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| {{About|the physics of the hydrogen atom|a chemical description |hydrogen|hydrogen with neutrons|isotopes of hydrogen}}
| | When these ligaments are torn, they will fall and make the penis appear better hung. Attendees help carry a giant phallus through the streets to the shrine. A man’s virility used to find out his placement from the culture inside earlier instances and never much has modified since then a man using a little penis is often ridiculed within the culture. Moreover, since it's an herbal product, there are no side effects which are a regular feature in the other commercial products. This means any man's penis will fill her vagina completely, unless you happen to be one of those rare guys with an erect penile length of less than four inches. <br><br>This is the herb that has helped the 60, 70, 80 even 90 year olds in Brazil have a healthy sex life naturally. A penis that is too short sometimes can lead you to failing to satisfy your partner which is very risky in any relationship. Females carry XX chromosomes, and a defect in one X chromosome may be corrected by another X chromosome which is basically normal. Who says that we didn't learn anything watching Real Housewives of Atlanta. Obviously, the "hypnotherapy session" won't help you to change your old habits overnight. <br><br>Squeezing works much better if the penis is made erect. s a big possibility of scar retraction after you undergo the procedure. Adult males may test the male organ exercises like kegel and extend the length and girth of the penis. Apart from the penis enlargement, they can get substantial erections also. The quality of the programs, the clarity and comprehensiveness of information and the levels of client support provided do vary considerably however. <br><br>You still light up: Smoking is one of the main causes of most health problems, including declining sexual performance. What they are designed to do is in the interim elevate the flow of blood to your penis. So how do we do this without using spam, adwords or the latest advertising wave adsense. Yohimbine helps impotence caused by poor circulation and emotional problems resulting in raising your testosterone levels and your libido. The science behind these penis enlargement devices consist of enlarging your corpora cavernosa (set of chambers in the penis), so that it can hold more blood, hence making your penis bigger, wider, and harder during erections. <br><br>The results of this program are unparalleled gain in width, length, pleasure and rock hard erections. As mentioned earlier, the skin of the penis covers an intricate network of nerve tissue. When it comes to the ease of application, creams would certainly score higher than other methods like massage and exercising. Many private tests with members of the medical community have shown these methods to produce permanent results. It fits around the penis and allows the user to create a vacuum by using an attached pump.<br><br>If you have just about any inquiries relating to in which and tips on how to work with [http://matteopompoli.info/sitemap/ penis enlargement before after], it is possible to e mail us in our website. |
| {{Use dmy dates|date=July 2012}}
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| <!-- Here is the template for this nuclide; skip past it to edit the text. -->
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| {{Infobox stable isotope
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| |isotope_name = Hydrogen-1
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| |isotope_filename = Hydrogen 1.svg
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| |alternate_names = protium
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| |mass_number = 1
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| |symbol = H
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| |num_neutrons = 0
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| |num_protons = 1
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| |abundance = 99.985%
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| |mass = 1.007825
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| |spin = ½+
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| |excess_energy = 7288.969
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| |error1 = 0.001
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| |binding_energy = 0.000
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| |error2 = 0.0000
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| }}
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| [[File:hydrogen atom.svg|thumb|200px|right|Depiction of a hydrogen atom showing the diameter as about twice the [[Bohr model]] radius. (Image not to scale)]]
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| A '''hydrogen atom''' is an [[atom]] of the chemical element [[hydrogen]]. The [[Electric charge|electrically]] neutral atom contains a single positively charged [[proton]] and a single negatively charged [[electron]] bound to the nucleus by the [[Coulomb force]]. '''Atomic hydrogen''' constitutes [[abundance of the chemical elements|about 75% of the elemental]] ([[baryon]]ic) mass of the universe.<ref>
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| {{cite web
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| |last=Palmer|first=D.
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| |title=Hydrogen in the Universe
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| |url=http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/971113i.html
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| |publisher=[[NASA]]
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| |date=13 September 1997
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| |accessdate=5 February 2008
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| }}</ref>
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| | |
| In everyday life on Earth, isolated hydrogen atoms (usually called "atomic hydrogen" or, more precisely, "monatomic hydrogen") are extremely rare. Instead, hydrogen tends to combine with other atoms in compounds, or with itself to form ordinary ([[diatomic]]) hydrogen gas, H<sub>2</sub>. "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).
