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A hydrogen atom is an
atom Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, a ...
of the
chemical element A chemical element is a species of atoms that have a given number of protons in their atomic nucleus, nuclei, including the pure Chemical substance, substance consisting only of that species. Unlike chemical compounds, chemical elements canno ...
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-to ...
. The
electrically Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
neutral atom contains a single positively charged
proton A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mass ...
and a single negatively charged
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 n ...
bound to the nucleus by the
Coulomb force Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
. Atomic hydrogen constitutes about 75% of the
baryon In particle physics, a baryon is a type of composite subatomic particle which contains an odd number of valence quarks (at least 3). Baryons belong to the hadron family of particles; hadrons are composed of quarks. Baryons are also classifie ...
ic mass of the universe. In everyday life on Earth, isolated hydrogen atoms (called "atomic hydrogen") are extremely rare. Instead, a hydrogen atom tends to combine with other atoms in compounds, or with another hydrogen atom to form ordinary (
diatomic Diatomic molecules () are molecules composed of only two atoms, of the same or different chemical elements. If a diatomic molecule consists of two atoms of the same element, such as hydrogen () or oxygen (), then it is said to be homonuclear. O ...
) hydrogen gas, H2. "Atomic hydrogen" and "hydrogen atom" in ordinary English use have overlapping, yet distinct, meanings. For example, a water molecule contains two hydrogen atoms, but does not contain atomic hydrogen (which would refer to isolated hydrogen atoms).
Atomic spectroscopy Atomic spectroscopy is the study of the electromagnetic radiation absorbed and emitted by atoms. Since unique elements have characteristic (signature) spectra, atomic spectroscopy, specifically the electromagnetic spectrum or mass spectrum, is app ...
shows that there is a discrete infinite set of states in which a hydrogen (or any) atom can exist, contrary to the predictions of
classical physics Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
. Attempts to develop a theoretical understanding of the states of the hydrogen atom have been important to the
history of quantum mechanics The history of quantum mechanics is a fundamental part of the history of modern physics. Quantum mechanics' history, as it interlaces with the history of quantum chemistry, began essentially with a number of different scientific discoveries: the ...
, since all other atoms can be roughly understood by knowing in detail about this simplest atomic structure.


Isotopes

The most abundant
isotope Isotopes are two or more types of atoms that have the same atomic number (number of protons in their nuclei) and position in the periodic table (and hence belong to the same chemical element), and that differ in nucleon numbers ( mass num ...
, hydrogen-1, protium, or light hydrogen, contains no
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the atomic nucleus, nuclei of atoms. Since protons and ...
s and is simply a
proton A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mass ...
and an
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 n ...
. Protium is
stable A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...
and makes up 99.985% of naturally occurring hydrogen atoms.
Deuterium Deuterium (or hydrogen-2, symbol or deuterium, also known as heavy hydrogen) is one of two stable isotopes of hydrogen (the other being protium, or hydrogen-1). The nucleus of a deuterium atom, called a deuteron, contains one proton and one ...
contains one neutron and one proton in its nucleus. Deuterium is stable and makes up 0.0156% of naturally occurring hydrogen and is used in industrial processes like
nuclear reactor A nuclear reactor is a device used to initiate and control a fission nuclear chain reaction or nuclear fusion reactions. Nuclear reactors are used at nuclear power plants for electricity generation and in nuclear marine propulsion. Heat fr ...
s and
Nuclear Magnetic Resonance Nuclear magnetic resonance (NMR) is a physical phenomenon in which nuclei in a strong constant magnetic field are perturbed by a weak oscillating magnetic field (in the near field) and respond by producing an electromagnetic signal with a ...
.
Tritium Tritium ( or , ) or hydrogen-3 (symbol T or H) is a rare and radioactive isotope of hydrogen with half-life about 12 years. The nucleus of tritium (t, sometimes called a ''triton'') contains one proton and two neutrons, whereas the nucleus of ...
contains two neutrons and one proton in its nucleus and is not stable, decaying with a
half-life Half-life (symbol ) is the time required for a quantity (of substance) to reduce to half of its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo radioactive decay or how long stable ...
of 12.32 years. Because of its short half-life, tritium does not exist in nature except in trace amounts. Heavier isotopes of hydrogen are only created artificially in
particle accelerators A particle accelerator is a machine that uses electromagnetic fields to propel charged particles to very high speeds and energies, and to contain them in well-defined beams. Large accelerators are used for fundamental research in particle ...
and have half-lives on the order of 10−22 seconds. They are unbound
resonances Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillat ...
located beyond the
neutron drip line The nuclear drip line is the boundary beyond which atomic nuclei decay by the emission of a proton or neutron. An arbitrary combination of protons and neutrons does not necessarily yield a stable nucleus. One can think of moving up and/or to ...
; this results in prompt emission of a neutron. The formulas below are valid for all three isotopes of hydrogen, but slightly different values of the
Rydberg constant In spectroscopy, the Rydberg constant, symbol R_\infty for heavy atoms or R_\text for hydrogen, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to the electromagnetic spectra of an atom. The constant first aro ...
(correction formula given below) must be used for each hydrogen isotope.


