Hydrogen-like atom

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Hydrogen-like atom

A hydrogen-like atom is an atom

Atom

|-! bgcolor=gray | Properties|-||}The atom is a basic unit of matter consisting of a dense, central atomic nucleus surrounded by a electron cloud of electric charge electrons....
with one electron

Electron

The electron is a subatomic particle that carries a negative electric charge. It has elementary particle and is believed to be a point particle....
and thus is isoelectronic with hydrogen

Hydrogen

Hydrogen is the chemical element with atomic number 1. It is represented by the chemical symbol H. At standard temperature and pressure, hydrogen is a colorless, odorless, nonmetallic, tasteless, highly combustion and explosive Diatomic molecule gas with the molecular formula H2....
. Except for the hydrogen

Hydrogen

Atomic number

In chemistry and physics, the atomic number is the number of protons found in the atomic nucleus of an atom. It is conventionally represented by the symbol Z....
of the atom. Examples of hydrogen-like ions are He⁺

Helium

Helium is a colorless, odorless, tasteless, non-toxic, inert monatomic chemical element that heads the noble gas group in the periodic table and whose atomic number is 2....
, Li²⁺

Lithium

Lithium is a chemical element with the symbol Li and atomic number 3. It is a soft alkali metal with a silver-white color. Under standard conditions for temperature and pressure, it is the lightest metal and the least dense solid element....
, Be³⁺

Beryllium

Beryllium is a chemical element with the symbol Be and atomic number 4.A Bivalent element, beryllium is found naturally only combined with other elements in minerals....
and B⁴⁺

Boron

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Encyclopedia

A hydrogen-like atom is an atom

Atom

Electron

The electron is a subatomic particle that carries a negative electric charge. It has elementary particle and is believed to be a point particle....
and thus is isoelectronic with hydrogen

Hydrogen

Atomic number

Helium

Helium is a colorless, odorless, tasteless, non-toxic, inert monatomic chemical element that heads the noble gas group in the periodic table and whose atomic number is 2....
, Li²⁺

Lithium

Beryllium

Beryllium is a chemical element with the symbol Be and atomic number 4.A Bivalent element, beryllium is found naturally only combined with other elements in minerals....
and B⁴⁺

Boron

Boron is a chemical element with atomic number 5 and the chemical symbol B. Boron is a trivalent metalloid element which occurs abundantly in the evaporite ores borax and ulexite....
. Because hydrogen-like atoms are two-particle systems with an interaction depending only on the distance between the two particles, their (non-relativistic) Schrödinger equation can be solved in analytic form. The solutions are one-electron functions and are referred to as hydrogen-like atomic orbitals.

Hydrogen-like atomic orbitals are eigenfunctions of the one-electron angular momentum operator l and its z component l_z. The energy eigenvalues do not depend on the corresponding quantum numbers, but solely on the principal quantum number n. Hence, a hydrogen-like atomic orbital is uniquely identified by the values of: principal quantum number

Principal quantum number

In atomic physics, the principal quantum number symbolized as n is the firstof a set of quantum numbers of an atomic orbital. The quantum number n labels the energy levels of hydrogenic atoms....
n, angular momentum quantum number

Azimuthal quantum number

The Azimuthal quantum number symbolized as l is a quantum number for an atomic orbital that determines its orbital angular momentum. The azimuthal quantum number is the second of a set of quantum numbers which describe the unique quantum state of an electron and is designated by the letter l....
l, and magnetic quantum number

Magnetic quantum number

In atomic physics, the magnetic quantum number is the third of a set of quantum numbers which describe the unique quantum state of an electron and is designated by the letter m....
m. To this must be added the two-valued spin quantum number

Spin quantum number

In atomic physics, the spin quantum number is a quantum number that parameterizes the intrinsic angular momentum of a given Elementary particle....
m_s = ±½ in application of the Aufbau principle

Aufbau principle

The Aufbau principle is used to determine the electron configuration of an atom, molecule or ion. The principle postulates a hypothetical process in which an atom is "built up" by progressively adding electrons....
. This principle restricts the allowed values of the four quantum numbers in electron configuration

Electron configuration

In atomic physics and quantum chemistry, electron configuration is the arrangement of electrons in an atom, molecule, or other physical structure....
s of more-electron atoms. In hydrogen-like atoms all degenerate orbitals of fixed n and l, l_z and s varying between certain values (see below) form an atomic shell

Electron shell

File:Periodic Table of Elements showing Electron Shells.svgAn electron shell may be crudely thought of as an orbit followed by electrons around an atom Atomic nucleus....
.