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| ==Production and reactivity==
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| The H–H bond is one of the toughest [[chemical bond|bonds]] in [[chemistry]], with a bond dissociation [[enthalpy]] of 435.88 kJ/mol at {{convert|298|K}}. As a consequence of this strong bond, H<sub>2</sub> dissociates to only a minor extent until higher temperatures. At {{convert|3000|K}}, the degree of dissociation is just 7.85%:<ref>Greenwood, N. N.; & Earnshaw, A. (1997). Chemistry of the Elements (2nd Edn.), Oxford:Butterworth-Heinemann. ISBN 0-7506-3365-4.</ref>
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| :H<sub>2</sub> ⇌ 2 H
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| Hydrogen atoms are so [[reactivity (chemistry)|reactive]] that they combine with almost all [[chemical element|elements]].
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| | |
| ==Isotopes==
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| The most [[Abundance of the chemical elements|abundant]] [[isotope]], '''hydrogen-1''', '''protium''', or '''light hydrogen''', contains no [[neutron]]s; other [[isotopes of hydrogen]], such as [[deuterium]] or [[tritium]], contain one or more neutrons. The formulas below are valid for all three isotopes of hydrogen, but slightly different values of the [[Rydberg constant]] (correction formula given below) must be used for each hydrogen isotope.
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| | |
| ==Quantum theoretical analysis==
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| The hydrogen atom has special significance in [[quantum mechanics]] and [[quantum field theory]] as a simple [[two-body problem]] physical system which has yielded many simple [[closed-form expression|analytical]] solutions in closed-form.
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| | |
| In 1913, [[Niels Bohr]] obtained the spectral frequencies of the hydrogen atom after making a number of simplifying assumptions. These assumptions, the cornerstones of the [[Bohr model]], were not fully correct but did yield fairly correct energy answers (with a relative error in the ground state ionization energy of around [[fine structure constant|α]]<sup>2</sup>/4 or around 10<sup>-5</sup>). Bohr's results for the frequencies and underlying energy values were duplicated by the solution to the [[Schrödinger equation]] in 1925–1926. The solution to the Schrödinger equation for hydrogen is [[analytical expression|analytical]], giving a simple expressions for the hydrogen [[energy levels]] and thus the frequencies of the hydrogen [[spectral line]]s. The solution of the Schrödinger equation goes much further than the Bohr model, because it also yields the shape of the electron's wave function ("orbital") for the various possible quantum-mechanical states, thus explaining the [[anisotropic]] character of atomic bonds.
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| The Schrödinger equation also applies to more complicated atoms and [[molecule]]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.
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| The Schrödinger equation is not fully accurate. The next improvement was the [[Dirac equation]] (see below).
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| | |
| === Solution of Schrödinger equation: Overview of results ===
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| The solution of the Schrödinger equation (wave equations) for the hydrogen atom uses the fact that the [[Coulomb's law|Coulomb potential]] produced by the nucleus is [[isotropic]] (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 [[Spherical coordinate system|angular coordinates]] follows completely generally from this isotropy of the underlying potential: the [[eigenstates]] of the [[Hamiltonian (quantum mechanics)|Hamiltonian]] (that is, the energy eigenstates) can be chosen as simultaneous eigenstates of the [[angular momentum operator]]. This corresponds to the fact that angular momentum is conserved in the [[orbital motion (quantum)|orbital motion]] of the electron around the nucleus. Therefore, the energy eigenstates may be classified by two angular momentum [[quantum number]]s, ''ℓ'' and ''m'' (both are integers). The angular momentum quantum number {{nowrap|''ℓ'' {{=}} 0, 1, 2, ...}} determines the magnitude of the angular momentum. The magnetic quantum number {{nowrap|''m'' {{=}} −''ℓ'', ..., +''ℓ''}} determines the projection of the angular momentum on the (arbitrarily chosen) ''z''-axis.
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| 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 ''r''). This leads to a third quantum number, the principal quantum number {{nowrap|''n'' {{=}} 1, 2, 3, ...}}. The principal quantum number in hydrogen is related to the atom's total energy.
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| 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. {{nowrap|''ℓ'' {{=}} 0, 1, ..., ''n'' − 1}}.
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| 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, states of the same ''n'' but different ''ℓ'' are also [[degenerate energy levels|degenerate]] (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 a (effective) potential differing from the form 1/''r'' (due to the presence of the inner electrons shielding the nucleus potential).
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| Taking into account the [[spin (physics)|spin]] 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]] 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 [[quantum superposition|superposition]] of these states. This explains also why the choice of ''z''-axis for the directional [[quantization (physics)|quantization]] of the angular momentum vector 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''.
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| | |
| === Alternatives to the Schrödinger theory ===
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| In the language of Heisenberg's matrix mechanics, the hydrogen atom was first solved by [[Wolfgang Pauli]]<ref name="pauli_1926">
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| {{cite journal
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| | last = Pauli | first = W | authorlink = Wolfgang Pauli
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| | year = 1926
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| | title = Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik
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| | journal = Zeitschrift für Physik
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| | volume = 36 | pages = 336–363
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| | doi = 10.1007/BF01450175
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| |bibcode = 1926ZPhy...36..336P }}</ref> using a rotational symmetry in four dimension [O(4)-symmetry] generated by the angular momentum
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| and the [[Laplace–Runge–Lenz vector]]. By extending the symmetry group O(4) to the [[dynamical group]] O(4,2),
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| the entire spectrum and all transitions were embedded in a single irreducible group representation.<ref>
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| {{cite journal
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| | title = Group Dynamics of the Hydrogen Atom
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| | author = Kleinert H.
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| | journal = Lectures in Theoretical
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| Physics, edited by W.E. Brittin and A.O. Barut, Gordon and
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| Breach, N.Y. 1968
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| | volume =
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| | pages = 427–482
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| | year = 1968
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| | doi =
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| | url = http://www.physik.fu-berlin.de/~kleinert/kleiner_re4/4.pdf
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| }}</ref>
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| In 1979 the (non relativistic) hydrogen atom was solved for the first time within [[R.P. Feynman|Feynman's]] [[path integral formulation]]
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| of [[quantum mechanics]].<ref>
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| {{cite journal
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| | title = Solution of the path integral for the H-atom
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| | author = Duru I.H., Kleinert H.
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| | journal = Physics Letters B
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| | volume = 84
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| | issue = 2
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| | pages = 185–188
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| | year = 1979
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| | doi = 10.1016/0370-2693(79)90280-6
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| | url = http://www.physik.fu-berlin.de/~kleinert/kleiner_re65/65.pdf
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| |bibcode = 1979PhLB...84..185D }}</ref><ref>
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| {{cite journal
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| | title = Quantum Mechanics of H-Atom from Path Integrals
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| | author = Duru I.H., Kleinert H.
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| | journal = Fortschr. Phys
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| | volume = 30
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| | issue = 2
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| | pages = 401–435
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| | year = 1982
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| | doi = 10.1002/prop.19820300802
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| | url = http://www.physik.fu-berlin.de/~kleinert/kleiner_re83/83.pdf
| |
| }}</ref> This work greatly extended the range of applicability of [[R.P. Feynman|Feynman's]] method.
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| | |
| === Mathematical summary of eigenstates of hydrogen atom ===
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| {{main|Hydrogen-like atom}}
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| In 1928, [[Paul Dirac]] found [[Dirac equation|an equation]] that was fully compatible with [[Special Relativity]], and (as a consequence) made the wave function a 4-component "[[Dirac spinor]]" 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.
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| ====Energy levels====
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| The energy levels of hydrogen, including [[fine structure]] (excluding [[Lamb shift]] and [[hyperfine structure]]), are given by the Sommerfeld expression:
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| ::<math>\begin{array}{rl} E_{j\,n} & = -m_\text{e}c^2\left[1-\left(1+\left[\dfrac{\alpha}{n-j-\frac{1}{2}+\sqrt{\left(j+\frac{1}{2}\right)^2-\alpha^2}}\right]^2\right)^{-1/2}\right] \\ & \approx -\dfrac{m_\text{e}c^2\alpha^2}{2n^2} \left[1 + \dfrac{\alpha^2}{n^2}\left(\dfrac{n}{j+\frac{1}{2}} - \dfrac{3}{4} \right) \right] , \end{array}</math>
| |
| | |
| where ''α'' is the [[fine-structure constant]] and ''j'' is the "total angular momentum" [[quantum number]], which is equal to |''ℓ'' ± 1/2| depending on the direction of the electron spin. 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]]).
| |
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| The value | |
| ::<math>\frac{m_{\text{e}} c^2\alpha^2}{2} = \frac{0.51\,\text{MeV}}{2 \cdot 137^2} = 13.6 \,\text{eV} </math>
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| is called the [[Rydberg constant]] and was first found from the Bohr model as given by
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| ::<math>-13.6 \,\text{eV} = -\frac{m_{\text{e}} e^4}{8 h^2 \varepsilon_0^2},</math>
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| where ''m''<sub>e</sub> is the [[electron mass]], ''e'' is the [[elementary charge]], ''h'' is the [[Planck constant]], and ''ε''<sub>0</sub> is the [[vacuum permittivity]].