Hydrogen ion

Lone neutral hydrogen atoms are rare under normal conditions. However, neutral hydrogen is common when it is covalently bound to another atom, and hydrogen atoms can also exist in cationic and anionic forms. If a neutral hydrogen atom loses its electron, it becomes a cation. The resulting ion, which consists solely of a proton for the usual isotope, is written as "H+" and sometimes called '' hydron''. Free protons are common in the
interstellar medium In astronomy, the interstellar medium is the matter and radiation that exist in the space between the star systems in a galaxy. This matter includes gas in ionic, atomic, and molecular form, as well as dust and cosmic rays. It fills interstella ...
, and
solar wind The solar wind is a stream of charged particles released from the upper atmosphere of the Sun, called the corona. This plasma mostly consists of electrons, protons and alpha particles with kinetic energy between . The composition of the sol ...
. In the context of
aqueous solution An aqueous solution is a solution in which the solvent is water. It is mostly shown in chemical equations by appending (aq) to the relevant chemical formula. For example, a solution of table salt, or sodium chloride (NaCl), in water would be r ...
s of classical Brønsted–Lowry acids, such as
hydrochloric acid Hydrochloric acid, also known as muriatic acid, is an aqueous solution of hydrogen chloride. It is a colorless solution with a distinctive pungent smell. It is classified as a strong acid. It is a component of the gastric acid in the dige ...
, it is actually
hydronium In chemistry, hydronium (hydroxonium in traditional British English) is the common name for the aqueous cation , the type of oxonium ion produced by protonation of water. It is often viewed as the positive ion present when an Arrhenius acid ...
, H3 O+, that is meant. Instead of a literal ionized single hydrogen atom being formed, the acid transfers the hydrogen to H2O, forming H3O+. If instead a hydrogen atom gains a second electron, it becomes an anion. The
hydrogen anion The hydrogen anion, H−, is a negative ion of hydrogen, that is, a hydrogen atom that has captured an extra electron. The hydrogen anion is an important constituent of the atmosphere of stars, such as the Sun. In chemistry, this ion is calle ...
is written as "H" and called ''
hydride In chemistry, a hydride is formally the anion of hydrogen( H−). The term is applied loosely. At one extreme, all compounds containing covalently bound H atoms are called hydrides: water (H2O) is a hydride of oxygen, ammonia is a hydride ...
''.


Theoretical analysis

The hydrogen atom has special significance in
quantum mechanics 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, ...
and
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
as a simple
two-body problem In classical mechanics, the two-body problem is to predict the motion of two massive objects which are abstractly viewed as point particles. The problem assumes that the two objects interact only with one another; the only force affecting each ...
physical system which has yielded many simple analytical solutions in closed-form.


