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Klein–Gordon equation

September 13th, 2017

The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second order in space and time and manifestly Lorentz covariant. It is a quantized version of the relativistic energy–momentum relation. Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. Its theoretical relevance is similar to that of the Dirac equation. Electromagnetic interactions can be incorporated, forming the topic of scalar electrodynamics, but because common spinless particles like the pi mesons are unstable and also experience the strong interaction (with unknown interaction term in the Hamiltonian), the practical utility is limited.

The equation can be put into the form of a Schrödinger equation. In this form it is two coupled differential equations, each of first order in time. The solutions have two components, reflecting the charge degree of freedom in relativity. It admits a conserved quantity, but this is not positive definite. The wave function cannot therefore be interpreted as a probability amplitude. The conserved quantity is instead interpreted as electric charge and the norm squared of the wave function is interpreted as a charge density. The equation describes all spinless particles with positive, negative as well as zero charge. Within the Feynman–Stueckelberg interpretation, particles and antiparticles are treated mathematically as if they propagate forward and backward in time respectively, the advanced propagator (as opposed to the retarded propagator) is employed for antiparticles. Physically, all particles move forward in time.

Any solution of the free Dirac equation is componentwise a solution of the free Klein–Gordon equation.

The equation does not form the basis for a consistent quantum relativistic one-particle theory. There is no known such theory for particles of any spin. For full reconciliation of quantum mechanics with special relativity quantum field theory is needed, in which the Klein–Gordon equation reemerges as the equation obeyed by the components of all free quantum fields. In quantum field theory, the solutions of the free (noninteracting) versions of the original equations still play a role. They are needed to build the Hilbert space (Fock space) and to express quantum field by using complete sets (spanning sets of Hilbert space) of wave functions.

The Klein–Gordon equation with mass parameter





m




{\displaystyle m}


is

Solutions of the equation are complex-valued functions





ψ



(


t


,




x




)




{\displaystyle \psi (t,{\mathbf {x}})}


of the time variable





t




{\displaystyle t}


and space variables







x






{\displaystyle {\mathbf {x}}}


; the Laplacian











2






{\displaystyle \nabla ^{2}}


acts on the space variables only.

The equation is often abbreviated as

where μ = mc/ħ and is the d’Alembert operator, defined by

(We are using the (−, +, +, +) metric signature.)

The Klein–Gordon equation is often written in natural units:

The form of the Klein–Gordon equation is derived by requiring that plane wave solutions of the equation:

obey the energy momentum relation of special relativity:

Unlike the Schrödinger equation, the Klein–Gordon equation admits two values of ω for each k, one positive and one negative. Only by separating out the positive and negative frequency parts does one obtain an equation describing a relativistic wavefunction. For the time-independent case, the Klein–Gordon equation becomes

which is formally the same as the homogeneous screened Poisson equation.

The equation was named after the physicists Oskar Klein and Walter Gordon, who in 1926 proposed that it describes relativistic electrons. Other authors making similar claims in that same year were Vladimir Fock, Johann Kudar, Théophile de Donder and Frans-H. van den Dungen, and Louis de Broglie. Although it turned out that modeling the electron’s spin required the Dirac equation, the Klein–Gordon equation correctly describes the spinless relativisitic composite particles, like the pion. On July 4, 2012 CERN announced the discovery of the Higgs boson. Since the Higgs boson is a spin-zero particle, it is the first observed ostensibly elementary particle to be described by the Klein–Gordon equation. Further experimentation and analysis is required to discern whether the Higgs boson observed is that of the Standard Model, or a more exotic, possibly composite, form.

The Klein–Gordon equation was first considered as a quantum wave equation by Schrödinger in his search for an equation describing de Broglie waves. The equation is found in his notebooks from late 1925, and he appears to have prepared a manuscript applying it to the hydrogen atom. Yet, because it fails to take into account the electron’s spin, the equation predicts the hydrogen atom’s fine structure incorrectly, including overestimating the overall magnitude of the splitting pattern by a factor of 4n/2n − 1 for the n-th energy level. The Dirac equation relativistic spectrum is, however, easily recovered if the orbital momentum quantum number is replaced by total angular momentum quantum number j. In January 1926, Schrödinger submitted for publication instead his equation, a non-relativistic approximation that predicts the Bohr energy levels of hydrogen without fine structure.

In 1926, soon after the Schrödinger equation was introduced, Vladimir Fock wrote an article about its generalization for the case of magnetic fields, where forces were dependent on velocity, and independently derived this equation. Both Klein and Fock used Kaluza and Klein’s method. Fock also determined the gauge theory for the wave equation. The Klein–Gordon equation for a free particle has a simple plane wave solution.

