Hydrogen-like atom - Citizendia

A hydrogen-like atom is an atom with one electron and thus is isoelectronic with hydrogen. History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J Two or more molecular entities ( Atoms Molecules Ions are described as being isoelectronic with each other if they have the same number of Hydrogen (ˈhaɪdrədʒən is the Chemical element with Atomic number 1 Except for the hydrogen atom itself (which is neutral) these atoms carry the positive charge e(Z-1), where Z is the atomic number of the atom. Hydrogen (ˈhaɪdrədʒən is the Chemical element with Atomic number 1 See also List of elements by atomic number In Chemistry and Physics, the atomic number (also known as the proton Examples of hydrogen-like ions are He⁺, Li²⁺, Be³⁺ and B⁴⁺. Helium ( He) is a colorless odorless tasteless non-toxic Inert Monatomic Chemical Lithium (ˈlɪθiəm is a Chemical element with the symbol Li and Atomic number 3 Beryllium (bəˈrɪliəm is a Chemical element with the symbol Be and Atomic number 4 Boron (ˈbɔərɒn is a Chemical element with Atomic number 5 and the chemical symbol B. Because hydrogen-like atoms are two-particle systems with an interaction depending only on the distance between the two particles, their (non-relativistic) Schrödinger equation can be solved in analytic form. The solutions are one-electron functions and are referred to as hydrogen-like atomic orbitals. ^[1]

Hydrogen-like atomic orbitals are eigenfunctions of the one-electron angular momentum operator l and its z component l_z. The energy eigenvalues do not depend on the corresponding quantum numbers, but solely on the principal quantum number n. Hence, a hydrogen-like atomic orbital is uniquely identified by the values of: principal quantum number n, angular momentum quantum number l, and magnetic quantum number m. In Atomic physics, the principal quantum number symbolized as n is the first of a set of Quantum numbers (which includes the principal quantum The Azimuthal quantum number (or orbital angular momentum quantum number, second quantum number) symbolized as l (lower-case L is a Quantum number In Atomic physics, the magnetic quantum number is the third of a set of Quantum numbers (the Principal quantum number, the Azimuthal quantum number To this must be added the two-valued spin quantum number m_s = ±½ in application of the Aufbau principle. In Atomic physics, the spin quantum number is a Quantum number that parameterizes the intrinsic Angular momentum (or spin angular momentum or simply The Aufbau principle (from the German Aufbau meaning "building up construction" also Aufbau rule or building-up principle) is This principle restricts the allowed values of the four quantum numbers in electron configurations of more-electron atoms. In Atomic physics and Quantum chemistry, electron configuration is the arrangement of Electrons in an Atom, Molecule, or other In hydrogen-like atoms all degenerate orbitals of fixed n and l, l_z and s varying between certain values (see below) form an atomic shell. An electron shell may be crudely thought of as an Orbit followed by Electrons around an Atom nucleus.

The Schrödinger equation of atoms or atomic ions with more than one electron has not been solved analytically, because of the computational difficulty imposed by the Coulomb interaction between the electrons. Numerical methods must be applied in order to obtain (approximate) wavefunctions or other properties from quantum mechanical calculations. Due to the spherical symmetry (of the Hamiltonian), the total angular momentum L of an atom is a conserved quantity. Many numerical procedures start from products of atomic orbitals that are eigenfunctions of the one-electron operators l and l_z. The radial parts of these atomic orbitals are sometimes numerical tables or are sometimes Slater orbitals. Slater-type orbitals (STOs are functions used as Atomic orbitals in the Linear combination of atomic orbitals molecular orbital method. By angular momentum coupling many-electron eigenfunctions of L² (and possibly S²) are constructed. In Quantum mechanics, the procedure of constructing Eigenstates of total angular momentum out of eigenstates of separate angular momenta is called angular momentum coupling

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. ^[2]

1 Mathematical characterization
2 Notes
3 See also
4 References

Mathematical characterization

The atomic orbitals of hydrogen-like atoms are solutions to the Schrödinger equation in a spherically symmetric potential. In Quantum mechanics, the particle in a spherically symmetric potential describes the dynamics of a particle in a Potential which has spherical symmetry In this case, the potential term is the potential given by Coulomb's law:

$V(r) = -\frac{1}{4 \pi \epsilon_0} \frac{Ze^2}{r}$

where

ε₀ is the permittivity of the vacuum,
Z is the atomic number (charge of the nucleus),
e is the elementary charge (charge of an electron),
r is the distance of the electron from the nucleus. The Mathematical study of potentials is known as Potential theory; it is the study of Harmonic functions on Manifolds This mathematical ---- Bold text Coulomb's law', developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form Permittivity is a Physical quantity that describes how an Electric field affects and is affected by a Dielectric medium and is determined by the ability See also List of elements by atomic number In Chemistry and Physics, the atomic number (also known as the proton The elementary charge, usually denoted e, is the Electric charge carried by a single Proton, or equivalently the negative of the electric charge carried

After writing the wave function as a product of functions:

$\psi(r, \theta, \phi) = R(r)Y_{lm}(\theta,\phi)\,$

(in spherical coordinates), where $Y l m$ are spherical harmonics, we arrive at the following Schrödinger equation:

$\left[ - \frac{\hbar^2}{2\mu} \left({1 \over r^2}{\partial \over \partial r}\left(r^2 {\partial R(r)\over \partial r}\right) - {l(l+1)R\over r^2} \right) + V(r)R(r) \right]= E R(r),$

where $μ$ is, approximately, the mass of the electron. In Mathematics, the spherical coordinate system is a Coordinate system for representing geometric figures in three dimensions using three coordinates the radial In Mathematics, the spherical harmonics are the angular portion of an Orthogonal set of solutions to Laplace's equation represented in a system of Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J More accurately, it is the reduced mass of the system consisting of the electron and the nucleus. Reduced mass is the "effective" Inertial mass appearing in the Two-body problem of Newtonian mechanics.

Different values of l give solutions with different angular momentum, where l (a non-negative integer) is the quantum number of the orbital angular momentum. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position Quantum numbers describe values of conserved numbers in the dynamics of the Quantum system. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position The magnetic quantum number m (satisfying $-l\le m\le l$ ) is the (quantized) projection of the orbital angular momentum on the z-axis. In Atomic physics, the magnetic quantum number is the third of a set of Quantum numbers (the Principal quantum number, the Azimuthal quantum number See here for the steps leading to the solution of this equation. In Quantum mechanics, the particle in a spherically symmetric potential describes the dynamics of a particle in a Potential which has spherical symmetry

Non-relativistic Wave function and energy

In addition to l and m, a third integer n > 0, emerges from the boundary conditions placed on R. The functions R and Y that solve the equations above depend on the values of these integers, called quantum numbers. Quantum numbers describe values of conserved numbers in the dynamics of the Quantum system. 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:

$\psi_{nlm} = R_{nl}(r)\, Y_{lm}(\theta,\phi)$

$R_{nl} (r) = \sqrt {{\left ( \frac{2 Z}{n a_{\mu}} \right ) }^3\frac{(n-l-1)!}{2n[(n+l)!]} } e^{- Z r / {n a_{\mu}}} \left ( \frac{2 Z r}{n a_{\mu}} \right )^{l} L_{n-l-1}^{2l+1} \left ( \frac{2 Z r}{n a_{\mu}} \right )$

where:

$L_{n-l-1}^{2l+1}$ are the generalized Laguerre polynomials in the definition given here. In Mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 &ndash 1886 are the Canonical solutions of Laguerre's equation In Mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 &ndash 1886 are the Canonical solutions of Laguerre's equation
$a_{\mu} = {{4\pi\varepsilon_0\hbar^2}\over{\mu e^2}}$

Note that $a_{\mu}\,$ is approximately equal to $a_0\,$ (the Bohr radius). In the Bohr model of the structure of an Atom, put forward by Niels Bohr in 1913 Electrons orbit a central nucleus. If the mass of the nucleus is infinite then $\mu = m_e\,$ and $a_\mu = a_0\,$ .

$E_{n} = -\left(\frac{Z^2 \mu e^4}{32 \pi^2\epsilon_0^2\hbar^2}\right)\frac{1}{n^2}$ .
$Y_{lm} (\theta,\phi)\,$ function is a spherical harmonic. In Mathematics, the spherical harmonics are the angular portion of an Orthogonal set of solutions to Laplace's equation represented in a system of

Quantum numbers

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

$n=1,2,3,4, \dots\,$

$l=0,1,2,\dots,n-1\,$

$m=-l,-l+1,\ldots,0,\ldots,l-1,l\,$

See for a group theoretical interpretation of these quantum numbers this article. Among other things, this article gives group theoretical reasons why $l < n\,$ and $-l \le m \le \,l$ .

Angular momentum

Each atomic orbital is associated with an angular momentum l. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position It is a vector operator, and the eigenvalues of its square l² ≡ l_x² + l_y² + l_z² are given by:

$l^2 Y_{lm} = \hbar^2 l(l+1) Y_{lm}$

The projection of this vector onto an arbitrary direction is quantized. In Physics, quantization is a procedure for constructing a Quantum field theory starting from a classical field theory. If the arbitrary direction is called z, the quantization is given by:

$l_z Y_{lm} = \hbar m Y_{lm},$

where m is restricted as described above. Note that l² and l_z commute and have a common eigenstate, which is in accordance with Heisenberg's uncertainty principle. In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain Since l_x and l_y do not commute with l_z, it is not possible to find a state which is an eigenstate of all three components simultaneously. Hence the values of the x and y components are not sharp, but are given by a probability function of finite width. The fact that the x and y components are not well-determined, implies that the direction of the angular momentum vector is not well determined either, although its component along the z-axis is sharp.

These relations do not give the total angular momentum of the electron. For that, electron spin must be included. In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin

This quantization of angular momentum closely parallels that proposed by Niels Bohr (see Bohr model) in 1913, with no knowledge of wavefunctions. Niels Henrik David Bohr (nels ˈb̥oɐ̯ˀ in Danish 7 October 1885 – 18 November 1962 was a Danish Physicist who made fundamental contributions to understanding In Atomic physics, the Bohr model created by Niels Bohr depicts the Atom as a small positively charged nucleus surrounded by Electrons Year 1913 ( MCMXIII) was a Common year starting on Wednesday (link will display the full calendar of the Gregorian calendar (or a Common

Including spin-orbit interaction

For more details on this topic, see Azimuthal quantum number#Addition of quantized angular momenta. The Azimuthal quantum number (or orbital angular momentum quantum number, second quantum number) symbolized as l (lower-case L is a Quantum number

In a real atom the spin interacts with the magnetic field created by the electron movement around the nucleus, a phenomenon known as spin-orbit interaction. In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges In Quantum physics, the spin-orbit interaction (also called spin-orbit effect or spin-orbit coupling) is any interaction of a particle's spin When one takes this into account, the spin and angular momentum are no longer conserved, which can be pictured by the electron precessing. In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J Precession refers to a change in the direction of the axis of a rotating object Therefore one has to replace the quantum numbers l, m and the projection of the spin m_s by quantum numbers which represent the total angular momentum (including spin), j and m_j, as well as the quantum number of parity. In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin Quantum numbers describe values of conserved numbers in the dynamics of the Quantum system. In Physics, a parity transformation (also called parity inversion) is the flip in the sign of one Spatial Coordinate.

Notes

^ In quantum chemistry an orbital is synonymous with "a one-electron function", a square integrable function of x, y, and z.
^ This was observed as early as 1929 by E. A. Hylleraas, Z. f. Physik vol. 48, p. 469 (1929). English translation in H. Hettema, Quantum Chemistry, Classic Scientific Papers, p. 81, World Scientific, Singapore (2000). Later it was pointed out again by H. Shull and P. -O. Löwdin, J. Chem. Phys. vol. 23, p. 1362 (1955).

References

Tipler, Paul & Ralph Llewellyn (2003). A Rydberg atom is an excited atom with one or more Electrons that have a very high Principal quantum number. Positronium ( Ps) is a system consisting of an Electron and its anti-particle, a Positron, bound together into an " Exotic atom An exotic atom is a normal Atom in which one or more sub-atomic particles have been replaced by other particles of the same charge Modern Physics (4th ed. ). New York: W. H. Freeman and Company. ISBN 0-7167-4345-0

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