The Schrödinger equation of atoms or atomic ions with more than one electron has not been solved analytically, because of the computational difficulty imposed by the Coulomb interaction between the electrons. Numerical methods must be applied in order to obtain (approximate) wavefunctions or other properties from quantum mechanical calculations. Due to the spherical symmetry (of the Hamiltonian

Hamiltonian

Hamiltonian may refer toIn mathematics:* Hamiltonian system* Hamiltonian path, in graph theory* Hamiltonian group, in group theory* Hamiltonian ...
), the total angular momentum L of an atom is a conserved quantity. Many numerical procedures start from products of atomic orbitals that are eigenfunctions of the one-electron operators l and l_{z. The radial parts of these atomic orbitals are sometimes numerical tables or are sometimes Slater orbitals. By angular momentum couplingAngular momentum coupling

In quantum mechanics, the procedure of constructing eigenstates of total angular momentum out of eigenstates of separate angular momenta is called angular momentum coupling....
many-electron eigenfunctions of L² (and possibly S²) are constructed.

In quantum chemical calculations hydrogen-like atomic orbitals cannot serve as an expansion basis, because they are not complete. The non-square-integrable continuum (E > 0) states must be included to obtain a complete set, i.e., to span all of one-electron Hilbert space.

Mathematical characterization

The atomic orbitals of hydrogen-like atoms are solutions to the Schrödinger equation in a spherically symmetric potentialParticle in a spherically symmetric potential

In quantum mechanics, the particle in a spherically symmetric potential describes the dynamics of a particle in a potential which has spherical symmetry....
. In this case, the potentialPotential

*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds. This mathematical formulation arises from the fact that, in physics, the scalar potential is irrotational, and thus has a vanishing Laplacian ? the very definition of a harmonic function....
term is the potential given by Coulomb's lawCoulomb's law

Coulomb's law, sometimes called the Coulomb law, is an equation describing the electrostatic force between electric charges. It was developed in the 1780s by French physicist Charles Augustin de Coulomb and was essential to the development of the classical electromagnetism....
:

where
e₀ is the permittivityPermittivity

Permittivity is a physical quantity that describes how an electric field affects, and is affected by a dielectric medium, and is determined by the ability of a material to polarization in response to the field, and thereby reduce the total electric field inside the material....
of the vacuum,
Z is the atomic numberAtomic number

In chemistry and physics, the atomic number is the number of protons found in the atomic nucleus of an atom. It is conventionally represented by the symbol Z....
(number of protons in the nucleus),
e is the elementary chargeElementary charge

The elementary charge, usually denoted e, is the electric charge carried by a single proton, or equivalently, the negative of the electric charge carried by a single electron....
(charge of an electron),
r is the distance of the electron from the nucleus.

After writing the wave function as a product of functions:
(in spherical coordinates), where are spherical harmonicsSpherical harmonics

In mathematics, the spherical harmonics are the angular portion of an orthogonal set of solutions to Laplace's equation represented in a system of spherical coordinates....
, we arrive at the following Schrödinger equation:

where is, approximately, the massMass

In physical science, mass refers to the degree of acceleration a body acquires when subject to a force: bodies with greater mass are accelerated less by the same force....
of the electronElectron

The electron is a subatomic particle that carries a negative electric charge. It has elementary particle and is believed to be a point particle....
. More accurately, it is the reduced massReduced mass

Reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. This is a quantity with the Units_of_measurement of mass, which allows the two-body problem to be solved as if it were a one-body problem....
of the system consisting of the electron and the nucleus.

Different values of l give solutions with different angular momentumAngular momentum

In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
, where l (a non-negative integer) is the quantum numberQuantum number

Quantum numbers describe values of conserved numbers in the dynamics of the quantum system. They often describe specifically the energies of electrons in atoms, but other possibilities include angular momentum, Spin etc....
of the orbital angular momentumAngular momentum

In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
. The magnetic quantum numberMagnetic quantum number

In atomic physics, the magnetic quantum number is the third of a set of quantum numbers which describe the unique quantum state of an electron and is designated by the letter m....
m (satisfying ) is the (quantized) projection of the orbital angular momentum on the z-axis. See hereParticle in a spherically symmetric potential

In quantum mechanics, the particle in a spherically symmetric potential describes the dynamics of a particle in a potential which has spherical symmetry....
for the steps leading to the solution of this equation.

Non-relativistic Wave function and energy
In addition to l and m, a third integer n > 0, emerges from the boundary conditions placed on R. The functions R and Y that solve the equations above depend on the values of these integers, called quantum numberQuantum number

Quantum numbers describe values of conserved numbers in the dynamics of the quantum system. They often describe specifically the energies of electrons in atoms, but other possibilities include angular momentum, Spin etc....
s. It is customary to subscript the wave functions with the values of the quantum numbers they depend on. The final expression for the normalized wave function is:

where:
are the generalized Laguerre polynomialsLaguerre polynomials

In mathematics, the Laguerre polynomials, named after Edmond Laguerre ,are the canonical solutions of Laguerre's equation:which is a second-order linear differential equation....
in the definition given hereLaguerre polynomials

In mathematics, the Laguerre polynomials, named after Edmond Laguerre ,are the canonical solutions of Laguerre's equation:which is a second-order linear differential equation....
.

