History of Quantum Structures and IQSA
1. The Birth of Quantum Mechanics
It was June 7, 1925, that Werner Heisenberg left for the North Sea island of Helgoland wanting to find some rest after a bad attack of hay fever. Heisenberg was working at that time on the spectral lines of hydrogen, trying to find a manner to calculate these lines in a consistent way. In Helgoland, although he went there to rest, he got completely obsessed by the problem, and he hardly slept, deviding his time between working on his problem, engaging in mountain climbing and learning by heart poems from Goethe's West Osticher Divan.
It was one of these nights that Heisenberg 'invented' modern quantum mechanics. He wrote later in his book 'Der Teil und das Ganze' : "It was about three o' clock at night when the final result of the calculation lay before me. At first I was deeply shaken. I was so excited that I could not think of sleep. So I left the house and awaited the sunrise on the top of a rock."
On June 9 Heisenberg returned to Gottingen and sent a copy of his results to Wolfgang Pauli, commenting in the accompanying letter: "Everything is still vague and unclear to me, but it seems as if the electrons will nomore move on orbits".
On July 25, Heisenberg's paper announcing the invention of quantum mechanics is received by the Zeitschrift fur Physik . Before that he had also given a copy of the paper to Max Born commenting "that he had written a crazy paper and did not dare to send it in for publication, and that Born should read it and advice him on it."
Born mentiones that at first he was completely astonished by the strangeness of the calculations that Heisenberg proposes in the paper. But then, one morning, on July 10, Born suddenly realized that the type of calculation that Heisenberg proposes corresponds exactly to the matrix calculation that had been invented by mathematicians a long time before. Then Born reformulates, together with one of his students Pascual Jordan, Heisenberg's results in formal matrix language, to give rise to the first formal formulation of the new quantum mechanics .
It is amazing to know that shortly after Born received a copy of a paper written by a young British physicist that he did not know, Paul Adrien Dirac, which contained many of the results that he and Jordan just derived from Heisenberg's calculations . Dirac had already in this first paper on quantum mechanics introduced a much more abstract mathematical language than matrix mechanics, it were the first steps finally leading to von Neumann's abstract Hilbert space formulation.
When Heisenberg wrote the first paper on quantum mechanics he had not known about matrix mathematics, but rapidly caught up, and started to work together with Born and Jordan on elaborating further the mathematical aspects of the theory, and also Pauli got caught up in the new physics. In the fall of 1925 he derived for the first time the complete Balmer formula for the hydrogyn atom (the set of discrete energy levels of an electron bound in hydrogen) .
Erwin Schroedinger did not know anything of all these happenings. He was also working on the problem of the hydrogyn atom but starting from a completely different approach. Schroedinger was, already long before he came with the 'second' invention' of quantum mechanics, actively interested in the problem of the description of the atom. He was inspired in his approach by work of Louis de Broglie and Albert Einstein, considering the wave aspects of quantum particles. His goal was to formulate quantum mechanics as a part of classical wave mechanics, where the particle behavior of quantum entities would correspond to the behavior of singularities of the waves.
And so Schrodinger indeed manages to present a wave model of the atom and to also derive the complete Balmer formula, as Pauli did at the same time by means of matrix mechanics: the foundations of Schrodinger's wave mechanics was laid . During the next half year, Schrodingers paper on the foundations of wave mechanics was followed by three other papers, containing elaborations of the mathematical aspects of the formalism and applications to new problems [7, 8, 9]. It became clear that wave mechanincs and matrix mechanics gave identical results, also in problems other than the description of the hydrogen atom. And of course the question arose: what do these theories, founded on completely different conceptual assumptions, have in common? Schrodinger  investigated the similarities of matrix mechanics and wave mechanics, and could show that indeed they will lead to similar results in all conceivable situations.
The mystery of how such conceptually completely different theories, expressed in formalism formulated by means of a completely different mathematical appratus, could give rise to identical results was only completely understood however after John von Neumann formulated the operator algebra version of quantum mechanics in 1932 . That is also the place where Hilbert space, as a mathematical structure, was introduced into the formulation of quantum mechanics.
W. Heisenberg, "Der Teil un das Ganze", Piper, Munich, (1969).
W. Heisenberg, Zeitschr. Phys., 33, 879, (1925).
M. Born and P. Jordan, Zeitschr. Phys., 34, 858, (1925).
P.A.M. Dirac, Proc. Roy. Soc. A, 109642, (1925).
W. Pauli, Zeitschr. Phys., 36, 336, (1926).
E. Schrodinger, Ann. der Phys., 79361, (1926).
E. Schrodinger, Ann. der Phys., 79489, (1926).
E. Schrodinger, Ann. der Phys., 80437, (1926).
E. Schrodinger, Ann. der Phys., 81109, (1926).
E. Schrodinger, Ann. der Phys., 79734, (1926).
J. von Neumann, Mathematische Grundlagen der Quanten-Mechanik, Springer-Verlag, