A hydrogen-like atom (or hydrogenic atom) is any
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 ...
or
ion with a single
valence electron. These atoms are
isoelectronic
Isoelectronicity is a phenomenon observed when two or more molecules have the same structure (positions and connectivities among atoms) and the same electronic configurations, but differ by what specific elements are at certain locations in th ...
with
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 ...
. Examples of hydrogen-like atoms include, but are not limited to,
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 ...
itself, all
alkali metals such as
Rb and
Cs, singly ionized
alkaline earth metals such as
Ca+ and
Sr+ and other ions such as
He+,
Li2+, and
Be3+ and
isotopes
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 numbers ...
of any of the above. A hydrogen-like atom includes a positively charged core consisting of the
atomic nucleus
The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. After the discovery of the neutron ...
and any
core electrons Core electrons are the electrons in an atom that are not valence electrons and do not participate in chemical bonding. The nucleus and the core electrons of an atom form the atomic core. Core electrons are tightly bound to the nucleus. Therefore, u ...
as well as a single valence electron. Because helium is common in the universe, the spectroscopy of singly ionized helium is important in
EUV astronomy, for example, of DO
white dwarf
A white dwarf is a stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very dense: its mass is comparable to the Sun's, while its volume is comparable to the Earth's. A white dwarf's faint luminosity comes ...
stars.
The non-relativistic
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 ...
and relativistic
Dirac equation for the hydrogen atom can be solved analytically, owing to the simplicity of the two-particle physical system. The one-electron
wave function
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 ...
solutions are referred to as ''hydrogen-like atomic orbitals''. Hydrogen-like atoms are of importance because their corresponding orbitals bear similarity to the hydrogen atomic orbitals.
Other systems may also be referred to as "hydrogen-like atoms", such as
muonium
Muonium is an exotic atom made up of an antimuon and an electron,
which was discovered in 1960 by Vernon W. Hughes
and is given the chemical symbol Mu. During the muon's lifetime, muonium can undergo chemical reactions. Due to the mass diff ...
(an electron orbiting an
antimuon),
positronium (an electron and a
positron
The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. It has an electric charge of +1 '' e'', a spin of 1/2 (the same as the electron), and the same mass as an electron. When a positron collide ...
), certain
exotic 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 (muonic atoms ...
s (formed with other particles), or
Rydberg atom
A Rydberg atom is an excited atom with one or more electrons that have a very high principal quantum number, ''n''. The higher the value of ''n'', the farther the electron is from the nucleus, on average. Rydberg atoms have a number of peculi ...
s (in which one electron is in such a high energy state that it sees the rest of the atom effectively as a
point charge
A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take u ...
).
Schrödinger solution
In the solution to the Schrödinger equation, which is non-relativistic, hydrogen-like atomic orbitals are
eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
s of the one-electron angular momentum operator ''L'' and its ''z'' component ''L''
z. A hydrogen-like atomic orbital is uniquely identified by the values of 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 ...
''n'', the
angular momentum 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 th ...
''l'', and the
magnetic quantum number ''m''. The energy eigenvalues do not depend on ''l'' or ''m'', but solely on ''n''. To these must be added the two-valued
spin quantum number
In atomic physics, the spin quantum number is a quantum number (designated ) which describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. The phrase was originally used to describe t ...
''m
s'' = ±, setting the stage for the
Aufbau principle
The aufbau principle , from the German ''Aufbauprinzip'' (building-up principle), also called the aufbau rule, states that in the ground state of an atom or ion, electrons fill subshells of the lowest available energy, then they fill subshells ...
. This principle restricts the allowed values of the four quantum numbers in
electron configuration
In atomic physics and quantum chemistry, the electron configuration is the distribution of electrons of an atom or molecule (or other physical structure) in atomic or molecular orbitals. For example, the electron configuration of the neon ato ...
s of more-electron atoms. In hydrogen-like atoms all degenerate orbitals of fixed ''n'' and ''l'', ''m'' and ''s'' varying between certain values (see below) form an
atomic shell
In chemistry and atomic physics, an electron shell may be thought of as an orbit followed by electrons around an atom's nucleus. The closest shell to the nucleus is called the "1 shell" (also called the "K shell"), followed by the "2 shell" ( ...
.
The Schrödinger equation of atoms or 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), the total angular momentum ''J'' 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 coupling many-electron eigenfunctions of ''J''
2 (and possibly ''S''
2) 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.