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| | |
| This constant is often used in [[atomic physics]] in the form of the Rydberg unit of energy:
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| :<math>1 \,\text{Ry} \equiv h c R_\infty = 13.605\;692\;53(30) \,\text{eV}.</math><ref name="codata">P.J. Mohr, B.N. Taylor, and D.B. Newell (2011), "The 2010 CODATA Recommended Values of the Fundamental Physical Constants" (Web Version 6.0). This database was developed by J. Baker, M. Douma, and S. Kotochigova. Available: http://physics.nist.gov/constants. National Institute of Standards and Technology, Gaithersburg, MD 20899. [http://physics.nist.gov/cgi-bin/cuu/Value?ryd Link to R<sub>∞</sub>], [http://physics.nist.gov/cgi-bin/cuu/Value?rydhcev Link to hcR<sub>∞</sub>]</ref>
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| | |
| The exact value of the Rydberg constant above assumes that the nucleus is infinitely massive with respect to the electron. For hydrogen-1, hydrogen-2 ([[deuterium]]), and hydrogen-3 ([[tritium]]) the constant must be slightly modified to use the [[reduced mass]] of the system, rather than simply the mass of the electron. However, since the nucleus is much heavier than the electron, the values are nearly the same. The Rydberg constant ''R<sub>M</sub>'' for a hydrogen atom (one electron), ''R'' is given by:
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| <math>R_M = \frac{R_\infty}{1+m_{\text{e}}/M},</math>
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| where ''M'' is the mass of the atomic nucleus. For hydrogen-1, the quantity <math>m_{\text{e}}/M,</math> 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.
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| | |
| ====Wavefunction====
| |
| The normalized position [[wavefunction]]s, given in [[spherical coordinates]] are:
| |
| :<math> \psi_{n\ell m}(r,\vartheta,\varphi) = \sqrt {{\left ( \frac{2}{n a_0} \right )}^3\frac{(n-\ell-1)!}{2n(n+\ell)!} } e^{- \rho / 2} \rho^{\ell} L_{n-\ell-1}^{2\ell+1}(\rho) Y_{\ell}^{m}(\vartheta, \varphi ) </math>
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| [[Image:Hydrogen eigenstate n4 l3 m1.png|thumb|right|3D illustration of the eigenstate <math>\psi_{4,3,1}</math>. Electrons in this state are 45% likely to be found within the solid body shown.]]
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| where:
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| :<math> \rho = {2r \over {na_0}} </math>,
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| :<math> a_0 </math> is the [[Bohr radius]],
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| :<math> L_{n-\ell-1}^{2\ell+1}(\rho) </math> is a [[Laguerre polynomial#Generalized Laguerre polynomials|generalized Laguerre polynomial]] of degree {{nowrap|''n'' − ''ℓ'' − 1}}, and
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| :<math> Y_{\ell}^{m}(\vartheta, \varphi ) \,</math> is a [[spherical harmonic]] function of degree ''ℓ'' and order ''m''. Note that the [[Laguerre polynomial#Generalized Laguerre polynomials|generalized Laguerre polynomials]] are defined differently by different authors. The usage here is consistent with the definitions used by Messiah,<ref>{{cite book|last=Messiah|first=Albert|title=Quantum Mechanics|year=1999|publisher=Dover|location=New York|isbn=0-486-40924-4|pages=1136}}</ref> and Mathematica.<ref>[http://reference.wolfram.com/mathematica/ref/LaguerreL.html LaguerreL]. Wolfram Mathematica page</ref> In other places, the Laguerre polynomial includes a factor of <math>(n+\ell)!</math>,<ref>{{cite book|last=Griffiths|first=David|title=Introduction to Quantum Mechanics|year=1995|publisher=Pearson Education, Inc.|location=New Jersey|isbn=0-13-111892-7|pages=152}}</ref> or the generalized Laguerre polynomial appearing in the hydrogen wave function is <math> L_{n+\ell}^{2\ell+1}(\rho)</math> instead. <ref>{{cite book|last=Condon and Shortley|title=The Theory of Atomic Spectra|year=1963|publisher=Cambridge|location=London|pages=441}}</ref>
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| The quantum numbers can take the following values:
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| :<math> n=1,2,3,\ldots </math>
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| :<math>\ell=0,1,2,\ldots,n-1</math>
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| :<math>m=-\ell,\ldots,\ell.</math>
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| | |
