Hydrogen atom

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Hydrogen atom

A hydrogen atom is an atom of the chemical element 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....
. The electrically

Electric charge

Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. Electrically charged matter is influenced by, and produces, electromagnetic fields....
neutral atom contains a single positively-charged proton

Proton

The proton is a subatomic particle with an electric charge of +1 elementary charge. It is found in the nucleus of each atom but is also stable by itself and has a second identity as the hydrogen ion, H+....
and a single negatively-charged 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....
bound to the nucleus by the Coulomb force. The most abundant isotope

Isotope

Isotopes are any of the different types of atoms of the same chemical element, each having a different atomic mass . Isotopes of an element have atomic nucleus with the same number of protons but different numbers of neutron....
, hydrogen-1, protium, or light hydrogen, contains no neutron

Neutron

The neutron is a subatomic particle with no net electric charge and a mass slightly larger than that of a proton.Neutrons are usually found in atomic nucleus....
s; other isotopes contain one or more neutrons. This article primarily concerns hydrogen-1.

The hydrogen atom has special significance in quantum mechanics

Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
and quantum field theory as a simple two-body problem

Two-body problem

In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, two stars orbiting each other , and a classical electron orbiting an atomic nucleus....
physical system which has yielded many simple analytical

Closed-form expression

In mathematics, an expression is said to be a closed-form expression if, and only if, it can be expressed analytically in terms of a bounded number of certain "well-known" function s....
solutions in closed-form.

In 1914, Niels Bohr

Niels 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....
obtained the spectral frequencies of the hydrogen atom after making a number of simplifying assumptions.

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Encyclopedia

A hydrogen atom is an atom of the chemical element hydrogen

Hydrogen

Electric charge

Proton

Electron

Isotope

Neutron

Quantum mechanics

Two-body problem

Closed-form expression

Niels Bohr

Bohr 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....
, were not fully correct but did yield the correct energy answers. Bohr's results for the frequencies and underlying energy values were confirmed by the full quantum-mechanical analysis which uses the Schrödinger equation, as was shown in 1925/26. The solution to the Schrödinger equation

Schrödinger equation

In physics, especially quantum mechanics, the Schr?dinger equation is an equation that describes how the quantum state of a physical system changes in time....
for hydrogen is analytical

Closed-form expression

Spectral line

A spectral line is a dark or bright line in an otherwise uniform and continuous optical spectrum, resulting from an excess or deficiency of photons in a narrow frequency range, compared with the nearby frequencies....
s can be calculated. The solution of the Schrödinger equation goes much further than the Bohr model however, because it also yields the shape of the electron's wave function ("orbital") for the various possible quantum-mechanical states, thus explaining the anisotropic character of atomic bonds.

The Schrödinger equation also applies to more complicated atoms and molecule

Molecule

In chemistry, a molecule is defined as a sufficiently stable, electric charge neutral group of at least two atoms in a definite arrangement held together by very strong chemical bonds....
s. However, in most such cases the solution is not analytical and either computer calculations are necessary or simplifying assumptions must be made.

Solution of Schrödinger equation: Overview of results

The solution of the Schrödinger equation (wave equations) for the hydrogen atom uses the fact that the Coulomb potential

Coulomb'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....
produced by the nucleus is isotropic (it is radially symmetric in space and only depends on the distance to the nucleus). Although the resulting energy eigenfunctions (the "orbitals") are not necessarily isotropic themselves, their dependence on the angular coordinates

Coordinates (mathematics)

Coordinates are numbers which describe the location of points in a plane or in space. For example, the height above sea level is a coordinate which is useful for describing points near the surface of the earth....
follows completely generally from this isotropy of the underlying potential: The eigenstates of the Hamiltonian

Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian H is the observable corresponding to the total energy of the system. As with all observables, the Spectrum of the Hamiltonian is the set of possible outcomes when one measures the total energy of a system....
(= energy eigenstates) can be chosen as simultaneous eigenstates of the angular momentum operator

Angular momentum operator

In quantum mechanics, the angular momentum operator is an operator analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic physics and other quantum problems involving rotational symmetry....
. This corresponds to the fact that angular momentum is conserved in the orbital motion

Orbital motion (quantum)

The article on magnetism states that the physical cause of an atomic magnetic dipole is two kinds of movement of electrons.This article clarifies that electrons don?t actually move in their orbitals....
of the electron around the nucleus. Therefore, the energy eigenstates may be classified by two angular momentum quantum number

Quantum 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, l and m (integer numbers). The "angular momentum" quantum number l = 0, 1, 2, ... determines the magnitude of the angular momentum. The "magnetic" quantum number m = −l, .., +l determines the projection of the angular momentum on the (arbitrarily chosen) z-axis.

In addition to mathematical expressions for total angular momentum and angular momentum projection of wavefunctions, an expression for the radial dependence of the wave functions must be found. It is only here that the details of the 1/r Coulomb potential enter (leading to Laguerre polynomials

Laguerre 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 r). This leads to a third quantum number, the principal quantum number n = 1, 2, 3, ... The principal quantum number in hydrogen is related to atom's total energy.