In the simplest model, the atomic orbitals of hydrogen-like atoms/ions are solutions to the
Schrödinger equation in a spherically symmetric potential. In this case, the
potential
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...
term is the potential given by
Coulomb's law
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 ...
:
where
* ''ε''
0 is the
permittivity
In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
of the vacuum,
* ''Z'' is the
atomic number
The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of an atomic nucleus. For ordinary nuclei, this is equal to the proton number (''n''p) or the number of protons found in the nucleus of ever ...
(number of protons in the nucleus),
* ''e'' is the
elementary 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 fundam ...
(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
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 ...
), where
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 ...
, we arrive at the following Schrödinger equation:
where
is, approximately, the
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
of 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 ...
(more accurately, it is 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 consisting of the electron and the nucleus), and
is the reduced
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 ...
.
Different values of ''l'' give solutions with different
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 ...
, where ''l'' (a non-negative integer) 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 ...
of 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 ...
. The
magnetic quantum number ''m'' (satisfying
) is the (quantized) projection of the orbital angular momentum on the ''z''-axis. See
here
Here is an adverb that means "in, on, or at this place". It may also refer to:
Software
* Here Technologies, a mapping company
* Here WeGo (formerly Here Maps), a mobile app and map website by Here
Television
* Here TV (formerly "here!"), a ...
for the steps leading to the solution of this equation.
Non-relativistic wavefunction 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 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''. 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 polynomials.
*
where
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 ...
. 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
(But for
positronium .)
is the Bohr radius.
*
*
function 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 ...
.
parity due to angular wave function is
.
Quantum numbers
The quantum numbers
,
and
are integers and can have the following values:
For a group-theoretical interpretation of these quantum numbers, see
this article. Among other things, this article gives group-theoretical reasons why
and
.
Angular momentum
Each atomic orbital is associated with an
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 ...
L. It is a
vector operator, and the eigenvalues of its square ''L''
2 ≡ ''L''
''x''2 + ''L''
''y''2 + ''L''
''z''2 are given by:
The projection of this vector onto an arbitrary direction is
quantized. If the arbitrary direction is called ''z'', the quantization is given by:
where ''m'' is restricted as described above. Note that ''L''
2 and ''L''
''z'' commute and have a common eigenstate, which is in accordance with Heisenberg's
uncertainty principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
. Since ''L''
''x'' and ''L''
''y'' do not commute with ''L''
''z'', it is not possible to find a state that 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
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 ...
must be included.
This quantization of angular momentum closely parallels that proposed by
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 ...
(see
Bohr model
In atomic physics, the Bohr model or Rutherford–Bohr model, presented by Niels Bohr and Ernest Rutherford in 1913, is a system consisting of a small, dense nucleus surrounded by orbiting electrons—similar to the structure of the Solar Syst ...
) in 1913, with no knowledge of wavefunctions.
Including spin–orbit interaction
In a real atom, 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 a moving electron can interact with the
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
of the nucleus through relativistic effects, a phenomenon known as
spin–orbit interaction
In quantum physics, the spin–orbit interaction (also called spin–orbit effect or spin–orbit coupling) is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin–orb ...
. When one takes this coupling 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 ...
and the
orbital angular momentum are no longer
conserved, which can be pictured by 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 ...
precess
Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In othe ...
ing. Therefore, one has to replace the quantum numbers ''l'', ''m'' and the projection of 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 ...
''m
s'' by quantum numbers that represent the total angular momentum (including
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 ...
), ''j'' and ''m
j'', as well as 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 ...
of
parity
Parity may refer to:
* Parity (computing)
** Parity bit in computing, sets the parity of data for the purpose of error detection
** Parity flag in computing, indicates if the number of set bits is odd or even in the binary representation of the ...
.
See the next section on the Dirac equation for a solution that includes the coupling.
Solution to Dirac equation
In 1928 in England
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 ...
. The equation was solved for hydrogen-like atoms the same year (assuming a simple Coulomb potential around a point charge) by the German
Walter Gordon. Instead of a single (possibly complex) function as in the Schrödinger equation, one must find four complex functions that make up a
bispinor
In physics, and specifically in quantum field theory, a bispinor, is a mathematical construction that is used to describe some of the fundamental particles of nature, including quarks and electrons. It is a specific embodiment of a spinor, spe ...
. The first and second functions (or components of the spinor) correspond (in the usual basis) to spin "up" and spin "down" states, as do the third and fourth components.