| Additionally, these wavefunctions are ''normalized'' (i.e., the integral of their modulus square equals 1) and [[Orthogonal functions|orthogonal]]:
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| :<math>\int_0^{\infty} r^2 dr\int_0^{\pi} \sin \vartheta d\vartheta \int_0^{2 \pi} d\varphi\; \psi^*_{n\ell m}(r,\vartheta,\varphi)\psi_{n'\ell'm'}(r,\vartheta,\varphi)=\langle n,\ell, m | n', \ell', m' \rangle = \delta_{nn'} \delta_{\ell\ell'} \delta_{mm'},</math>
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| where <math>| n, \ell, m \rangle</math> is the representation of the wavefunction <math> \psi_{n\ell m} </math> in [[Dirac notation]], and <math> \delta </math> is the [[Kronecker delta]] function.<ref>Introduction to Quantum Mechanics, Griffiths 4.89</ref>
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| | |
| ====Angular momentum====
| |
| The [[eigenvalue]]s for [[Angular momentum operator]]:
| |
| : <math> L^2\, | n, \ell, m\rangle = {\hbar}^2 \ell(\ell+1)\, | n, \ell, m \rang </math>
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| : <math> L_z\, | n, \ell, m \rang = \hbar m \,| n, \ell, m \rang. </math>
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| | |
| === Visualizing the hydrogen electron orbitals ===
| |
| {{main|atomic orbital}}
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| [[File:HAtomOrbitals.png|frame|Probability densities through the ''xz''-plane for the electron at different quantum numbers (''ℓ'', across top; ''n'', down side; ''m'' = 0)]]
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| The image to the right shows the first few hydrogen atom orbitals (energy eigenfunctions). These are cross-sections of the [[probability amplitude|probability density]] 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.
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| 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).
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| | |
| [[media:HAtomOrbitals2.png|An image with more orbitals]] is also available (up to higher numbers ''n'' and ''ℓ'').
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| 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]] that appear as a result of solving [[Schrödinger's equation]] in polar coordinates.)
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| The [[quantum number]]s determine the layout of these nodes.<ref>[http://www.physics.byu.edu/faculty/durfee/courses/Summer2009/physics222/AtomicQuantumNumbers.pdf Summary of atomic quantum numbers]. Lecture notes. 28 July 2006</ref> There are:
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| * <math>n-1</math> total nodes,
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| * <math>l</math> of which are angular nodes:
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| ** <math>m</math> angular nodes go around the <math>\phi</math> axis (in the xy plane). <small>(The figure above does not show these nodes since it plots cross-sections through the xz-plane.)</small>
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| ** <math>l-m</math> (the remaining angular nodes) occur on the <math>\theta</math> (vertical) axis.
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| * <math>n - l - 1</math> (the remaining non-angular nodes) are radial nodes.
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| | |
| === 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]], 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]], in the [[inertial frame]] of the moving electron, the electromagnetic field of the nucleus has a magnetic component. The spin of the electron has an associated [[magnetic moment]] 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]]'s [[orbital motion (quantum)|orbital motion]] around the nucleus, and its [[spin (physics)|spin]].
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| | |
| Both of these features (and more) are incorporated in the relativistic [[Dirac equation]], 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 quantum number|total angular momentum number]] ''j'' (arising through the coupling between [[electron spin]] and [[angular momentum operator|orbital angular momentum]]). States of the same ''j'' and the same ''n'' are still degenerate. Thus, direct analytical solution of [[Dirac equation]] predicts 2S(1/2) and 2P(1/2) levels of Hydrogen to have exactly the same energy, which is in a contradiction with observations ([[Lamb shift|Lamb-Retherford experiment]]).