The non-relativistic equation for the energy of a free particle is

By quantizing this, we get the non-relativistic Schrödinger equation for a free particle,

where

is the momentum operator ( being the del operator), and

is the energy operator.

The Schrödinger equation suffers from not being relativistically invariant, meaning that it is inconsistent with special relativity.

It is natural to try to use the identity from special relativity describing the energy:

Then, just inserting the quantum mechanical operators for momentum and energy yields the equation

The square root of a differential operator can be defined with the help of Fourier transformations, but due to the asymmetry of space and time derivatives, Dirac found it impossible to include external electromagnetic fields in a relativistically invariant way. So he looked for another equation that can be modified in order to describe the action of electromagnetic forces. In addition, this equation, as it stands, is nonlocal (see also ).

Klein and Gordon instead began with the square of the above identity, i.e.

which, when quantized, gives

which simplifies to

Rearranging terms yields

Since all reference to imaginary numbers has been eliminated from this equation, it can be applied to fields that are real valued as well as those that have complex values.

Rewriting the first two terms using the inverse of the Minkowski metric diag(−c2, 1, 1, 1), and writing the Einstein summation convention explicitly we get

Thus the Klein–Gordon equation can be written in a covariant notation. This often means an abbreviation in the form of

where

and

This operator is called the d’Alembert operator.

Today this form is interpreted as the relativistic field equation for spin-0 particles. Furthermore, any component of any solution to the free Dirac equation (for a spin-one-half particle) is automatically a solution to the free Klein–Gordon equation[why?]. This generalizes to particles of any spin due extension to the Bargmann–Wigner equations. Furthermore, in quantum field theory, every component of every quantum field must satisfy the free Klein–Gordon equation, making the equation a generic expression of quantum fields.

The Klein–Gordon equation can be generalized to describe a field in some potential V(ψ) as:

The conserved current associated to the U(1) symmetry of a complex field





ϕ



(


x


)







C





{\displaystyle \phi (x)\in \mathbb {C} }


satisfying the Klein–Gordon equation reads

The form of the conserved current can be derived systematically by applying Noether’s theorem to the U(1) symmetry. We will not do so here, but simply give a proof that this conserved current is correct.

From the Klein Gordon equation for a complex field





ϕ



(


x


)




{\displaystyle \phi (x)}


of mass





m




{\displaystyle m}


written in covariant notation

and its complex conjugate

we have, multiplying by the left respectively by






ϕ










(


x


)




{\displaystyle \phi ^{*}(x)}


and





ϕ



(


x


)




{\displaystyle \phi (x)}


(and omitting for brevity the explicit




x




{\displaystyle x}


dependence),

Subtracting the former from the latter we obtain

from which we obtain the conservation law for the Klein Gordon field:

The Klein–Gordon equation for a free particle can be written as

We look for plane wave solutions of the form

for some constant angular frequency ω ∈ ℝ and wave number k ∈ ℝ3. Substitution gives the dispersion relation:

Energy and momentum are seen to be proportional to ω and k:

So the dispersion relation is just the classic relativistic equation:

For massless particles best stainless steel water bottle, we may set m = 0, recovering the relationship between energy and momentum for massless particles:

The Klein–Gordon equation can also be derived via a variational method by considering the action:

where ψ is the Klein–Gordon field and m is its mass. The complex conjugate of ψ is written ψ. If the scalar field is taken to be real-valued, then ψ = ψ.

Applying the formula for the Hilbert stress–energy tensor to the Lagrangian density (the quantity inside the integral), we can derive the stress–energy tensor of the scalar field. It is

By integration of the time–time component T00 over all space, one may show that both the positive and negative frequency plane wave solutions can be physically associated with particles with positive energy. This is not the case for the Dirac equation and its energy–momentum tensor.

There is a simple way to make any field interact with electromagnetism in a gauge invariant way: replace the derivative operators with the gauge covariant derivative operators. This is because to maintain symmetry of the physical equations for the wavefunction





ϕ





{\displaystyle \phi }


under a local U(1) gauge transformation





ϕ








ϕ







=


e


x


p


(


i


θ



)


ϕ





{\displaystyle \phi \rightarrow \phi ‘=exp(i\theta )\phi }


where





θ



(


t


,




x




)




{\displaystyle \theta (t,{\textbf {x}})}


is a locally variable phase angle, which transformation redirects the wavefunction in the complex phase space defined by





e


x


p


(


i


θ



)