Failed classical description

Experiments by
Ernest Rutherford Ernest Rutherford, 1st Baron Rutherford of Nelson, (30 August 1871 – 19 October 1937) was a New Zealand physicist who came to be known as the father of nuclear physics. ''Encyclopædia Britannica'' considers him to be the greatest ...
in 1909 showed the structure of the atom to be a dense, positive nucleus with a tenuous negative charge cloud around it. This immediately raised questions about how such a system could be stable.
Classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical fie ...
had shown that any accelerating charge radiates energy, as shown by the
Larmor formula In electrodynamics, the Larmor formula is used to calculate the total power radiated by a nonrelativistic point charge as it accelerates. It was first derived by J. J. Larmor in 1897, in the context of the wave theory of light. When any charge ...
. If the electron is assumed to orbit in a perfect circle and radiates energy continuously, the electron would rapidly spiral into the nucleus with a fall time of: t_\text \approx \frac \approx 1.6 \times 10^ \text , where a_0 is the
Bohr radius The Bohr radius (''a''0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an ...
and r_0 is the
classical electron radius The classical electron radius is a combination of fundamental physical quantities that define a length scale for problems involving an electron interacting with electromagnetic radiation. It links the classical electrostatic self-interaction energ ...
. If this were true, all atoms would instantly collapse, however atoms seem to be stable. Furthermore, the spiral inward would release a smear of electromagnetic frequencies as the orbit got smaller. Instead, atoms were observed to only emit discrete frequencies of radiation. The resolution would lie in the development of
quantum mechanics 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, ...
.