Here, is the reduced mass of the nucleus-electron system, that is, where is the mass of the nucleus. Typically, the nucleus is much more massive than the electron, so .

.
function is a spherical harmonicSpherical Harmonic

Spherical Harmonic is a science fiction novel from the Saga of the Skolian Empire series of books by Catherine Asaro which tells the story of Pharaoh Dyhianna Selei , ruler of the Skolian Empire, after the Radiance War fought by the Imperialate and their enemy Eubians....
.

Quantum numbers

The quantum numbers n, l and m are integers and can have the following values:

See for a group theoretical interpretation of these quantum numbers this article. Among other things,
this article gives group theoretical reasons why and .

Angular momentum

Each atomic orbital is associated with an angular momentumAngular momentum

In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
l. It is a vector operator, and the eigenvalues of its square l² = l_x² + l_y² + l_z² are given by:

The projection of this vector onto an arbitrary direction is quantizedQuantization (physics)

In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field . This is a generalization of the procedure for building quantum mechanics from classical mechanics....
. If the arbitrary direction is called z, the quantization is given by:

where m is restricted as described above. Note that l² and l_z commute and have a common eigenstate, which is in accordance with Heisenberg's uncertainty principleUncertainty principle

In quantum physics, the Werner Heisenberg uncertainty principle states that certain physical quantities, like the position and momentum, cannot both have precise values at the same time....
. Since l_x and l_y do not commute with l_z, it is not possible to find a state which is an eigenstate of all three components simultaneously. Hence the values of the x and y components are not sharp, but are given by a probability function of finite width. The fact that the x and y components are not well-determined, implies that the direction of the angular momentum vector is not well determined either, although its component along the z-axis is sharp.

These relations do not give the total angular momentum of the electron. For that, electron spinSpin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
must be included.

This quantization of angular momentum closely parallels that proposed by Niels BohrNiels Bohr

Niels Henrik David Bohr was a Denmark physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922....
(see Bohr modelBohr model

In atomic physics, the Bohr model created by Niels Bohr depicts the atom as a small, positively charged atomic nucleus surrounded by electrons that travel in circular orbits around the nucleus—similar in structure to the solar system, but with electrostatic forces providing attraction, rather than gravity....
) in 1913, with no knowledge of wavefunctions.

Including spin-orbit interaction
In a real atom the spinSpin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
interacts with the magnetic fieldMagnetic field

A magnetism field is a vector field which can exert a magnetic force on moving electric charges and on magnetic dipoles . When placed in a magnetic field, magnetic dipoles tend to align their axes parallel to the magnetic field....
created by the electron movement around the nucleus, a phenomenon known as spin-orbit interactionSpin-orbit interaction

In quantum physics, the spin-orbit interaction is any interaction of a particle's spin with its motion. The first and best known example of this is that spin-orbit interaction causes shifts in an electron's energy level , due to electromagnetic interaction between the electron's spin and the nucleus's electric field, through which it moves...
. When one takes this into account, the spinSpin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
and angular momentumAngular momentum

In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
are no longer conservedAngular momentum

In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
, which can be pictured by the electronElectron

The electron is a subatomic particle that carries a negative electric charge. It has elementary particle and is believed to be a point particle....
precessing. Therefore one has to replace the quantum numbers l, m and the projection of the spinSpin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
m_s by quantum numbers which represent the total angular momentum (including spinSpin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
), j and m_j, as well as the quantum numberQuantum number

Quantum numbers describe values of conserved numbers in the dynamics of the quantum system. They often describe specifically the energies of electrons in atoms, but other possibilities include angular momentum, Spin etc....
of parityParity (physics)

In physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all spatial coordinates:...
.

See also
Rydberg atomRydberg atom

A Rydberg atom is an excited state with one or more electrons that have a very high principal quantum number. These atoms have a number of peculiar properties including an exaggerated response to Electric field and magnetic fields, long decay periods and electron wavefunctions that approximate, under some conditions, Classical physics orbits...
PositroniumPositronium

Positronium is a system consisting of an electron and its antimatter, a positron, bound together into an "exotic atom". The orbit of the two particles and the set of energy levels is similar to that of the hydrogen atom ....
Exotic atomExotic atom

An exotic atom is an otherwise normal atom in which one or more sub-atomic particles have been replaced by other particles of the same charge. For example, electrons may be replaced by other negatively charged particles such as muons or pions ....}