Angel "Java" Lopez en Blog

Publicado el 11 de Mayo, 2014, 16:23

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Venga hoy una nueva serie, que necesito escribir para poner en claro algunos puntos que estoy estudiando. En estos días me encuentro con un libro clásico: "Mathematical Foundations of Quantum Mechanics", de John Von Neumann. Paso en esta serie a compartir y comentar el prefacio. Hoy leo:

The object of this book is to present the new quantum mechanics in a unified representation which, so far as it is possible and useful, is mathematically rigorous. This new quantum mechanics has in recent years achieved in its essential parts what is presumably a definitive form: the so-called "transformation theory." Therefore the principal emphasis, shall be placed on the general and fundamental questions which have arisen in connection with this theory. In particular, the difficult problems of interpretation, many of which are even now not fully resolved, will be investigated in detail* In this context the relation of quantum mechanics to statistics and to the classical statistical mechanics is of special importance. However, we shall as a rule omit any discussion of the application of quantum mechanical methods to particular problems, as well as any discussion of special theories derived from the general theory — at least so far as this is possible without endangering the understanding of the general relationships. This seems the more advisable since several excellent treatments of these problems are either in print or in process of publication.

Para "transformation theory" leer:

The term transformation theory refers to a procedure and a "picture" used by P. A. M. Dirac in his early formulation of quantum theory, from around 1927.[1]

This "transformation" idea refers to the changes a quantum state undergoes in the course of time, whereby its vector "moves" between "positions" or "orientations" in itsHilbert space.[2] Time evolution, quantum transitions, and symmetry transformations in Quantum mechanics may thus be viewed as the systematic theory of abstract, generalized rotations in this space of quantum state vectors.

Remaining in full use today, it would be regarded as a topic in the mathematics of Hilbert space, although, technically speaking, it is somewhat more general in scope. While the terminology is reminiscent of rotations of vectors in ordinary space, the Hilbert space of a quantum object is more general, and holds its entire quantum state.

(The term further sometimes evokes the wave-particle duality, according to which aparticle (a "small" physical object) may display either particle or wave aspects, depending on the observational situation. Or, indeed, a variety of intermediate aspects, as the situation demands.)

Se refiere principalmente al trabajo de P.A.M. Dirac. Von Neumann es consciente de los problemas de interpretación a los que da lugar la teoría de la transformación. Es importante no olvidar ese punto. Por otro lado, von Neumann, como matemático, prefiere renuncia al tratamiento con matrices, y pasar a trabajar con operados. Leo:

On the other hand, a presentation of the mathematical tools necessary for the purposes of this theory will be given, i. e., a theory of Hilbert space and the so called Hermitean operators. For this end, an accurate introduction to unbounded operators is also necessary, i.e., an extension of the theory beyond its classical limits (developed by Hilbert and E. Hellinger, P. Riesz, E. Schmidt, 0. Toeplitz). The following may be said regarding the method employed in this mode of treatment: as a rule, calculations should be performed with the operators themselves (which represent physical quantities) and not with the matrices, which, after the introduction of a (special and arbitrary) coordinate system in Hilbert space, result from them. This "coordinate free, " i.e., invariant, method, with its strongly geometric language, possesses noticeable formal advantages.

Vemos que prefiere el "coordinate free", el trabajar sin unas coordenadas en particular, en un lenguaje geométrico. Gran parte de la física moderna lucha por liberarse de las coordenadas, y describir sus modelos en forma de una geometría sin "punto de origen" o "ejes coordenados". Tengo que estudiar los "unbounded operatros", que parecen necesitar una "extension of the theory".

Nos leemos!

Angel "Java" Lopez

Por ajlopez, en: Ciencia