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| | |
| * There are always [[quantum fluctuation|vacuum fluctuation]]s of the [[electromagnetic field]], according to quantum mechanics. Due to such fluctuations degeneracy between states of the same j but different l is lifted, giving them slightly different energies. This has been demonstrated in the famous [[Lamb shift|Lamb-Retherford experiment]] and was the starting point for the development of the theory of [[Quantum electrodynamics]] (which is able to deal with these vacuum fluctuations and employs the famous [[Feynman diagram]]s for approximations using [[perturbation theory (quantum mechanics)|perturbation theory]]). This effect is now called [[Lamb shift]].
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| 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.
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| Due to the high precision of the theory also very high precision for the experiments is needed, which utilize a [[frequency comb]].
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| ====Hydrogen ion====
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| Hydrogen is not found without its electron in ordinary chemistry (room temperatures and pressures), as ionized hydrogen is highly chemically reactive. When ionized hydrogen is written as "H<sup>+</sup>" as in the solvation of classical acids such as [[hydrochloric acid]], the [[hydronium ion]], [[hydrogen|H]]<sub>3</sub>[[oxygen|O]]<sup>+</sup>, is meant, not a literal ionized single hydrogen atom. In that case, the acid transfers the proton to H<sub>2</sub>O to form H<sub>3</sub>O<sup>+</sup>.
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| Ionized hydrogen without its electron, or free protons, are common in the [[interstellar medium]], and [[solar wind]].
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| ==See also==
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| {{Wikipedia books|Hydrogen}}
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| {{col-begin|width=auto}}
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| {{col-break}}
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| * [[Antihydrogen]]
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| * [[Atomic orbital]]
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| * [[Balmer series]]
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| * [[Helium atom]]
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| {{col-break}}
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| * [[Proton decay]]
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| * [[Quantum chemistry]]
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| * [[Quantum state]]
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| * [[Theoretical and experimental justification for the Schrödinger equation]]
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| * [[Trihydrogen cation]]
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| {{col-end}}
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| ==References==
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| {{reflist|2}}
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| ==Books==
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| *{{cite book
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| | first=David J.
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| | last=[[David J. Griffiths|Griffiths]]
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| | coauthors=
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| | title=Introduction to Quantum Mechanics
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| | publisher=Prentice Hall
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| | year=1995
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| |isbn=0-13-111892-7
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| }} Section 4.2 deals with the hydrogen atom specifically, but all of Chapter 4 is relevant.
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| *{{cite book
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| | first=B.H.
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| | last=Bransden
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| | coauthors=C.J. Joachain
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| | title=Physics of Atoms and Molecules
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| | publisher=Longman
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| | year=1983
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| | isbn=0-582-44401-2 tacos
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| }}
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| * [[Hagen Kleinert|Kleinert, H.]] (2009). ''Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets'', 4th edition, [http://www.worldscibooks.com/physics/7305.html Worldscibooks.com], World Scientific, Singapore (also available online [http://www.physik.fu-berlin.de/~kleinert/re.html#B8 physik.fu-berlin.de])
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| ==External links==
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| *[http://scienceworld.wolfram.com/physics/HydrogenAtom.html Physics of hydrogen atom on Scienceworld]
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| *[http://webphysics.davidson.edu/faculty/dmb/hydrogen/ Interactive graphical representation of orbitals]
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| *[http://www.falstad.com/qmatom/ Applet which allows viewing of all sorts of hydrogenic orbitals]
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| *[http://panda.unm.edu/Courses/Finley/P262/Hydrogen/WaveFcns.html The Hydrogen Atom: Wave Functions, and Probability Density "pictures"]
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| *[http://www.physics.drexel.edu/~tim/open/hydrofin Basic Quantum Mechanics of the Hydrogen Atom]
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| *[http://search.japantimes.co.jp/cgi-bin/nn20101105a1.html "Research team takes image of hydrogen atom" Kyodo News, Friday, 5 November 2010 – (includes image)]
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| {{Isotope|element=hydrogen
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| |lighter=(none, lightest possible)
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| |heavier=[[Deuterium|hydrogen-2]]
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| |before=[[Free neutron|neutronium-1]]<br>[[Diproton|helium-2]]
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| |after=Stable
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| }}
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| [[Category:Concepts in physics]]
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| [[Category:Atoms]]
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| [[Category:Quantum models]]
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| [[Category:Hydrogen]]
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| [[Category:Hydrogen physics]]
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| [[Category:Isotopes of hydrogen]]
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| {{Link GA|uk}}
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| [[pl:Wodór atomowy]]
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