Bohr–Sommerfeld Model

In 1913,
Niels Bohr Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922 ...
obtained the energy levels and spectral frequencies of the hydrogen atom after making a number of simple assumptions in order to correct the failed classical model. The assumptions included: # Electrons can only be in certain, discrete circular orbits or ''stationary states'', thereby having a discrete set of possible radii and energies. # Electrons do not emit radiation while in one of these stationary states. # An electron can gain or lose energy by jumping from one discrete orbit to another. Bohr supposed that the electron's angular momentum is quantized with possible values: L = n \hbar where n = 1,2,3,\ldots and \hbar is
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
over 2 \pi. He also supposed that the
centripetal force A centripetal force (from Latin ''centrum'', "center" and ''petere'', "to seek") is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous c ...
which keeps the electron in its orbit is provided by the
Coulomb force Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
, and that energy is conserved. Bohr derived the energy of each orbit of the hydrogen atom to be: E_n = - \frac \frac, where m_e is the
electron mass The electron mass (symbol: ''m''e) is the mass of a stationary electron, also known as the invariant mass of the electron. It is one of the fundamental constants of physics. It has a value of about or about , which has an energy-equivalent o ...
, e is the
electron charge The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundame ...
, \epsilon_0 is the
vacuum permittivity Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
, and n is the
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 ...
(now known as the
principal quantum number In quantum mechanics, the principal quantum number (symbolized ''n'') is one of four quantum numbers assigned to each electron in an atom to describe that electron's state. Its values are natural numbers (from 1) making it a discrete variable. A ...
). Bohr's predictions matched experiments measuring the hydrogen spectral series to the first order, giving more confidence to a theory that used quantized values. For n=1, the valueEite Tiesinga, Peter J. Mohr, David B. Newell, and Barry N. Taylor (2019), "The 2018 CODATA Recommended Values of the Fundamental Physical Constants" (Web Version 8.0). Database developed by J. Baker, M. Douma, and S. Kotochigova. Available at http://physics.nist.gov/constants, National Institute of Standards and Technology, Gaithersburg, MD 20899
Link to RLink to hcR
/ref> \frac =\frac = 1 \,\text = 13.605\;693\;122\;994(26) \,\text is called the Rydberg unit of energy. It is related to the
Rydberg constant In spectroscopy, the Rydberg constant, symbol R_\infty for heavy atoms or R_\text for hydrogen, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to the electromagnetic spectra of an atom. The constant first aro ...
R_\infty of
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 ...
by 1 \,\text \equiv h c R_\infty. The exact value of the Rydberg constant assumes that the nucleus is infinitely massive with respect to the electron. For hydrogen-1, hydrogen-2 (
deuterium Deuterium (or hydrogen-2, symbol or deuterium, also known as heavy hydrogen) is one of two stable isotopes of hydrogen (the other being protium, or hydrogen-1). The nucleus of a deuterium atom, called a deuteron, contains one proton and one ...
), and hydrogen-3 (
tritium Tritium ( or , ) or hydrogen-3 (symbol T or H) is a rare and radioactive isotope of hydrogen with half-life about 12 years. The nucleus of tritium (t, sometimes called a ''triton'') contains one proton and two neutrons, whereas the nucleus of ...
) which have finite mass, the constant must be slightly modified to use the
reduced mass In physics, the reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, however, that the mass ...
of the system, rather than simply the mass of the electron. This includes the kinetic energy of the nucleus in the problem, because the total (electron plus nuclear) kinetic energy is equivalent to the kinetic energy of the reduced mass moving with a velocity equal to the electron velocity relative to the nucleus. However, since the nucleus is much heavier than the electron, the electron mass and reduced mass are nearly the same. The Rydberg constant ''RM'' for a hydrogen atom (one electron), ''R'' is given by R_M = \frac, where M is the mass of the atomic nucleus. For hydrogen-1, the quantity m_/M, is about 1/1836 (i.e. the electron-to-proton mass ratio). For deuterium and tritium, the ratios are about 1/3670 and 1/5497 respectively. These figures, when added to 1 in the denominator, represent very small corrections in the value of ''R'', and thus only small corrections to all energy levels in corresponding hydrogen isotopes. There were still problems with Bohr's model: # it failed to predict other spectral details such as
fine structure In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the non-relativistic Schrödinger equation. It was first measured precisely for the hydrogen atom ...
and
hyperfine structure In atomic physics, hyperfine structure is defined by small shifts in otherwise degenerate energy levels and the resulting splittings in those energy levels of atoms, molecules, and ions, due to electromagnetic multipole interaction between the n ...
# it could only predict energy levels with any accuracy for single–electron atoms (hydrogen-like atoms) # the predicted values were only correct to \alpha^2 \approx 10^, where \alpha is the
fine-structure constant In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter ''alpha''), is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between el ...
. Most of these shortcomings were resolved by Arnold Sommerfeld's modification of the Bohr model. Sommerfeld introduced two additional degrees of freedom, allowing an electron to move on an elliptical orbit characterized by its
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
and
declination In astronomy, declination (abbreviated dec; symbol ''δ'') is one of the two angles that locate a point on the celestial sphere in the equatorial coordinate system, the other being hour angle. Declination's angle is measured north or south of th ...
with respect to a chosen axis. This introduced two additional quantum numbers, which correspond to the orbital
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
and its projection on the chosen axis. Thus the correct multiplicity of states (except for the factor 2 accounting for the yet unknown electron spin) was found. Further, by applying
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
to the elliptic orbits, Sommerfeld succeeded in deriving the correct expression for the fine structure of hydrogen spectra (which happens to be exactly the same as in the most elaborate Dirac theory). However, some observed phenomena, such as the anomalous
Zeeman effect The Zeeman effect (; ) is the effect of splitting of a spectral line into several components in the presence of a static magnetic field. It is named after the Dutch physicist Pieter Zeeman, who discovered it in 1896 and received a Nobel pr ...
, remained unexplained. These issues were resolved with the full development of quantum mechanics and the
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
. It is often alleged that the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
is superior to the Bohr–Sommerfeld theory in describing hydrogen atom. This is not the case, as most of the results of both approaches coincide or are very close (a remarkable exception is the problem of hydrogen atom in crossed electric and magnetic fields, which cannot be self-consistently solved in the framework of the Bohr–Sommerfeld theory), and in both theories the main shortcomings result from the absence of the electron spin. It was the complete failure of the Bohr–Sommerfeld theory to explain many-electron systems (such as helium atom or hydrogen molecule) which demonstrated its inadequacy in describing quantum phenomena.