The terms "spin up" and "spin down" are relative to a chosen direction, conventionally the z direction. An electron may be in a superposition of spin up and spin down, which corresponds to the spin axis pointing in some other direction. The spin state may depend on location.
An electron in the vicinity of a nucleus necessarily has non-zero amplitudes for the third and fourth components. Far from the nucleus these may be small, but near the nucleus they become large.
The
eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
s of the
Hamiltonian, which means functions with a definite energy (and which therefore do not evolve except for a phase shift), have energies characterized not by the quantum number ''n'' only (as for the Schrödinger equation), but by ''n'' and a quantum number ''j'', the
total angular momentum quantum number. The quantum number ''j'' determines the sum of the squares of the three angular momenta to be ''j''(''j''+1) (times ''ħ''
2, see
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 ...
). These angular momenta include both orbital angular momentum (having to do with the angular dependence of ψ) and spin angular momentum (having to do with the spin state). The splitting of the energies of states of the same
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 ...
''n'' due to differences in ''j'' is called
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 ...
. The total angular momentum quantum number ''j'' ranges from 1/2 to ''n''−1/2.
The orbitals for a given state can be written using two radial functions and two angle functions. The radial functions depend on both the principal quantum number ''n'' and an integer ''k'', defined as:
:
where ℓ is the
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 ...
that ranges from 0 to ''n''−1. The angle functions depend on ''k'' and on a quantum number ''m'' which ranges from −''j'' to ''j'' by steps of 1. The states are labeled using the letters S, P, D, F et cetera to stand for states with ℓ equal to 0, 1, 2, 3 et cetera (see
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 ...
), with a subscript giving ''j''. For instance, the states for ''n''=4 are given in the following table (these would be prefaced by ''n'', for example 4S
1/2):
These can be additionally labeled with a subscript giving ''m''. There are 2''n''
2 states with principal quantum number ''n'', 4''j''+2 of them with any allowed ''j'' except the highest (''j''=''n''−1/2) for which there are only 2''j''+1. Since the orbitals having given values of ''n'' and ''j'' have the same energy according to the Dirac equation, they form a
basis for the space of functions having that energy.
The energy, as a function of ''n'' and , ''k'', (equal to ''j''+1/2), is:
(The energy of course depends on the zero-point used.) Note that if were able to be more than 137 (higher than any known element) then we would have a negative value inside the square root for the S
1/2 and P
1/2 orbitals, which means they would not exist. The Schrödinger solution corresponds to replacing the inner bracket in the second expression by 1. The accuracy of the energy difference between the lowest two hydrogen states calculated from the Schrödinger solution is about 9
ppm (90 μ
eV too low, out of around 10 eV), whereas the accuracy of the Dirac equation for the same energy difference is about 3 ppm (too high). The Schrödinger solution always puts the states at slightly higher energies than the more accurate Dirac equation. The Dirac equation gives some levels of hydrogen quite accurately (for instance the 4P
1/2 state is given an energy only about eV too high), others less so (for instance, the 2S
1/2 level is about eV too low). The modifications of the energy due to using the Dirac equation rather than the Schrödinger solution is of the order of α
2, and for this reason α is called 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 ...
.
The solution to the Dirac equation for quantum numbers ''n'', ''k'', and ''m'', is:
where the Ωs are columns of the two
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 ...
functions shown to the right.
signifies a spherical harmonic function:
:
in which
is an
associated Legendre polynomial. (Note that the definition of Ω may involve a spherical harmonic that doesn't exist, like
, but the coefficient on it will be zero.)
Here is the behavior of some of these angular functions. The normalization factor is left out to simplify the expressions.
:
:
:
:
From these we see that in the S
1/2 orbital (''k'' = −1), the top two components of Ψ have zero orbital angular momentum like Schrödinger S orbitals, but the bottom two components are orbitals like the Schrödinger P orbitals. In the P
1/2 solution (''k'' = 1), the situation is reversed. In both cases, the spin of each component compensates for its orbital angular momentum around the ''z'' axis to give the right value for the total angular momentum around the ''z'' axis.
The two Ω spinors obey the relationship:
:
To write the functions
and
let us define a scaled radius ρ:
:
with
:
where E is the energy (
) given above. We also define γ as:
:
When ''k'' = −''n'' (which corresponds to the highest ''j'' possible for a given ''n'', such as 1S
1/2, 2P
3/2, 3D
5/2...), then
and
are:
:
:
where ''A'' is a normalization constant involving the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers excep ...
