Information & explanations, latest texts & monographs on
Hydrogen_atom (including recent related patents.)
Hydrogen atomThe Hydrogen atom is composed of a single negatively charged electron, moving around the positively charged proton which is the nucleus of the hydrogen atom. The electron is bound to the proton by the Coulomb force. The hydrogen atom has special significance in quantum mechanics as a simple physical system for which an exact solution to the Schrödinger equation exists, from which the experimentally observed frequencies and intensities of the hydrogen spectral lines can be calculated. In 1913, Niels Bohr had deduced the spectral frequencies of the hydrogen atom making several assumptions (see The Bohr Model). The results of Bohr for the frequencies and underlying energy values are confirmed by the full quantum-mechanical analysis which uses the Schrödinger equation, as was shown in 1925/26. The full analysis goes further, because it also yields the shape of the electron's wave function ("orbital") for the different possible quantum-mechanical states. This allows to determine the intensity of spectral lines (which correspond to transitions between these states), among other things. In addition, the full analysis is applicable also to more complicated atoms with more than one electron, as well as molecules etc. However, in all of these cases approximations have to be made and computer calculations are usually necessary. Table of contents showTocToggle("show","hide") 1 Solution of Schrödinger equation: Overview of results 2 Picture of hydrogen orbitals 3 Features going beyond the Schrödinger solution 1 External links Solution of Schrödinger equation: Overview of results The solution of the Schrödinger equation for the hydrogen atom uses the fact that the Coulomb potential produced by the nucleus is isotropic (it 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 states are not only eigenstates of the Hamiltonian, but also eigenstates of the angular momentum operator. This corresponds to the fact that angular momentum is conserved in the motion of the electron around the nucleus. Therefore, the energy eigenstates may be classified by two angular momentum quantum numbers, l and m (integer numbers). The "angular momentum" quantum number l=0,1,2,... determines the magnitude of the angular momentum. The "magnetic" quantum number m=-l,..,+l determines the projection of the angular momentum on the (arbitrarily chosen) z-axis. In addition, the radial dependence of the wave functions has to be found. It is only here that the details of the 1/r Coulomb potential enter (leading to Laguerre polynomials in r). This leads to a third quantum number, the "main" quantum number n=1,2,3,... Note that the angular momentum quantum number can run only up to n-1, i.e. l=0,1,...,n-1. Due to angular momentum conservation, states of the same l but different m have the same energy (this holds for all problems with rotational symmetry). In addition, for the hydrogen atom, the states of the same n are also degenerate (i.e. they have the same energy); but this is a specialty and it 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 of the electron adds a last quantum number, the projection of the electrons spin along the z axis, which can take on two values. Therefore, any eigenstate of the electron in the hydrogen atom is described fully by four quantum numbers. According to the usual rules of quantum mechanics, the actual state of the electron may be any superposition of these states. This explains also why the choice of z-axis for the quantization of angular momentum is immaterial: An orbital of given l and m' obtained for another preferred axis z' can always be represented as a suitable superposition of the various states of different m (but same l) that have been obtained for z. Picture of hydrogen orbitals The picture below shows the first few hydrogen atom orbitals (energy eigenfunctions). These are cross-sections of the probability density that are color-coded (black=zero density, white=highest density). The angular momentum quantum number l is denoted in each column, using the usual spectroscopic letter code ("s" means l=0; "p": l=1; "d": l=2). The main 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 x-z plane (z is the vertical axis). The probability density in threedimensional space is obtained by rotating the one shown here around the z-axis. The "ground state", i.e. the state of lowest energy, in which the electron is usually found, is the first one, the "1s" state (n=1,l=0). Click here to view an image with more orbitals (up to higher numbers n and l). Note the number of black lines that occur in each but the first orbital. These are "nodal lines" (which are actually nodal surfaces in three dimensions). Their total number is always equal to n-1. 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:
This article is adapted from from Wikipedia All Wikipedia article text is available under the terms of the GNU Free Documentation License Recent Hydrogen_atom related patents From USPTO: 6714712: Radiation curable coating composition 6714296: Method and apparatus for inspecting photosensitive material for surface defects 6713781: Organic light-emitting device having phenanthroline-fused phenazine 6713644: Hydrosilation with platinum free neat copper containing catalyst 6713635: 2-oxo-1-pyrrolidine derivatives, process for preparing them and their uses 6713634: Pyrroloazepine derivatives 6713632: Process for the preparation of imidazole derivatives 6713628: Process for preparing pharmacologically acceptable salt of N-(1(S)-ethoxycarbonyl-3-phenylpropyl)-L-alanyl-amino acid 6713615: Process for producing erythromycin derivative 6713614: Dimeric azo pyridone colorants 6713612: Sulfonyldiazomethanes, photoacid generators, resist compositions, and patterning process 6713589: Phenyl, naphthly or fluorene cyclopentyl epoxy resins 6713577: Substituted pyridyl amine catalysts and processes for polymerizing and polymers 6713575: Method for producing highly productive supported ionic catalyst for gas phase polymerization 6713566: Organoboron derivatives and process for coupling organic compounds 6713564: Star block copolymer 6713562: Resin compositions and use of the same 6713555: Hydrolyzable and polymerizable silanes based on methylene dithiepane 6713554: Compositions for the manufacture of organo-mineral products, products obtained therefrom and their use 6713551: Resin composition for coating and coating composition for curing 6713528: Coloring composition, ink-jet ink and ink jet recording method 6713523: Photopolymerizable composition and photosensitive thermal recording material 6713516: Sulphonamide derivatives 6713511: Fatty acid derivatives 6713505: spla2 inhibitors 6713500: Agent for controlling animal diseases caused by parasites 6713496: 3-(imidazolyl)-2-alkoxypropanoic acids 6713492: N-acyloxylated cycloalkyl compounds, composition containing the same and methods of use therefor 6713489: 7-[(4'-trifluoromethyl-biphenyl-2-carbonyl)amino]-quinoline-3-carboxylic acid amides, and methods of inhibiting the secretion of apolipoprotein B 6713480: Phenylahistin and the phenylahistin analogs, a new class of anti-tumor compounds 6713477: Hydroxamic acid derivatives 6713473: Tricyclic compounds 6713468: Methods of using thiazolobenzoheterocycles 6713449: E2F activity inhibitory compounds 6713440: Resist and etching by-product removing composition and resist removing method using the same 6713434: Herbicidal compositions 6713426: Metallocene capable of being used for the process for the preparation of a syndiotactic polyolefin 6713425: One pot preparation of bimetallic catalysts for ethylene 1-olefin copolymerization 6713390: Barrier layer deposition using HDP-CVD 6713255: DNA chip, PNA chip, and their preparation methods 6713244: Silver halide emulsion 6713243: Silver halide photosensitive material 6713241: Thermally developable emulsions and imaging materials containing binder mixture 6713240: Black-and-white aqueous photothermographic materials containing mercaptotriazole toners 6713227: Color filter array having a green filter layer 6713226: High contrast photographic element containing a polyhydrazide nucleating agent 6713194: Organic electroluminescence element 6713193: Organic EL device 6713192: Organic electroluminescence device and organic light emitting medium |