Schrödinger equation

The Schrödinger equation allows one to calculate the stationary states and also the time evolution of quantum systems. Exact analytical answers are available for the nonrelativistic hydrogen atom. Before we go to present a formal account, here we give an elementary overview. Given that the hydrogen atom contains a nucleus and an electron, quantum mechanics allows one to predict the probability of finding the electron at any given radial distance r. It is given by the square of a mathematical function known as the "wavefunction," which is a solution of the Schrödinger equation. The lowest energy equilibrium state of the hydrogen atom is known as the ground state. The ground state wave function is known as the 1\mathrm wavefunction. It is written as: \psi_ (r) = \frac e^. Here, a_0 is the numerical value of the Bohr radius. The probability density of finding the electron at a distance r in any radial direction is the squared value of the wavefunction: , \psi_ (r) , ^2 = \frac e^. The 1 \mathrm wavefunction is spherically symmetric, and the surface area of a shell at distance r is 4 \pi r^2, so the total probability P(r) \, dr of the electron being in a shell at a distance r and thickness dr is P (r) \, dr = 4 \pi r^2 , \psi_ (r) , ^2 \, dr. It turns out that this is a maximum at r = a_0. That is, the Bohr picture of an electron orbiting the nucleus at radius a_0 corresponds to the most probable radius. Actually, there is a finite probability that the electron may be found at any place r, with the probability indicated by the square of the wavefunction. Since the probability of finding the electron ''somewhere'' in the whole volume is unity, the integral of P(r) \, dr is unity. Then we say that the wavefunction is properly normalized. As discussed below, the ground state 1 \mathrm is also indicated by the
quantum numbers 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 k ...
(n = 1, \ell = 0, m = 0). The second lowest energy states, just above the ground state, are given by the quantum numbers (2, 0, 0), (2, 1, 0), and (2, 1, \pm 1). These n = 2 states all have the same energy and are known as the 2 \mathrm and 2 \mathrm states. There is one 2 \mathrm state: \psi_ = \frac \left( 2 - \frac \right) e^, and there are three 2 \mathrm states: \psi_ = \frac \frac e^ \cos \theta, \psi_ = \mp \frac \frac e^ \sin \theta ~ e^. An electron in the 2 \mathrm or 2 \mathrm state is most likely to be found in the second Bohr orbit with energy given by the Bohr formula.


Wavefunction

The Hamiltonian of the hydrogen atom is the radial kinetic energy operator and Coulomb attraction force between the positive proton and negative electron. Using the time-independent Schrödinger equation, ignoring all spin-coupling interactions and using the
reduced mass In physics, the reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, however, that the mass ...
\mu = m_e M/(m_e + M), the equation is written as: \left( -\frac \nabla^2 - \frac \right) \psi (r, \theta, \varphi) = E \psi (r, \theta, \varphi) Expanding the
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
in spherical coordinates: -\frac \left \frac \frac \left( r^2 \frac \right) + \frac \frac \left( \sin \theta \frac \right) + \frac \frac \right- \frac \psi = E \psi This is a separable,
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...
which can be solved in terms of special functions. When the wavefunction is separated as product of functions R(r), \Theta(\theta), and \Phi(\varphi) three independent differential functions appears with A and B being the separation constants: * radial: \frac\left(r^2\frac\right) + \frac \left(E+\frac\right)R - AR = 0 * polar: \frac\frac\left(\sin\theta\frac\right)+A\sin^2\theta- B = 0 * azimuth: \frac \frac+B=0. The normalized position
wavefunction A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
s, given in
spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
are: \psi_(r, \theta, \varphi) = \sqrt e^ \rho^ L_^(\rho) Y_\ell^m (\theta, \varphi) where: * \rho = , * a^*_0 is the reduced Bohr radius, a^*_0 = , * L_^(\rho) is a generalized Laguerre polynomial of degree n - \ell - 1, and * Y_\ell^m (\theta, \varphi) is a
spherical harmonic In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form ...
function of degree \ell and order m. Note that the generalized Laguerre polynomials are defined differently by different authors. The usage here is consistent with the definitions used by Messiah, and Mathematica. In other places, the Laguerre polynomial includes a factor of (n + \ell) !, or the generalized Laguerre polynomial appearing in the hydrogen wave function is L_^ (\rho) instead. The quantum numbers can take the following values: * n = 1, 2, 3, \ldots (
principal quantum number In quantum mechanics, the principal quantum number (symbolized ''n'') is one of four quantum numbers assigned to each electron in an atom to describe that electron's state. Its values are natural numbers (from 1) making it a discrete variable. A ...
) * \ell = 0, 1, 2, \ldots, n - 1 (
azimuthal quantum number The azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers that describe ...
) * m=-\ell, \ldots, \ell (
magnetic quantum number In atomic physics, the magnetic quantum number () is one of the four quantum numbers (the other three being the principal, azimuthal, and spin) which describe the unique quantum state of an electron. The magnetic quantum number distinguishes the ...
). Additionally, these wavefunctions are ''normalized'' (i.e., the integral of their modulus square equals 1) and
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
: \int_0^ r^2 \, dr \int_0^ \sin \theta \, d\theta \int_0^ d\varphi \, \psi^*_ (r, \theta, \varphi) \psi_ (r, \theta, \varphi) = \langle n, \ell, m , n', \ell', m' \rangle = \delta_ \delta_ \delta_, where , n, \ell, m \rangle is the state represented by the wavefunction \psi_ in
Dirac notation Distributed Research using Advanced Computing (DiRAC) is an integrated supercomputing facility used for research in particle physics, astronomy and cosmology in the United Kingdom. DiRAC makes use of multi-core processors and provides a variety of ...
, and \delta is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...
function. The wavefunctions in momentum space are related to the wavefunctions in position space through a Fourier transform \varphi (p, \theta_p, \varphi_p) = (2 \pi \hbar)^ \int e^ \psi (r, \theta,\varphi) \, dV, which, for the bound states, results in \varphi (p, \theta_p, \varphi_p) = \sqrt n^2 2^ \ell! \frac C_^ \left( \frac \right) Y_\ell^m (\theta_p, \varphi_p), where C_N^\alpha (x) denotes a
Gegenbauer polynomial In mathematics, Gegenbauer polynomials or ultraspherical polynomials ''C''(''x'') are orthogonal polynomials on the interval minus;1,1with respect to the weight function (1 − ''x''2)''α''–1/2. They generalize Legendre polynom ...
and p is in units of \hbar / a^*_0. The solutions to the Schrödinger equation for hydrogen are analytical, giving a simple expression for the hydrogen
energy levels A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The ...
and thus the frequencies of the hydrogen
spectral line A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to ident ...
s and fully reproduced the Bohr model and went beyond it. It also yields two other quantum numbers and the shape of the electron's wave function ("orbital") for the various possible quantum-mechanical states, thus explaining the
anisotropic Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
character of atomic bonds. The Schrödinger equation also applies to more complicated atoms and
molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and b ...
s. When there is more than one electron or nucleus the solution is not analytical and either computer calculations are necessary or simplifying assumptions must be made. Since the Schrödinger equation is only valid for non-relativistic quantum mechanics, the solutions it yields for the hydrogen atom are not entirely correct. The
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
of relativistic quantum theory improves these solutions (see below).