Note that the maximum value of the angular momentum quantum number is limited by the principal quantum number: it can run only up to n − 1, i.e. l = 0, 1, ..., n − 1.

Due to angular momentum conservation, states of the same l but different m have the same energy (this holds for all problems with rotational symmetry

Rotational symmetry

File:The armoured triskelion on the flag of the Isle of Man.svgGenerally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation....
). In addition, for the hydrogen atom, states of the same n but different l are also degenerate (i.e. they have the same energy). However, this is a specific property of hydrogen and is no longer true for more complicated atoms which have a (effective) potential differing from the form 1/r (due to the presence of the inner electrons shielding the nucleus potential).

Taking into account the spin

Spin (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 ....
of the electron adds a last quantum number, the projection of the electron's spin angular momentum along the z axis, which can take on two values. Therefore, any eigenstate of the electron in the hydrogen atom is described fully by four quantum numbers. According to the usual rules of quantum mechanics, the actual state of the electron may be any superposition

Quantum superposition

Quantum superposition is the fundamental law of quantum mechanics. It defines the allowed state space of a quantum mechanical system.In Probability theory, every possible event has a non-negative real number between zero and one associated to it, the probability, which gives the chance that it happens....
of these states. This explains also why the choice of z-axis for the directional quantization

Quantization (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....
of the angular momentum vector is immaterial: An orbital of given l and m' obtained for another preferred axis z' can always be represented as a suitable superposition of the various states of different m (but same l) that have been obtained for z.

Mathematical summary of eigenstates of hydrogen atom

Energy levels

The energy levels of hydrogen, including fine structure

Fine structure

In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to first order relativistic corrections.The gross structure of line spectra is the line spectra predicted by non-relativistic electrons with no spin....
are given by

where

is the fine-structure constant

Fine-structure constant

In physics, the fine-structure constant, usually denoted is the characterizing the strength of the electromagnetic interaction. A fundamental physical constant and a dimensionless quantity, its numerical value is the same in all system of units....

j is a number which is the total angular momentum eigenvalue, i.e. l ± 1/2 depending on the direction of the electron spin. The value of -13.6 eV can be found from the simple Bohr model

Bohr model

It is even more elegantly connected to fine-structure constant:

Wavefunction

The normalized position wavefunction

Wavefunction

A wave function or wavefunction is a mathematical tool used in quantum mechanics to describe any physical system. It is a function from a mathematical space that maps the possible states of the system into the complex numbers....
s, given in spherical coordinates are: where: is the Bohr radius

Bohr radius

In the Bohr model of the structure of an atom, put forward by Niels Bohr in 1913, electrons orbit a central atomic nucleus. The model says that the electrons orbit only at certain distances from the nucleus, depending on their energy....
. are the generalized Laguerre polynomials of degree n-l-1. is a spherical harmonic

Spherical 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....
.

Angular momentum

The eigenvalues for Angular momentum operator

Angular momentum operator

Visualizing the hydrogen electron orbitals

The image to the right shows the first few hydrogen atom orbitals (energy eigenfunctions). These are cross-sections of the probability density

Probability amplitude

In quantum mechanics, a probability amplitude is a complex number whose Absolute value squared represents a probability or probability density. For example, the values taken by a normalised wave function are amplitudes, since gives the probability density at position ....
that are color-coded (black=zero density, white=highest density). The angular momentum (orbital) quantum number l is denoted in each column, using the usual spectroscopic letter code ("s" means l = 0; "p": l = 1; "d": l = 2). The main (principal) quantum number n (= 1, 2, 3, ...) is marked to the right of each row. For all pictures the magnetic quantum number m has been set to 0, and the cross-sectional plane is the xz-plane (z is the vertical axis). The probability density in three-dimensional space is obtained by rotating the one shown here around the z-axis.

The "ground state", i.e. the state of lowest energy, in which the electron is usually found, is the first one, the "1s" state (principal quantum level n = 1, l = 0).

An image with more orbitals is also available (up to higher numbers n and l).

Note the number of black lines that occur in each but the first orbital. These are "nodal lines" (which are actually nodal surfaces in three dimensions). Their total number is always equal to n − 1, which is the sum of the number of radial nodes (equal to n - l - 1) and the number of angular nodes (equal to l).

Features going beyond the Schrödinger solution

There are several important effects that are neglected by the Schrödinger equation and which are responsible for certain small but measurable deviations of the real spectral lines from the predicted ones:

Although the mean speed of the electron in hydrogen is only 1/137th of the speed of light
Speed of light

The speed of light in an free space is an important physical constant usually written as c, with a value of 299,792,458 metres per second....
, many modern experiments are sufficiently precise that a complete theoretical explanation requires a fully relativistic treatment of the problem. A relativistic treatment results in an momentum increase of about one part in 37,000 for the electron. Since the electron's wavelength is determined by its momentum, orbitals containing higher speed electrons show contraction due to smaller wavelengths.

