vec {r_{1}}}-{vec {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 { extstyle {frac {hbar ^{2}}{M}} abla _{r_{1}}cdot abla _{r_{2}}} disappears. 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 ( r → 1 , r → 2 ) {displaystyle ({vec {r}}_{1},,{vec {r}}_{2})} . In this approximation (Pauli approximation) the wave function is a second order spinor with 4 components ψ i j ( r → 1 , r → 2 ) {displaystyle psi _{ij}({vec {r}}_{1},,{vec {r}}_{2})} , where the indices i , j = ↑ , ↓ {displaystyle i,j=,uparrow ,downarrow } describe the spin projection of both electrons (z-direction up or down) in some coordinate system. It has to obey the usual normalization condition ∑ i j ∫ d r → 1 d r → 2 | ψ i j | 2 = 1 {displaystyle sum _{ij}int d{vec {r}}_{1}d{vec {r}}_{2}|psi _{ij}|^{2}=1} . This general spinor can be written as 2x2 matrix ψ = ( ψ ↑↑ ψ ↑↓ ψ ↓↑ ψ ↓↓ ) {displaystyle {oldsymbol {psi }}={egin{pmatrix}psi _{uparrow uparrow }&psi _{uparrow downarrow }\psi _{downarrow uparrow }&psi _{downarrow downarrow }end{pmatrix}}} and consequently also as linear combination of any given basis of four orthogonal (in the vector-space of 2x2 matrices) constant matrices σ k i {displaystyle {oldsymbol {sigma }}_{k}^{i}} with scalar function coefficients and three symmetric matrices (with total spin S = 1 {displaystyle S=1} , 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 ( σ x , σ y , σ z ) {displaystyle (sigma _{x},sigma _{y},sigma _{z})} ,with the three components Since all spin interaction terms between the four components of ψ {displaystyle {oldsymbol {psi }}} in the above (scalar) Hamiltonian are neglected (e.g. an external magnetic field, or relativistic effects, like angular momentum coupling), the four Schrödinger equations can be solved independently. The spin here only comes into play through the Pauli exclusion principle, which for fermions (like electrons) requires antisymmetry under simultaneous exchange of spin and coordinates