Results of Schrödinger equation

The solution of the Schrödinger equation (wave equation) for the hydrogen atom uses the fact that the
Coulomb potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
produced by the nucleus is
isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also used to describ ...
(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 follows completely generally from this isotropy of the underlying potential: the eigenstates of the Hamiltonian (that is, the energy eigenstates) can be chosen as simultaneous eigenstates of the
angular momentum operator In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum p ...
. This corresponds to the fact that angular momentum is conserved in the
orbital motion In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a ...
of the electron around the nucleus. Therefore, the energy eigenstates may be classified by two angular momentum
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 ...
s, \ell and m (both are integers). The angular momentum quantum number \ell = 0, 1, 2, \ldots determines the magnitude of the angular momentum. The magnetic quantum number m = -\ell, \ldots, +\ell 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 In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation: xy'' + (1 - x)y' + ny = 0 which is a second-order linear differential equation. This equation has nonsingular solutions on ...
in r). This leads to a third quantum number, the principal quantum number n = 1, 2, 3, \ldots. The principal quantum number in hydrogen is related to the 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., \ell = 0, 1, \ldots, n - 1. Due to angular momentum conservation, states of the same \ell but different m have the same energy (this holds for all problems with
rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which ...
). In addition, for the hydrogen atom, states of the same n but different \ell are also
degenerate Degeneracy, degenerate, or degeneration may refer to: Arts and entertainment * Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed * Degenerate art, a term adopted in the 1920s by the Nazi Party i ...
(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 an (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 or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally ...
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 In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
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 of these states. This explains also why the choice of z-axis for the directional quantization of the angular momentum vector is immaterial: an orbital of given \ell 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 \ell) that have been obtained for z.


Mathematical summary of eigenstates of hydrogen atom

In 1928,
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
found an equation that was fully compatible with
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
, and (as a consequence) made the wave function a 4-component "
Dirac spinor In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain co ...
" including "up" and "down" spin components, with both positive and "negative" energy (or matter and antimatter). The solution to this equation gave the following results, more accurate than the Schrödinger solution.


=Energy levels

= The energy levels of hydrogen, including
fine structure In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the non-relativistic Schrödinger equation. It was first measured precisely for the hydrogen atom ...
(excluding
Lamb shift In physics, the Lamb shift, named after Willis Lamb, is a difference in energy between two energy levels 2''S''1/2 and 2''P''1/2 (in term symbol notation) of the hydrogen atom which was not predicted by the Dirac equation, according to which th ...
and
hyperfine structure In atomic physics, hyperfine structure is defined by small shifts in otherwise degenerate energy levels and the resulting splittings in those energy levels of atoms, molecules, and ions, due to electromagnetic multipole interaction between the n ...
), are given by the Sommerfeld fine-structure expression:GermanEnglish
/ref> \begin E_ = & -\mu c^2 \left 1_-_\left(_1_+_\left[_\frac_\right2_\right)^_\right.html" ;"title="\frac_\right.html" ;"title="1 - \left( 1 + \left[ \frac \right">1 - \left( 1 + \left[ \frac \right2 \right)^ \right">\frac_\right.html" ;"title="1 - \left( 1 + \left[ \frac \right">1 - \left( 1 + \left[ \frac \right2 \right)^ \right\\ \approx & -\frac \left[ 1 + \frac \left( \frac - \frac \right) \right], \end where \alpha is the
fine-structure constant In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter ''alpha''), is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between el ...
and j is the
total angular momentum quantum number In quantum mechanics, the total angular momentum quantum number parametrises the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin). If s is the particle's sp ...
, which is equal to \left, \ell \pm \tfrac \, depending on the orientation of the electron spin relative to the orbital angular momentum. This formula represents a small correction to the energy obtained by Bohr and Schrödinger as given above. The factor in square brackets in the last expression is nearly one; the extra term arises from relativistic effects (for details, see #Features going beyond the Schrödinger solution). It is worth noting that this expression was first obtained by A. Sommerfeld in 1916 based on the relativistic version of the old Bohr theory. Sommerfeld has however used different notation for the quantum numbers.


Coherent states

The
coherent state In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state which has dynamics most closely resembling the oscillatory behavior of a classical h ...
s have been proposed as , s, \gamma, \bar \rangle \equiv M (s^2) \sum_^\infty (s^n e^ / \sqrt) , n, \bar \rangle, which satisfies d\bar \equiv \sin \bar \, d\bar \, d\bar \, d\bar / 8 \pi^2 and takes the form \begin \langle r, \theta, \varphi \mid s, \gamma, \bar \rangle = & e^ \sum_^\infty (s^n e^ / \sqrt) \\ & \times \, \sum_^n u_^\ell(r) \sum_^\ell \left \frac \right \left( \sin \right)^ \left( \cos \right)^ \\ & \times\, e^ Y_ (\theta, \varphi) \sqrt. \end


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 In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
that are color-coded (black represents zero density and white represents the 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'' means ''ℓ'' = 1, ''d'' means ''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 The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...
", i.e. the state of lowest energy, in which the electron is usually found, is the first one, the 1''s'' state ( principal quantum level ''n'' = 1, ''ℓ'' = 0). Black lines occur in each but the first orbital: these are the nodes of the wavefunction, i.e. where the probability density is zero. (More precisely, the nodes are
spherical harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form ...
that appear as a result of solving the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
in spherical coordinates.) The
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 ...
s determine the layout of these nodes. There are: * n-1 total nodes, * \ell of which are angular nodes: ** m angular nodes go around the \varphi axis (in the ''xy'' plane). (The figure above does not show these nodes since it plots cross-sections through the ''xz''-plane.) ** \ell-m (the remaining angular nodes) occur on the \theta (vertical) axis. * n - \ell - 1 (the remaining non-angular nodes) are radial nodes.