=


c


o


s


θ



+


i


s


i


n


θ





{\displaystyle exp(i\theta )=cos\theta +isin\theta }


, it is required that ordinary derivatives











μ







{\displaystyle \partial _{\mu }}







D



μ





=








μ









i


e



A



μ







{\displaystyle D_{\mu }=\partial _{\mu }-ieA_{\mu }}


while the gauge fields transform as





e



A



μ









e



A



μ








=


e



A



μ





+








μ





θ





{\displaystyle eA_{\mu }\rightarrow eA’_{\mu }=eA_{\mu }+\partial _{\mu }\theta }


. The Klein Gordon equation therefore becomes:

in natural units, where A is the vector potential. While it is possible to add many higher order terms, for example,

these terms are not renormalizable in 3+1 dimensions.

The field equation for a charged scalar field multiplies by i,[clarification needed] which means the field must be complex. In order for a field to be charged, it must have two components that can rotate into each other, the real and imaginary parts.

The action for a charged scalar is the covariant version of the uncharged action:

In general relativity, we include the effect of gravity by replacing partial with covariant derivatives and the Klein–Gordon equation becomes (in the mostly pluses signature)

or equivalently

where gαβ is the inverse of the metric tensor that is the gravitational potential field, g is the determinant of the metric tensor, μ is the covariant derivative and Γσμν is the Christoffel symbol that is the gravitational force field.

Norrlands universitetssjukhus

May 18th, 2017

Koordinater:

Norrlands universitetssjukhus (Nus) i Umeå er et svensk sykehus som drives av Västerbottens läns landsting. Sykehuset er nabo med Umeå universitet og har også et nært samarbeid med universitetets medisinske fakultet.

Sykehuset er det største i Norrland og blant de største i landet. Dets tre hovedoppgaver er å drive kvalifisert behandling, forskning og undervisning. Norrlands universitetssjukhus er gjennom sitt nære samarbeid med universitetet et nav for medisinsk forskning og utvikling i Nord-Sverige football uniforms cheap. Legeutdanningen er sentral for å dekke behovet i regionen.

Akuttmottaket har en helikopterlandingsplass på taket exercises for soccer goalies. Dit kommer pasienter i helikopter fra hele regionen med alvorlige skader eller sykdommer for rask behandling og intensivpleie. Länets ambulansehelikopter er stasjonert i Lycksele.

Sykehuset er regionsykehus for Norra sjukvårdsregionen. Den omfatter landstingen i Västernorrlands, Jämtlands, Västerbottens og Norrbottens län. Totalt bor ca 900 000 mennesker i regionen (2010) spredd over halve Sveriges areal 48 blade meat tenderizer.

Den 22. januar 2014 utpekte Dagens Medicin sykehuset til Sveriges beste universitetssykehus i 2013.

Norrlands universitetssjukhus har sju satsningsområder:

På Norrlands universitetssjukhus finnes de fleste spesialiteter utenom transplantasjons- og barnehjertekirurgi, med unntak av hornhinnetransplantasjoner som blir gjort her. Sykehusets senter for kardiovaskulær genetikk er landets eneste og utfører både gentester og behandling av unge med arvlig hjertesykdom.

Opprinnelsen til Norrlands universitetssjukhus var det sykehuset som Gustav III grunnla i 1784, i Storgatan 28, rett utenfor byens vestre bytollport brazil football uniform. Sykehuset hadde fire rom som hvert hadde to sengeplasser, samt to celler med gitter for sinnssyke pasienter.

Da Umeå etter hvert vokste seg større var det ved århundreskiftet et skrikende behov for et større og mer moderne sykehus, og Landstinget besluttet å bygge et helt nytt sykehus på Ålidbacken midt i Umeå. Det ble innviet i 1907 (og hadde 100-årsjubileum 2007) med 134 sengeplasser, hvorav ca 20 for sinnssyke.

I 1918 ble sykehuset ytterligere utbygd. Øyeavdelingen stod klar i 1926, og fire år senere var reumatikkavdelingen ferdig. I 1937 kom en avdeling spesielt for barn, og året etter kom avdelingen for øre-, nese- og halssykdommer. Kvinneklinikken kom i 1953 og i 1957 fikk sykehuset en klinikk for plastisk kirurgi. Når det senere ble bestemt at NUS skulle bli regionsykehus for Norrland, ble det en kraftig økning i antall spesialistenheter og ikke minst i lokaler som nå måtte kunne huse flere pasienter.

