Angel "Java" Lopez en Blog

Publicado el 24 de Septiembre, 2014, 14:44

Ya apareció en este blog el tema operadores, de forma leve, en:

Operadores en Mecánica Cuántica, por Richard Feynman
Operadores en Mecánica Cuántica, por Roger Penrose

y va a seguir apareciendo, seguramente en mi series Matemáticas y Física Cuántica y Ecuaciones Diferenciales. Uno de los que contribuyó a la teoría de los operadores fue John von Neumann, al que estoy citando en Fundamentos Matemáticos de la Mecánica Cuántica, por John Von Neumann.

Ayer cité al excelente  "Quantum Field Theory I: Basics in Mathematics and Physics", de Eberhard Zeidler. Y hoy lo vuelvo a citar, porque encuentro dos referencias al trabajo de von Neumann, la primera de Dieudonne:

In the fall 1926, the young John von Neumann (1903-1957) arrived in Gottingen to take up his duties as Hilbert's assistant. These were the  hectic years during which quantum mechanics was developing with breakneck speed, with a new idea popping up every few weeks from all over the  horizon. The theoretical physicists Born, Dirac, Heisenberg, Jordan, Pauli, and Shrodinger who were developing the new theory were groping for adequate athematical tools. It finally dawned upon them that their 'observables' had properties which made them look like Hermitean operators in Hilbert space...

Tengo que escribir sobre los espacios de Hilbert y cómo los observables terminan siendo operadores lineales en ese espacio. La condición de hermíticos es la que asegura que la aplicación de operador en un producto definido sobre un vector tenga resultado real, no complejo. Notablemente, Hilbert había desarrollado sus ideas décadas antes de la necesidad de usarlas en la naciente mecánica cuántica.

... and that by an extraordinary coincidence, the 'spectrum' of Hilbert (which he had chosen around 1900 from a superficial analogy) was to be the central conception in the explanation of the 'spectra' of atoms. It was therefore natural that they should enlist Hilbert's help to put some mathematical sense in their formal computations. With the assistance of Nordheim and von Neumann, Hilbert first tried integral operators in the space L2, but that needed the use of the Dirac delta function 5, a concept which was for the mathematicians of that time self-contradictory. John von Neumann therefore resolved to try another approach.

Jean Dieudonne (1906-1992)
History of Functional Analysis



La segunda es de un matemático que también contribuyó al tema, Marshall Harvey Stone:

Stimulated by an interest in quantum mechanics, John von Neumann began the work in operator theory which he was to continue as long as he lived. Most of the ideas essential for an abstract theory had already been developed by the Hungarian mathematician Pryges Riesz...

Conozco levemente el trabajo de Riesz, justamente leyendo algún libro de teoría funcional.

... who had established the spectral theory for bounded Hermitean operators in a form very much like as regarded now standard. Von Neumann saw the need to  extend Riesz's treatment to unbounded operators and found a clue to doing this in Carleman's highly original work on integral operators with singular kernels...

Desconocía el trabajo sobre operadores "unbounded", y su importancia. Tema para estudiar. 

The result was a paper von Neumann submitted for publication to the Mathematische Zeitschrift but later withdrew. The reason for this  withdrawal was that in 1928 Erhard Schmidt and myself, independently, saw the role which could be played in the theory by the concept of the adjoint operator, and the importance which should be attached to self-adjoint operators. When von Neumann learned from Professor Schmidt of this  observation, he was able to rewrite his paper in a much more satisfactory and complete form... Incidentally, for permission to withdraw the paper, the publisher exacted from Professor von Neumann a promise to write a book on quantum mechanics. The book soon appeared and has become one of the classics of modern physics.

Marshall Harvey Stone (1903-1989)

A primera impresión, el libro de von Neumann es más árido que el clásico de Dirac. Pero será cuestión de tener perservarncia y leer ambos, muy interesantes los dos.

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
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Por ajlopez, en: Ciencia