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 The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
, many modern experiments are sufficiently precise that a complete theoretical explanation requires a fully relativistic treatment of the problem. A relativistic treatment results in a momentum increase of about 1 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 A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
, in the
inertial frame In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration ...
of the moving electron, the electromagnetic field of the nucleus has a magnetic component. The spin of the electron has an associated
magnetic moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electroma ...
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 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 n ...
's
orbital motion In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a ...
around the nucleus, and its
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally ...
. Both of these features (and more) are incorporated in the relativistic
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
, 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 (arising through the coupling between electron spin and orbital angular momentum). States of the same and the same are still degenerate. Thus, direct analytical solution of
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
predicts 2S() and 2P() levels of hydrogen to have exactly the same energy, which is in a contradiction with observations ( Lamb–Retherford experiment). * There are always vacuum fluctuations of the
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
, according to quantum mechanics. Due to such fluctuations degeneracy between states of the same but different is lifted, giving them slightly different energies. This has been demonstrated in the famous Lamb–Retherford experiment and was the starting point for the development of the theory of
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
(which is able to deal with these vacuum fluctuations and employs the famous
Feynman diagram In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
s for approximations using
perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
). This effect is now called
Lamb shift In physics, the Lamb shift, named after Willis Lamb, is a difference in energy between two energy levels 2''S''1/2 and 2''P''1/2 (in term symbol notation) of the hydrogen atom which was not predicted by the Dirac equation, according to which th ...
. 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.


Alternatives to the Schrödinger theory

In the language of Heisenberg's
matrix mechanics Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of quantum j ...
, the hydrogen atom was first solved by
Wolfgang Pauli Wolfgang Ernst Pauli (; ; 25 April 1900 – 15 December 1958) was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics ...
using a
rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which ...
in four dimensions (4)-symmetrygenerated by the
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
and the
Laplace–Runge–Lenz vector In classical mechanics, the Laplace–Runge–Lenz (LRL) vector is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star or a planet revolving around a star. For t ...
. By extending the symmetry group O(4) to the dynamical group O(4,2), the entire spectrum and all transitions were embedded in a single irreducible group representation. In 1979 the (non-relativistic) hydrogen atom was solved for the first time within Feynman's
path integral formulation The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional i ...
of
quantum mechanics 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, ...
by Duru and Kleinert. This work greatly extended the range of applicability of Feynman's method.


See also

*
Antihydrogen Antihydrogen () is the antimatter counterpart of hydrogen. Whereas the common hydrogen atom is composed of an electron and proton, the antihydrogen atom is made up of a positron and antiproton. Scientists hope that studying antihydrogen may shed ...
*
Atomic orbital In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any ...
*
Balmer series The Balmer series, or Balmer lines in atomic physics, is one of a set of six named series describing the spectral line emissions of the hydrogen atom. The Balmer series is calculated using the Balmer formula, an empirical equation discovered b ...
*
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 ...
*
Lithium atom A lithium atom is an atom of the chemical element lithium. Stable lithium is composed of three electrons bound by the electromagnetic force to a nucleus containing three protons along with either three or four neutrons, depending on the isotope, h ...
*
Hydrogen molecular ion The dihydrogen cation or hydrogen molecular ion is a cation (positive ion) with formula . It consists of two hydrogen nuclei (protons) sharing a single electron. It is the simplest molecular ion. The ion can be formed from the ionization of a ne ...
*
Proton decay In particle physics, proton decay is a hypothetical form of particle decay in which the proton decays into lighter subatomic particles, such as a neutral pion and a positron. The proton decay hypothesis was first formulated by Andrei Sakharov ...
*
Quantum chemistry Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...
*
Quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
*
Theoretical and experimental justification for the Schrödinger equation The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relat ...
*
Trihydrogen cation The trihydrogen cation or protonated molecular hydrogen is a cation (positive ion) with formula , consisting of three hydrogen nuclei (protons) sharing two electrons. The trihydrogen cation is one of the most abundant ions in the universe. It is ...
*
List of quantum-mechanical systems with analytical solutions Much insight in quantum mechanics can be gained from understanding the closed-form solutions to the time-dependent non-relativistic Schrödinger equation. It takes the form : \hat \psi\left(\mathbf, t\right) = \left - \frac \nabla^2 + V\left(\mat ...


References


Books

* Section 4.2 deals with the hydrogen atom specifically, but all of Chapter 4 is relevant. * Kleinert, H. (2009). ''Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets'', 4th edition
Worldscibooks.com
World Scientific, Singapore (also available onlin


External links



{{Authority control Atoms Quantum models Hydrogen Hydrogen physics Isotopes of hydrogen pl:Wodór atomowy