·

Belagerung von Zaatcha

January 11th, 2017

Die Expedition zur Belagerung und Einnahme der Wüstenstadt Zaatcha (Algerien) wurde im Jahre 1849 durch französischen Truppen unter Général Émile Herbillon durchgeführt. Verteidiger der befestigten Anlage waren berberische und algerischen Krieger des Scheich Bouziane. Es war Teil des Heiligen Krieges, mit dem die Franzosen vertrieben werden sollten.

Um ihre Vormachtstellung nicht zu gefährden und die eroberten Bereiche zu schützen zogen die Franzosen mit mehr als 7.000 Mann vor den stark verteidigten Ksar.

In dieser Zeit befand sich der Emir Abd el-Kader und Anführer der gegen die Franzosen gerichteten Aufstände bereits in französischem Gewahrsam, was aber nicht bedeutete, dass in den exponierten Gebieten Ruhe eingekehrt wäre. Die französischstämmigen Siedler sahen die Bemühungen der Kolonialmacht mit gemischten Gefühlen und es entstand ein allgemeiner Drang zur Rückkehr nach Frankreich.

Im Mai 1849 nahm der Scheich Bouzian die Erhöhung der Dattelpalmensteuer zum Vorwand, die Bevölkerung aufzuhetzen. Er behauptete, dass er eine göttliche Botschaft erhalten habe, auf Grund derer er berufen sei, die französischen Neusiedler zu verjagen. Nachdem ein Leutnant aus dem Büro für arabische Angelegenheiten versucht hatte, den Scheich zu entführen, rief dieser den heiligen Krieg aus. Das „2e régiment étranger d’infanterie“ (2. Infanterieregiment der Fremdenlegion) das sich auf eine Sicherungsaktion bei Batna und Sétif befand, wurde nach Zaatcha beordert.

Die Legionäre des Colonel Jean-Luc Carbuccia, sowie das „3e bataillon d’Afrique“ (3. Afrikanisches Bataillon) erschienen am 16. Juli vor dem Ksar. Die sofort eingeleiteten Angriffe wurden verlustreich zurückgewiesen, da die Oase von einem Gewirr von Mauern und Gräben umgeben war, Zaatcha verfügte über eine krenelierte Mauer und einen Wassergraben, der den Angreifern den Weg versperrte.

Am 7. Oktober erschien Général Herbillon auf der Szene, mit ihm ein Expeditionscorps aus 4.000 Mann, die auch Belagerungsmaterial mit sich führten. Nach einer Artillerievorbereitung griff das 2. Regiment der Fremdenlegion eine Häusergruppe im Norden der Palmenplantage an und besetzte diese. Eine Einnahme der gesamten Oase war zu diesem Zeitpunkt jedoch nicht möglich.

Die französischen Truppen begannen Feldverschanzungen anzulegen, aus denen heraus versucht wurde, ein Bresche in die Mauer des Ksar zu schlagen. Am 20. Oktober begannen die Fremdenlegionäre und das 43e régiment d’infanterie de ligne einen Sturm, der von den Pionieren unterstützt wurde brazil football uniform. Diesen konnten die gut verschanzten Verteidiger zurückschlagen und den Angreifern schwere Verluste zufügen. Gleichzeitig konnten die Franzosen ständig feindliche Truppen abweisen, die versuchten als Verstärkung der Belagerten zu Hilfe zu kommen.

Am 8. November erreichte der Colonel François Certain Canrobert mit zwei Bataillonen Zuaven die Belagerer. Am 12. November kam als letzte Einheit das 8e bataillon de chasseurs à pied (8. Jägerbataillon zu Fuß) dazu, was die französischen Truppen auf einen Gesamtbestand von 7.000 Mann brachte.

Allerdings brachten die Zuaven die Cholera mit, was am Ende große Verluste verursachte.

Am 24. November machten die Belagerten, den Wachwechsel ausnutzend, einen überraschenden Ausfall. Die Berberischen Krieger mitsamt ihren Frauen stürmten in die Gräben der Belagerer. Die Jäger, verstärkt durch die Tirailleurs (Schützen) des Lieutenant-colonel Bourbaki wiesen diese blutig zurück.

In der Nacht vom 25. auf den 26. November wurden drei Breschen in die Verteidigungsmauer geschlagen und dabei der vorgelagerte Graben zugeschüttet. Um 07:00 Uhr traten drei, je dreihundert Mann starke Angriffskolonnen an, die unabhängig voneinander operierten. Sie standen unter dem Kommando der Colonels Canrobert, Barral und Lourmel. Die Tirailleurs von Commandant Bourbaki starteten gleichzeitig an anderer Stelle einen Ablenkungsangriff.

Der Angriff selbst war ein einziges Gemetzel, die engen Straßen waren vollgestopft mit zäh kämpfenden Verteidigern water bottle price, die jedoch nicht verhindern konnten, dass die Franzosen unter schweren Verlusten ein Haus nach dem anderen eroberten.

Das anschließende, von der Armée d’Afrique durchgeführte Massaker sollte wohl dazu dienen, zukünftigen Aufständischen den Mut zu nehmen, es sollte kein Akt der Rache sein. Was allerdings nunmehr die Grausamkeiten betrifft, die diesem Angriff folgten, so schrieb Alfred Nettement:

„Die Hartnäckigkeit der Verteidiger (von Zaatcha) hatte die Zuaven verärgert. Unser Sieg wurde durch kriminelle Exzesse entehrt. […] Nichts war ihnen heilig, nicht das Geschlecht, nicht das Alter. Das Blut, der Pulverdampf und der Kampfesrausch führten zu diesen schrecklichen Verletzungen der Humanität – Mitleid und Moral existierten nicht mehr. Kindern wurden die Köpfe an den Mauern zerschmettert, Frauen wurden so lange geschändet, bis der Tod für sie eine Erlösung war. Die militärischen Bulletins bestanden jedoch später darauf, dass dieses Vorgehen, sowie die Nachricht von der Zerstörung von Zaachta und den dort geschehenen Greueltaten, die sich in Windeseile über alle Oasen verbreitet hatte, seinen Zweck erfüllt habe.“

Die Scheichs Bouzian, Moussa und Lahcène wurden von den Zuaven des Commandant Lavarande gefangengenommen. Auf Befehl von Général Herbillon wurden sie exekutiert und ihre Köpfe in Biskra zur Schau gestellt. Es geschah dies um ihre angebliche Unverwundbarkeit zu widerlegen und der Rebellion in dieser Gegend die Motivation zu nehmen.

Bereits einen Tag nach dem französischen Sieg erschienen die ersten Vertreter von Stämmen vor Général Herbillon um sich zu unterwerfen.

Die 52 Tage andauernde Belagerung hatte die Armée d’Afrique mehr als 2.000 Mann an Verlusten gekostet, davon waren 600 Mann an der Cholera gestorben.

Alle Einwohner von Zaatcha, Männer, Frauen und Kinder wurden ohne Ausnahme getötet.

Metrophanes III of Constantinople

October 3rd, 2016

Metrophanes III of Byzantium (Greek: Μητροφάνης Γ΄ o Βυζάντιος, 1520–1580) was Ecumenical Patriarch of Constantinople two times, from 1565 to 1572 and from 1579 to 1580.

Metrophanes was born in 1520 to a Bulgarian merchant father in the village of Agia Paraskevi (now part of Istanbul), from where he took the sobriquet Byzantios (“of Byzantium”). His original name is variously given as Manuel or George.

In 1546 he was appointed Metropolitan of Caesarea by his personal friend Patriarch Dionysius II, who sent him to Venice mainly to raise funds, but Metrophanes went also to Rome and met the Pope. In 1548 this news caused a great concern in a part of the Greek population of Constantinople, with riots and an attempt to murder Dionysius who was considered as guilty as Metrophanes. Dionysius was on the point of being deposed, but no actions was taken against him because he enjoyed the support of Suleiman the Magnificent. Metrophanes was deposed from his See of Caesarea, but in 1551 he was forgiven and he went to live in the Monastery of the Holy Trinity in the island of Chalki where he took care and enlarged the library.

He was elected Patriarch the first time in January or February 1565 supported by the rich and influential Michael Cantacuzene. He reigned for seven years, and tried to improve the finances of the Patriarchate also through a trip in Moldavia. He was an open-minded man of letters, and well disposed towards the Westerners, both Catholic and Protestant.

He was deposed on 4 May 1572 when Michael Cantacuzene transferred his support to the young and brilliant Jeremias II Tranos. After his deposition, to grant him a financial revenue, he was appointed bishop eis zoarkeian (i.e. without pastoral obligations) of Larissa and Chios buy baseball jerseys, and he returned to live in the Monastery of the Holy Trinity in the island of Chalki, near the capital.

After his attempts to return to the throne brazil football uniform, in 1573 he was exiled to Mount Athos. Six years later, after the execution of Michael Cantacuzene and the murder of the Great Vizier Mehmed, Jeremias lost his supporters and Metrophanes was successfully restored on the throne on 25 November 1579. He died a few months later, on 9 August 1580, and was buried in the Pammakaristos Church, at the time the patriarchal cathedral.

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