:
:
Notice that because of the factor Zα, ''f''(''r)'' is small compared to ''g''(''r''). Also notice that in this case, the energy is given by
:
and the radial decay constant ''C'' by
:
In the general case (when ''k'' is not −''n''),
are based on two
generalized Laguerre polynomials of order
and
:
:
:
with ''A'' now defined as
:
Again ''f'' is small compared to ''g'' (except at very small ''r'') because when ''k'' is positive the first terms dominate, and α is big compared to γ−''k'', whereas when ''k'' is negative the second terms dominate and α is small compared to γ−''k''. Note that the dominant term is quite similar to corresponding the Schrödinger solution – the upper index on the Laguerre polynomial is slightly less (2γ+1 or 2γ−1 rather than 2ℓ+1, which is the nearest integer), as is the power of ρ (γ or γ−1 instead of ℓ, the nearest integer). The exponential decay is slightly faster than in the Schrödinger solution.
The normalization factor makes the integral over all space of the square of the absolute value equal to 1.
1S orbital
Here is the 1S
1/2 orbital, spin up, without normalization:
:
Note that γ is a little less than 1, so the top function is similar to an exponentially decreasing function of ''r'' except that at very small ''r'' it theoretically goes to infinity. But the value of the
only surpasses 10 at a value of ''r'' smaller than
which is a very small number (much less than the radius of a proton) unless is very large.
The 1S
1/2 orbital, spin down, without normalization, comes out as:
:
We can mix these in order to obtain orbitals with the spin oriented in some other direction, such as:
:
which corresponds to the spin and angular momentum axis pointing in the x direction. Adding ''i'' times the "down" spin to the "up" spin gives an orbital oriented in the y direction.
2P1/2 and 2S1/2 orbitals
To give another example, the 2P
1/2 orbital, spin up, is proportional to:
:
(Remember that
. ''C'' is about half what it is for the 1S orbital, but γ is still the same.)
Notice that when ρ is small compared to α (or ''r'' is small compared to
) the "S" type orbital dominates (the third component of the bispinor).
For the 2S
1/2 spin up orbital, we have:
:
Now the first component is S-like and there is a radius near ρ = 2 where it goes to zero, whereas the bottom two-component part is P-like.
Negative-energy solutions
In addition to bound states, in which the energy is less than that of an electron infinitely separated from the nucleus, there are solutions to the Dirac equation at higher energy, corresponding to an unbound electron interacting with the nucleus. These solutions are not normalizable, but solutions can be found which tend toward zero as goes to infinity (which is not possible when
except at the above-mentioned bound-state values of ). There are similar solutions with
These negative-energy solutions are just like positive-energy solutions having the opposite energy but for a case in which the nucleus repels the electron instead of attracting it, except that the solutions for the top two components switch places with those for the bottom two.
Negative-energy solutions to Dirac's equation exist even in the absence of a Coulomb force exerted by a nucleus. Dirac hypothesized that we can consider almost all of these states to be already filled. If one of these negative-energy states is not filled, this manifests itself as though there is an electron which is ''repelled'' by a positively-charged nucleus. This prompted Dirac to hypothesize the existence of positively-charged electrons, and his prediction was confirmed with the discovery of the
positron
The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. It has an electric charge of +1 '' e'', a spin of 1/2 (the same as the electron), and the same mass as an electron. When a positron collide ...
.
Beyond Gordon's solution to the Dirac equation
The Dirac equation with a simple Coulomb potential generated by a point-like non-magnetic nucleus was not the last word, and its predictions differ from experimental results as mentioned earlier. More accurate results include the
Lamb shift (radiative corrections arising from
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 ...
)
[For the radiative correction, see Nendzig, opus citatum.] 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 ...
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See also
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Rydberg atom
A Rydberg atom is an excited atom with one or more electrons that have a very high principal quantum number, ''n''. The higher the value of ''n'', the farther the electron is from the nucleus, on average. Rydberg atoms have a number of peculi ...
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Positronium
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Exotic 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 (muonic atoms ...
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Two-electron atom
In atomic physics, a two-electron atom or helium-like ion is a quantum mechanical system consisting of one nucleus with a charge of ''Z e'' and just two electrons. This is the first case of many-electron systems where the Pauli exclusion principle ...
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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 ...
Notes
References
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* Tipler, Paul & Ralph Llewellyn (2003). ''Modern Physics'' (4th ed.). New York: W. H. Freeman and Company.
{{Atomic models
Atoms
Quantum mechanics
Hydrogen