Definicion de covariante y de contravariante.

Тензоры ранга 1, 2 и 3 визуализируются с ковариантными и контравариантными компонентами.
se visualizan tensores de rango 1, 2 y 3 con componentes covariantes y contravariantes
definicion de covariante y de contravariante.

Assumptions in Heisenberg’s 1925 paper

    http://physics.stackexchange.com/questions/18519/assumptions-in-heisenbergs-1925-paper


    http://theorie2.physik.uni-erlangen.de/index.php/Papers_from_the_beginning_of_quantum_mechanics


    My first query is why does he claim the position and period of an electron to be unobservable “in principle”? There was theoretically no reason (at THAT time) to doubt that these quantities could be measured, though certainly they were indeterminate practically.

Werner Heisenberg obviously disagreed with this assumption of yours and it just happened that his ability to disagree made him a founder of quantum mechanics.

He has spent several years by trying to develop “quantized planetary” models of the helium atom etc. before he understood that this failing project is failing for fundamental reasons. Such a helium with well-defined positions would be described by a chaotic 3-body problem and there would be no way how it could be consistent with the known regular behavior of the helium atom (and other atoms and other coherent systems), including the sharp spectral lines.

So Heisenberg was able to see in 1925 something that you can’t see now: that the electrons can’t be going along any particular trajectories while they’re in the atoms. Instead, what is observed is that they have a totally sharp energy from a possible list, the spectrum – something we can really observe via the photons that atoms emit or absorb. To conclude that electrons can’t be going along particular classical trajectories in the atoms, he didn’t have to wait for measuring apparatuses that would be sufficiently accurate. He was able to make this conclusion out of the available data by “pure thought”, and he was right.

    Secondly, just because a theory dealing with those quantities is inconsistent, or not general enough, why does it imply that we cannot define or measure quantities that that theory deals with? We may be able to measure some quantities perfectly, but still formulate an incorrect theory around them.

Many combinations of options would be possible in a generic hypothetical world and you’re right that the combination of options you mentioned would be logically possible in another world but Heisenberg was talking about our world. He learned his message from special relativity that one shouldn’t talk about things that can’t be operationally defined – such as the simultaneity of events (which is observer-dependent) and tried to maximally apply this positivist mode of reasoning to the world of atoms. His analysis dictated that he may assume that the electron in the atom has a particular energy for a long time but it can’t have a well-defined position or velocity. So he reformulated physics around the notion of the energy which is measurable and found out the first formulation of quantum mechanics in the energy eigenstate basis Heisenberg picture.

    Finally, is there any ad-hoc basis to decide what these “uncertain” quantities are? More specifically, how could Heisenberg pinpoint position of an electron as an uncertain parameter and not any other quantity (like some electric field, etc.)?

You are mixing apples with oranges here. Heisenberg’s paper wasn’t discussing the electromagnetic field. It was discussing the general logical framework underlying physics and the examples he took were those from mechanics – rigid rotator and anharmonic oscillator – that were meant to be later generalized to a theory of atoms in particular just by a new choice of the potential energy formula.

There’s no observable concept of “electric fields” in the description of an atom or anharmonic oscilator at all. Even in classical physics, one deals with functions of positions and momenta. He figured out that not all functions are equally observable: energy (a particular function of positions and momenta) is much more observable and stable.

The underlying logic he has developed was later (soon) applied to other systems in mechanics such as atoms and molecules as well as field theory such as electromagnetism. But the essence isn’t in describing which degrees of freedom are there (they’re kept as close to those in the corresponding classical theory as possible); the essence of quantum mechanics is in the totally new set of postulates and methods to make predictions.

He realized that the right goal wasn’t just to find another classical theory just with some new degrees of freedom, which is the intrinsic, fundamental, and completely flawed assumption of your whole question from the beginning to the end. He realized that the new insights force physicists to formulate a completely new theory – and he (and others) has (have) already used the completely new term “quantum theory” for it – and he just did so, discovering some of the new explicit quantum formulae for nontrivial predictions (beyond the spectrum of the Hydrogen atom that was “explained” by Bohr’s toy model).

You may repeat many times that a complete conceptual revolution in physics (switching from the classical to the quantum) wasn’t needed and one should have only discussed new classical models with new variables (paying no attention to whether or not they may be actually observed) except that Heisenberg knew that it was needed and the months (and a few years) that followed his discovery made his assumption unquestionable.




1

Thanks for the detailed explanation. But I wanted to confirm the following – Heisenberg did not propose the indeterminacy of position/velocity due to some experimental results, rather, just as special relativity challenged the ad-hoc concept of time (which was used as a parameter for evolution of position, momentum and other quantities in classical mech), Heisenberg challenged the absolute determinacy of position/momentum (which were in turn parameters to describe fields, energy, etc.). And so in this sense it was a theoretical analogy to special rel? Is that correct? –  

2

I see… well definitely the helium model failure was a motivation as well. On a side note, instead of just studying quantum mechanics, I additionally intend to focus on such fundamental matters and questions underlying it. In other words I actually want to study the “Physics” of it, rather than just the mathematical framework blindly (excuse me if I’m being obscure), and understand how each aspect of the theory fits into the physical world. Do you have any suggestions as to how to go about it, and whether studying the original pioneering papers would help in this regard? – 

3

I meant that sometimes during the course of going through the mathematical formalism, it is possible to unknowingly ignore the physical interpretation of some steps taken, or the physical meaning behind some results. So as far as possible, I don’t want to ignore any of that. (Also which “good textbooks” are you referring to?) Again, sorry for the trouble. –

4

That’s very good you don’t want to ignore the physical interpretation and indeed, not too many words are being said about it in most cases. However, what is even more important than to appreciate the right physical interpretation of the formulae is to avoid a wrong interpretation of them – such as a classical or “visualizable” interpretation: none of it is ever right in QM. Certain things just don’t have any “easy to imagine” content and the calculated probabilities (and cross sections and allowed eigenvalues etc.) are really everything that is meaningful & “real” from a physical vantage point – 

5

The online reference to link Darrengol could be useful to see the problems H. was addressing at that time. Also a read of Sommerfeld paper, to see the “elipses” of the relativistic atom and how problematics they were, can be illuminating. –


Heisenberg’s paper is deriving its results from an assumption which is stated only obliquely in the paper, and which is central for all the conclusions. This assumption is explained more clearly on Wikipedia.

Heisenberg is dealing with the orbit of an electron in the atom. Let us assume that this orbit is precise, so that the electron has a position on the m-th Bohr orbit as a function of time is Xm(t). The motion is periodic, so you can Fourier transform this motion to get a Fourier series for the electron’s position

X(t)=∑neinωtXmn

The quantity Xmn is the n-th Fourier coefficient of the m-th Bohr orbit. This quantity is associated with the frequency nω where ω=2π/T is the classical orbit (radian) frequency and T is the classical orbital period. Notice that the classical Fourier frequencies are multiples of a least common multiple, which is (2π times) the reciprocal period.

The fundamental reason Heisenberg rejects this description (which is very close to Bohr’s original idea, and developed by Kramers and Heisenberg) is the fact that these integer spaced frequencies nω are not observed in atomic transitions.

the frequencies that you do observe are the quantum frequencies, which are the difference in energy between the n-th Bohr orbit and the m-th Bohr orbit. There is a fundamental mismatch between the classical orbital description with its integer tower of frequencies and the observed electromagnetic wave emission of the atom, which has a completely different non-integerly spaced collection of frequencies.

The quantum frequencies are given by En−Em, the difference in energy of the n-th and m-th orbit, which however do become integer spaced when n and m are both large. In this limit, called the correspondence limit, En−Em=∂E∂J(n−m) where the partial derivative is of the classical energy with respect to the classical action variable J.

So in the correspondence limit, the classical orbit description is valid, because the frequencies you observe in atomic transitions match the frequencies you would deduce by Fourier transforming a sharp classical orbit.

But what about at smaller quantum numbers? Here Heisenberg makes a radical new assumption. He takes the quantities Xnm, which are the n-th Fourier coefficient of the m-th Bohr orbit, and says that they appear in quantum mechanics with the frequency En−Em, not with the frequency 2πnT! This idea is already present in Bohr to some extent, even in 1913 Bohr states that the transition from orbit n to orbit m should correspond to the classical Fourier component of motion somehow, but Bohr does not develop this idea fully, leaving it vague.

Heisenberg then states that if X_{nm} are quantum Fourier coefficients, then it is immediate that their time development should be

Xnm(t)=ei(En−Em)tXnm(0)

Here you can recognize the Heisenberg equation of motion for the matrix elements of X. This is required by the correspondence principle, to match the frequency of classical Fourier coefficients for large orbits. It is also incompatible with the picture of sharp orbits, because the X matrix elements no longer make integer-spaced towers which can be used to reconstruct a periodic classical orbit. Further, the coefficients with opposite frequencies are complex conjugates of each other Xmn=X∗nm, in the classical picture, it would be Xm,n=X∗m,−n.

Part of the difference is a trivial shifting: the classical n=0 point is shifted to n=m in the matrix description, just because the near-diagonal part is the classical motion, not the 0 column. This shifting is expressed by the correspondence rule that Xclm,n=Xm(m+n), where the left hand side is the classical Fourier coefficients, and the right hand side is the quantum matrix elements. But even with this shifting, the conjugation relations are off.

The complex conjugation in QM reflects along the diagonal of the matrix, it doesn’t reflect the horizontal row along a vertical line running down the middle. You can see how the classical limit emerges by looking at large m,m+p in the matrix, The reflection to m+p,m is p units away from the diagonal to the left, while the original position is p units to the right. So when the rows become continuous and the columns stay discrete, the complex conjugation relations reproduce those of classical mechanics on the Fourier coefficients.

But things are not quite right, because the stuff to the left of the midpoint in a given row is not the complex conjugate of the right. This means that if you try to write down the classical orbit as a function of time, you will fail, producing complex quantities which are not periodic, just some nonsense.

It is important to see Heisenberg’s intuition here— he was sure that the quantum Xmn is a complete description of the quantum motion, but it does not include the classical orbits. His conviction is that the orbit was not a part of the description, that it was a redundant classical idea that was no longer useful, and the fact that his description did not allow you to reproduce the orbit was a positive sign, not an incompleteness.

Other stuff in the paper

The next step is to derive the multiplication law. This is explained on Wikipedia, but it is pretty obvious from the classical law for multiplying Fourier series by convolution. The result is matrix multiplication.

Heisenberg then derives the on-diagonal part of the canonical commutation relations from some complicated radiation sum-rules he did with Kramers. The derivation on Wikipedia is more elementary, but uses essentially the same ingredients, without relying on Kramers-Heisenberg sum rules, and without doing ad-hoc tricks like differentiating with respect to n. The derivation of the on-diagonal canonical commutation relation is the main hurdle that makes this paper magical— it is difficult to follow, you need to do it a different way today.

Why uncertainty?

The uncertainty principle, although only explicitly formulated in 1927, is already present in 1925 to a large extent, except not stated in terms of complementary variables.

Heisenberg’s matrices only allow you to reconstruct a fuzzy orbit, it is only a classical periodic orbit to the extent the the frequencies are integer spaced. So for Heisenberg, the quantities which are “uncertain” are not uncertain yet in a statistical sense (that comes later, after Born’s interpretation of the wavefunction), but they are uncertain in the sense that they cannot be reconstructed in a quantum system.

Heisenberg would have said that the momentum is also uncertain, because the momentum fourier series cannot be reconstructed from the matrix elements of P. The energy would be certain, because the energy levels are precise in the description (ignoring back-reaction from the EM field emissions).

This is an artifact of the fact that Heisenberg was working in frequency space, so that the Hamiltonian was diagonal. In this picture, every quantity which does not commute with H would be considered uncertain, because it would necessarily have off-diagonal matrix elements that do not allow you to reconstruct it’s time variation precisely.

This concept of uncertainty is not the same as the 1927 uncertainty, which came after further developments clarified the notion of state. In 1925, Heisenberg wan’t sure how to describe the state, he could only describe the analogs of classical motion in the Bohr orbits.

So the notion of fuzziness of quantity in the 1925 paper should be considered an ill-definedness of the classical quantity as a function of time, not as a statistical statement about the values of observation of that quantity (at least not yet).

El advenimiento del microcosmos

Название книги: Microcosm The Quantum Revolution In Economics And Technology . Pagina 30

Автор: Gilder, George -

Ключевые слова: Filosofia Ensayo

The “informativeness” of subatomic matter is the key to modem electronics. Because the world of the microcosm consists not of inert and opaque solids but of vibrant, complex, and comprehensible fields, it constitutes a useful arena for modern information technology. It is because there is so much analogical information in the microcosm that the microcosm is a uniquely powerful medium for information.The quantum-regulated movement of electrons across quantum-mapped crystalline paths epitomizes information technology, and in-formation technology epitomizes the quantum era.

In the atom of information, this era acquires its definitive symbol. What was once a blank solid is now revealed in part as information, what was once an inert particle now shines with patterns and probabilities, what was once opaque and concrete is now a transparent tracery of physical laws. Far from plunging reality into clouds, quantum theory makes the universe radically more intelligible. The new science does not estrange human beings from their environment. Since thought is the most distinctly human power, the quantum world is actually more anthropomorphic than the world of Newtonian masses and forces.

A more intelligible universe, penetrable by the human mind, endows people with greater power to create wealth. But it also radically changes the way wealth is created. Throughout previous human history, the creation of wealth depended chiefly upon the extraction, transport, combination, and modification of heavy materials against the resistance of gravity, the constraints of entropy, and the constrictions of time and space. When things are large and approached out-side-in, it is expensive to move and manipulate them. Their costs derive from the weight, rarity, entropy, and resistance of their matter. But small things, virtually devoid of matter, move less like weights than like thoughts. In the microcosm, the costs of fuel and materials decline drastically; the expense devolves from matter to mind. Just as quantum science overthrew Newtonian matter in the explanation of the universe, the quantum economy overthrows Newtonian matter in the creation of new wealth.

El transistor (bardeen y los colaboradores en el premio nobel)

El transistor (bardeen y los colaboradores en el premio nobel)

En el centro de todo esta el electron.

Most interested people understand much of what electrons do. But very few have any clear idea of what an electron actually is, or its implications for the concept of matter and its overthrow in the world economy.

From the telephone to the human brain, from the television set to the computer, information mostly flows in the form of electrons. This function of electrons has quantum roots. As in Planck’s black body radiation, electrons do not respond to applied energy in a continuous, proportional, or linear way. They are non-linear; they have quantum thresholds and resonances. These quantum functions shape their electrical properties. In order to move through a solid, electrons must be freed from their atoms, jumping from one energy state to a free state across measurable energy “band gaps” in strict accordance with quantum rules. These rules give electrons identifiable and controllable features that can be used to convey information.

With controlled pulses of electrons down wires, computers could be interconnected around the world. With controlled flows of electrons in and out of tiny capacitors, computer memories could be constantly read, written, and restored.

Crossing decisively into the microcosm, Heisenberg declared that the waves which Bohr had examined in recreating the atom were not conventional waves at all. Designated “probability amplitudes,” they were waves or fields that defined the statistical likelihood of finding an electron at any particular location. This was a climactic step in the overthrow of materialism in physics. With the electron itself depicted as a wave and the wave depicted as a probability field, the specific particle in this theory had disappeared into a cloud. With it disappeared the last shreds of Newtonian logic and mechanistic solidity.

As Bohr put it, quantum theory required “a final renunciation of the classical idea of causality and a radical revision of our attitude toward the problem of physical reality.”

Microcosm The Quantum Revolution In Economics And Technology .  Pagina 25

Los encargados de aplicar en forma práctica la teoría cuántica, fueron los inventores del transistor en 1948, lo cual significó un hito en la historia del desarrollo de las tecnologias de información.

John Bardeen (1908-1991), William B. Shockley (1910-1989), and Walter H. Brattain (1902-1987)

Bardeen en su Nobel Lecture de 1947 establece que en la raiz de toda la investigacion que condujo al desarrollo del primer transistor estuvo la Wilson’s quantum mechanical theory, based on the energy band model, and describing conduction in terms of excess electrons and holes. It is fundamental to all subsequent developments. The theory shows how the concentration of carriers depends on the temperature and on impurities.

JO H N BA R D E E N Semiconductor research leading to the point contact transistor

Nobel Lecture, December 11, 1956

Del mecanicismo causistico a lo probabilistico

Del mecanicismo causistico a lo probabilistico

Today most sophisticated people imagine that they have transcended Newton and have come to terms with the findings of modern science. But they have not. As an intellectual faith, materialist logic still prevails.

We still believe that the solid world we see and feel—governed by determinate chains of cause and effect, rooted in Newtonian masses and forces—is real and in some sense definitive. The atom may not be ultimate, but they assume some other particle is, perhaps the quark.

At the foundations of the physical world, so it is supposed, are physical solids—”building blocks”—that resemble in some way the solids we see. They link together in causal chains of mechanical logic like a set of cogs and levers. These solids are deemed to comprise all matter, from atoms and billiard balls to bricks and the human brain.

Announced in 1913 and proved for the single electron of the hydrogen atom, the Bohr model was the first great vindication of quantum theory. One test of scientific advance is whether it extends the realms of human understanding and control.

The established physics could not explain the effectiveness of chemistry, let alone extend it to atoms. Unlike a solar system, atoms do not exist in majestic isolation. Ceaselessly in movement, they endlessly jiggle together in what is called Brownian motion. We even step on them. In a world of Newtonian continuities, electron orbits would vary continually as atoms collided with one another. Constantly knocked loose in these collisions, electrons in a conductor should flow far more copiously and respond to heat more massively than experiments showed.

Reunifying chemistry and physics in the microcosm, the new model of the atom explained the apparent solidity of the physical world. Establishing a gap, called a band gap, between an electron in its ground state and an electron excited to a higher energy level, the new physics showed why the constant collisions of atoms do not cause the atomic structure to collapse. A small collision will not affect an atom. An electron will not respond to any small disturbance. It will react only if it receives its necessary quantum of energy, defined by its resonant frequency times Planck’s constant.

Microcosm The Quantum Revolution In Economics And Technology .  Pagina 21

Heisenberg y los observables

Heisenberg y los observables

Generalmente es reconocido el hecho de que el papel publicado en julio de 1925 por Werner Heisenberg es el que di fin a la “Teora Cuntica Vieja”, y que con la exposicin de su Mecnica Matricial se di entrada a la Mecnica Cuntica tal como se conoce y se practica en la actualidad.

Heisenber mismo, en su famoso articulo fundacional de la nueva mecánica cuántica, resaltaria el hecho de que “it is well known that the formal rules which are used in quantum theory for calculating observable quantities such as the energy of the hydrogen atom may be seriously criticzised on the grounds that they contain, as basic element, relationships between quantities that are apparently unobservable in principle, e.g., position and period of revolution of electron... Experience however shows that only the hydrogen atom and its Stark effect are amenable to treatment by these formal rules of quantum teory”.

Heisenberg también critica el principio de correspondencia: “It has become the practice to characterize this failure of the quantum-theoretical rules as a deviation from classical mechanics, since the rules themselves were essentially derived from classical mechanics.”

Y en concordancia con Husserl:

On doit s'accommoder du fait que ce n'est qu' travers le processus de connaissance lui-mme que se dcide ce qu'on doit entendre par "connaissance". [...] Toute formulation dans le langage est toujours, non seulement une saisie de la ralit, mais aussi une manire de la mettre en forme et de l'idaliser [...] La connaissance n'est sans doute en dernire instance rien d'autre que l'agencement non pas l'agencement de quelque chose qui serait dj disponible en tant qu'objet de notre conscience ou de notre perception, mais plutt l'agencement de quelque chose qui ne devient un veritable contenu de conscience ou un processus perceptif qu' travers cet agencement meme (Whm, 363-364)

"Philosophie. Le manuscrit de 1942"

Heisenberg, W. Philosophie. Le manuscrit de 1942. Introduction et traduction par C. Chevalley (490 p.). Editions du Seuil, 1998. Premire dition en allemand : Ordnung der Wirklichkeit, 1989. Seconde dition francaise : Arla, 2003, 2010 (173 p.)

da el paso fundamental que destraba el callejon sin salida a que habia llegado la mecanica cuantica.

“In this situation it seems sensible to discard all hope of observing hitherto unobservable quantites, such as the position and period of electron, and to concede that the partial agreement of the quantum rules with experience is more or less fortuitous. Instead it seems more reasonable to try to establish a theoretical quantum mechanics, analogous to classical mechanics, but in which only relations between observable quantities occur.”

Dando a luz la mecánica matricial. Y abriendo una nueva dimension para la fisica.

Quantum-theoretical re-interpretation of kinematic and mechanical relations. W.Heisenberg

As hitherto defined, quantum mechanics enables the radiation

emitted by the atom, the energy values of the stationary states, and other

parameters characteristic for the stationary states to be treated. The theory

hence complies with the experimental data contained in atomic spectra. In all those cases, however, where a visual description is required of a transient event, e.g. When interpreting Wilson photographs, the formalism of the theory does not seem to allow an adequate representation of the experimental state of affairs. At this point Schrödinger’s wave mechanics, menawhile developed on the basis of the de Broglie’s theses, came to the assistance of quantum mechanics.

Tal como Heisenberg lo considera, el cambio de la mecánica clásica a la mecánica cuántica fue así:

In classical physics the aim of research was to investigate objective processes occurring in space and time, and to discover the laws governing their progress from the initial conditions. In classical physics a problem was considered solved when a particular phenomenon had been proved to occur objectively in space and time, and it have been shown to obey the general rules of classical physics as formulated by differential equations.

The manner in which the knowledge of each process had been acquired, what observations may possibly have led to its experimental determination, was completely immaterial, and it was also immaterial for the consequences of the classical theory, which possible observations were to verify the predicitions of the theory. In the quantum theory, however, the situation is completely different. The very fact that the formalism of quantum mechanics cannot be interpreted as visual description of a phenomenom occurring in space and time shows that quantum mechanics is in no way concerned with the objective determination of space - time phenomena. On the contrary, the formalism of quantum mechanics should be used in such a way that the probability for the outcome of a further experiment may be concluded from the determination of an experimental situation in an atomic system, providing that the system is subject to no perturbations other than those necessitated by performing the two experiments.

Nobel Lecture 1939 Heisenberg.

La catástrofe del átomo de Bohr.

La catastrofe del atomo de Bohr.

Bohr cuantizó las orbitas planetarias enunciando el principio, más fundamental de que el ímpetu angular del sistema es un múltiplo entero de la constante de Planck (entre 2p); pero, a pesar de su gran éxito en los átomos hidrogenoides, finalmente se debió concluir que esta cuantización es incorrecta.

En general, el modelo de Bohr:

1. No explica de donde surge la relación mvr = nh/2p ; con lo cual sus resultados son asombrosamente congruentes con los hechos experimentales.
   2. Solo es aplicable para el átomo de hidrógeno, para átomos más complejos sus ecuaciones resultan insatisfactorias
   3. Cuando el espectro del átomo de sodio se examina con un espectroscopio de alta resolución, la línea original se descompone en dos, lo que no se explica en su teoría.
   4. La teoría de Bohr no explica por que cuando los espectros de emisión atómica se observan en presencia de un campo magnético surge una multiplicidad de líneas espectrales (efecto Zeeman).
   5. No explica porque algunas líneas espectrales son mas brillantes que otras.
   6. Existía una incoherencia lógica en su teoría , pues al lado de los principios fundamentales de la Física Clásica y el electromagnetismo, se introdujeron postulados nuevos (momento angular y condición de frecuencia) que entraban en contradicción con los principios de los cuales partía.

Las reglas de la cuantización (momento angular y condición de frecuencia) se añadierón a la Física Clásica sin ninguna liga lógica. Bohr mismo hizo un examen crítico de su teoría a fin de mostrar a los jóvenes físicos la necesidad de buscar los principios de la teoría de los fenómenos atómicos

Con el advenimiento de la mecánica ondulatoria de De Broglie y la ecuación de Schrodinger los estados dinámicos quedaron limitados automáticamente a la serie postulada por Bohr y sólo se abandona el concepto clásico del electrón "planetario".

QG-Mendeleiev, Genaro carmona.

Modelo atómico de Niels Bohr

http://wwwgcmendeleiev.blogspot.ru/2009/02/modelo-atomico-de-niels-bohr.html

El modelo atomico de Bohr y el principio de correspondencia.

3. De las magnitudes fundamentales a los observables.

El modelo atomico de Bohr y el principio de correspondencia.

Niels Bohr en su artículo “On the constitution of atoms and molecules” de Julio de 1913 aplicó por primera vez la hipótesis cuántica a la estructura atómica, a la vez que buscó una explicación a los espectros discontinuos de la luz emitida por los elementos gaseosos. Todo ello llevó a formular un nuevo modelo de la estructura electrónica de los átomos partiendo del modelo de su maestro Rutherford.