Even when there is no external magnetic field
Magnetic 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....
, in the inertial frame of the moving electron, the electromagnetic field of the nucleus has a magnetic component. The spin of the electron has an associated magnetic moment
Magnetic moment

In physics, astronomy, chemistry, and electrical engineering, the term magnetic moment of a system usually refers to its magnetic dipole moment, and is a measure of the strength of the system's net Magnetism....
which interacts with this magnetic field. This effect is also explained by special relativity, and it leads to the so-called spin-orbit coupling, i.e., an interaction between the 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....
's orbital motion
Orbital motion (quantum)

The article on magnetism states that the physical cause of an atomic magnetic dipole is two kinds of movement of electrons.This article clarifies that electrons don?t actually move in their orbitals....
around the nucleus, and its spin
Spin (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 ....
.

Both of these features (and more) are incorporated in the relativistic Dirac equation

Dirac equation

In physics, the Dirac equation is a theory of relativity quantum mechanics wave equation formulated by British physicist Paul Dirac in 1928 and provides a description of elementary particle spin-? particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity....
, with predictions that come still closer to experiment. Again the Dirac equation may be solved analytically in the special case of a two-body system, such as the hydrogen atom. The resulting solution quantum states now must be classified by the total angular momentum number j (arising through the coupling between electron spin and orbital angular momentum). States of the same j and the same n are still degenerate.

There are always vacuum fluctuations of the electromagnetic field
Electromagnetic field

The electromagnetic field is a physical field produced by electric charge. It affects the behavior of charged objects in the vicinity of the field....
, according to quantum mechanics. Due to such fluctuations degeneracy between states of the same j but different l is lifted, giving them slightly different energies. This has been demonstrated in the famous Lamb-Retherford experiment
Lamb shift

In physics, the Lamb shift, named after Willis Lamb , is a small difference in energy between two energy levels and of the hydrogen atom in quantum mechanics....
and was the starting point for the development of the theory of Quantum electrodynamics
Quantum electrodynamics

Quantum electrodynamics is a relativity theory quantum field theory of electrodynamics. QED was developed by a number of physicists, beginning in the late 1920s....
(which is able to deal with these vacuum fluctuations and employs the famous Feynman diagram
Feynman diagram

In quantum field theory a Feynman diagram is an intuitive graphical representation of a contribution to the transition amplitude or correlation function of a quantum mechanical or statistical field theory....
s for approximations using perturbation theory
Perturbation theory (quantum mechanics)

In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation theory for describing a complicated quantum system in terms of a simpler one....
). This effect is now called Lamb shift
Lamb shift

In physics, the Lamb shift, named after Willis Lamb , is a small difference in energy between two energy levels and of the hydrogen atom in quantum mechanics....
.

For these developments, it was essential that the solution of the Dirac equation for the hydrogen atom could be worked out exactly, such that any experimentally observed deviation had to be taken seriously as a signal of failure of the theory.

Due to the high precision of the theory also very high precision for the experiments is needed, which utilize a frequency comb

Frequency comb

A frequency comb is the graphic representation of the spectrum of a modelocking. An octave spanning comb can be used for mapping radio frequencies into the optical frequency range or it can be used to steer a piezoelectric mirror within a carrier envelope phase correcting feedback loop....
.

Hydrogen ion

Hydrogen is not found without its electron in ordinary chemistry (room temperatures and pressures), as ionized hydrogen is highly chemically reactive. When ionized hydrogen is written as "H⁺" as in the solvation of classical acids such hydrochloric acid

Hydrochloric acid

Hydrochloric acid is the solution of hydrogen chloride in water. It is a highly corrosive, strong acid mineral acid and has major industrial uses....
, the hydronium ion, H

Hydrogen

Oxygen

Oxygen no O2 produced; 2) O2 produced, but absorbed in oceans & seabed rock; 3) O2 starts to gas out of the oceans, but is absorbed by land surfaces and formation of ozone layer; 4-5) O2 sinks filled and the gas accumulates]]...
⁺, is meant, not a literal ionized single hydrogen atom. In that case, the acid transfers the proton to H₂O to form H₃O⁺.

Ionized hydrogen without its electron, or free protons, are common in the interstellar medium

Interstellar medium

In astronomy, the interstellar medium is the gas and cosmic dust that pervade interstellar space: the matter that exists between the stars within a galaxy....
, and solar wind

Solar wind

The solar wind is a Electric current—a Plasma —ejected from the stellar atmosphere of the sun. It consists mostly of electrons and protons with energies of about 1 electron volt....
.

Hydrogen atom

Solution of Schrödinger equation: Overview of results

Mathematical summary of eigenstates of hydrogen atom

Energy levels

Wavefunction

Angular momentum

Visualizing the hydrogen electron orbitals

Features going beyond the Schrödinger solution

Hydrogen ion

See also

External links