atomic physics Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. Atomic physics typically refers to the study of atomic structure and the interaction between atoms. It is primarily concerned wit ...

, a two-electron atom or helium-like ion is a

quantum mechanical Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...

system consisting of one

nucleus Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to: *Atomic nucleus, the very dense central region of an atom *Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA Nucle ...

with a

charge Charge or charged may refer to: Arts, entertainment, and media Films * ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * ''Charge!!'', an album by The Aqua ...

of ''Z e'' and just two

electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...

s. This is the first case of many-electron systems where the

Pauli exclusion principle In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...

plays a central role. It is an example of a

three-body problem In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...

. The first few two-electron atoms are:

Schrödinger equation

The Schrödinger equation for any two-electron system, such as the neutral

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 ...

(He, ''Z'' = 2), the negative

Hydrogen Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic, an ...

ion An ion () is an atom or molecule with a net electrical charge. The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by conven ...

(H⁻, ''Z'' = 1), or the positive

Lithium Lithium (from el, λίθος, lithos, lit=stone) is a chemical element with the symbol Li and atomic number 3. It is a soft, silvery-white alkali metal. Under standard conditions, it is the least dense metal and the least dense solid el ...

ion (Li⁺, ''Z'' = 3) is:Physics of Atoms and Molecules, B.H. Bransden, C.J.Joachain, Longman, 1983, For a more rigorous mathematical derivation of Schrödinger's equation, see also.

\psi

where r₁ is the position of one electron (''r''₁ = , r₁, is its magnitude), r₂ is the position of the other electron (''r''₂ = , r₂, is the magnitude), ''r''₁₂ = , r₁₂, is the magnitude of the separation between them given by

, \mathbf_,  = , \mathbf_2 - \mathbf_1 ,

''μ'' is the two-body reduced mass of an electron with respect to the nucleus of mass ''M''

\mu = \frac

and ''Z'' is the

atomic number The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of an atomic nucleus. For ordinary nuclei, this is equal to the proton number (''n''p) or the number of protons found in the nucleus of every ...

for the element (not a

quantum number In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be kno ...

). The cross-term of two Laplacians

\frac\nabla_1\cdot\nabla_2

is known as the ''mass polarization term'', which arises due to the motion of

atomic nuclei The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. After the discovery of the neutron ...

. The wavefunction is a function of the two electron's positions:

\psi = \psi(\mathbf_1,\mathbf_2)

There is no closed form solution for this equation.

Spectrum

The optical spectrum of the two electron atom has two systems of lines. A para system of single lines, and an ortho system of triplets (closely spaced group of three lines). The energy levels in the atom for the single lines are indicated by ¹S₀ ¹P₁ ¹D₂ ¹F₃ etc., and for the triplets, some energy levels are split: ³S₁ ³P₂ ³P₁ ³P₀ ³D₃ ³D₂ ³D₁ ³F₄ ³F₃ ³F₂. Alkaline earths and

mercury Mercury commonly refers to: * Mercury (planet), the nearest planet to the Sun * Mercury (element), a metallic chemical element with the symbol Hg * Mercury (mythology), a Roman god Mercury or The Mercury may also refer to: Companies * Merc ...

also have spectra with similar features, due to the two outer valence electrons.

References

{{Reflist Atoms Quantum models

Schrödinger equation

Spectrum

See also

References