El modelo de Bohr implicaba los siguientes postulados:

1.- Mientras que en la mecánica clásica la energía del electrón podía tener cualquier valor, en la nueva mecánica el electrón tiene definidos ciertos estados estacionarios de movimiento (niveles de energía) que le son permitidos; cada uno de estos estados estacionarios tiene una energa fija y definida.

2.- Cuando un electrón está en uno de estos estados no irradia pero cuando cambia de estado absorbe o desprende energía.

3.- En cualquiera de estos estados, el electrón se mueve siguiendo una órbita elíptica (o circular, un caso especial de la elipse) alrededor del núcleo.

4.- Los estados de movimiento electrnico permitidos son aquellos en los cuales el momento angular del electrón (m v r ) son un múltiplo entero de h/2pi .

Con este modelo explicó el espectro discontinuo del hidrogeno, como la emisión de fotones emitidos por el electrón al brincar de un nivel energético a otro.

Sin embargo el modelo propuesto no explicaba los espectros de átomos más complejos.

“On the constitution of atoms and molecules”. Niels Bohr. Phylosophical Magazine. Series 6, Volume 26. July 1913, p. 1-25

Ese mismo año, en diciembre de 1913, en la Sociedad Física de Copenhague, Bohr expuso el principio de correspondencia entre la nueva teoría cuántica y la electrodinámica clásica. Postulando una relación entre las teorías cuántica y clásica mendiante alguna condición límite.

Las leyes de la mecánica cuántica describen objetos microscópicos tales como átomos y partículas elementales, mientras que una variedad de sistemas macroscópicos (sólidos rígidos, condensadores eléctricos, etc.) pueden ser descritos con exactitud por teorías clásicas tales como la mecánica clásica y el electromagnetismo. Por el contrario, es razonable creer que las máximas leyes de la Física deben de ser independientes del tamaño del objeto físico descrito. Así que la física clásica debe de emerger como una aproximación a la física cuántica a medida que los sistemas aumentan de tamaño.

Considérese dos estados estacionarios de un átomo con energías digamos, E₁ y E₂. Si ocurre una transición atómica entre ellos la radiación emitida tiene una frecuencia v, = h^-1 ( E₂-E₁), donde h es la constante de Planck. Si ahora nos desplazamos hacia la región del espectro de energías donde la separación entre dos niveles consecutivos es cada vez menor, la radiación que se emite tiene una frecuencia cuyo valor es cada vez más próximo al que se obtiene de las ecuaciones de la electrodinámica clásica al suponer que la trayectoria de una partícula cargada (el electrón) se curva suavemente hacia el interior de su órbita. Con esta idea, Bohr pudo conciliar los complejos problemas que se originaron por el descubrimiento del cuanto de luz y el del núcleo atómico de Rutherford cuando el cuanto de acción de Planck es muy pequeño comparado con la acción que aparece en el sistema por describirse, hay una reconciliación entre la descripción clásica de la naturaleza, que contiene la regla de que la naturaleza no "pega de brincos", con la forma discontinua en que el campo de radiación y un átomo intercambian energía.

El paradigma de esta culminación del principio de correspondencia es la ecuación de Schrodinger: se construye desde la ecuación clásica de ondas hamiltoniana, añadiendo la proporcionalidad cuántica entre la energía y la frecuencia de ondas introducida por Einstein. La forma de la ecuación es clásica (es una ecuacin diferencial con variables continuas), pero sus efectos son discretos (mide los niveles de energía de un átomo).

El principio de correspondencia. Ricardo Sanchez Ortiz de Urbina.

LA CONTRIBUCIÓN DE NIELS BOHR A LA MECÁNICA CUÁNTICA LEOPOLDO GARCÍA-COLÍN

http://www.123helpme.com/view.asp?id=101558

http://dwb.unl.edu/Teacher/NSF/C04/C04Links/www.fwkc.com/encyclopedia/low/articles/q/q021000030f.html

Pozos Cuanticos. Particula en un pozo de potencial, infinito, finito.

La energia de una particula atrapada en una caja de paredes energeticas de altura infinita, esta cuantizada.
Asi pasa cuando se atrapa una particula en una caja de paredes finitas, solo que la longitud de onda es mayor que la distancia entre las paredes, por el efecto tunel.

En estos videos se explica, y se ven ejemplos de esto. No es perdida de tiempo verlos.