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Hydrogenlike atomA hydrogenlike atom is an atom with one electron. Except for the hydrogen atom itself (which is neutral) these atoms carry the positive charge e(Z1), where Z is the atomic number of the atom. Because hydrogenlike atoms are twoparticle systems with an interaction depending only on the distance between the two particles, their (nonrelativistic) Schrödinger equation can be solved in analytic form. The solutions are oneelectron functions and are referred to as hydrogenlike atomic orbitals.^{[1]} Hydrogenlike atomic orbitals are eigenfunctions of the oneelectron 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 hydrogenlike atomic orbital is uniquely identified by the values of: principal quantum number n, angular momentum quantum number l, and magnetic quantum number m. To this must be added the twovalued spin quantum number m_{s} = ±½ in application of the Aufbau principle. This principle restricts the allowed values of the four quantum numbers in electron configurations of moreelectron atoms. In hydrogenlike atoms all degenerate orbitals of fixed n and l, l_{z} and s varying between certain values (see below) form an atomic shell. The Schrödinger equation of atoms or atomic ions with more than one electron cannot be solved analytically, because of 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 oneelectron 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 manyelectron eigenfunctions of L2 (and possibly S2) are constructed. } In quantum chemical calculations hydrogenlike atomic orbitals cannot serve as an expansion basis, because they are not complete. The nonsquareintegrable continuum (E > 0) states must be included to obtain a complete set, i.e., to span all of oneelectron Hilbert space.^{[2]} Additional recommended knowledge
Mathematical characterizationThe atomic orbitals of hydrogenlike atoms are solutions to the Schrödinger equation in a spherically symmetric potential. In this case, the potential term is the potential given by Coulomb's law: where
After writing the wave function as a product of functions: (in spherical coordinates), where Y_{lm} are spherical harmonics, we arrive at the following Schrödinger equation: where μ is, approximately, the mass of the electron. More accurately, it is the reduced mass of the system consisting of the electron and the nucleus. Different values of l give solutions with different angular momentum, where l (a nonnegative integer) is the quantum number of the orbital angular momentum. The magnetic quantum number m (satisfying ) is the (quantized) projection of the orbital angular momentum on the zaxis. See here for the steps leading to the solution of this equation. Nonrelativistic Wave function and energyIn 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. 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:
Relativistic Wave function and energyQuantum relativistic treatment of electrons uses Dirac equation. In this approach energy levels depends on n (principal quantum number) and m (magnetic quantum number)^{[3]}, energies allowed are: Where:
If we discount energy associated with rest mass of electron levels can be written as: Quantum numbersThe quantum numbers n, l and m are integers and can have the following values: See for a group theoretical interpretation of these quantum numbers this article. Among other things, this article gives group theoretical reasons why and . Angular momentumEach atomic orbital is associated with an angular momentum 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. 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 welldetermined, implies that the direction of the angular momentum vector is not well determined either, although its component along the zaxis is sharp. These relations do not give the total angular momentum of the electron. For that, electron spin must be included. This quantization of angular momentum closely parallels that proposed by Niels Bohr (see Bohr model) in 1913, with no knowledge of wavefunctions. Including spinorbit interaction
In a real atom the spin interacts with the magnetic field created by the electron movement around the nucleus, a phenomenon known as spinorbit interaction. When one takes this into account, the spin and angular momentum are no longer conserved, which can be pictured by the electron precessing. 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. Notes
See alsoReferences


This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Hydrogenlike_atom". A list of authors is available in Wikipedia. 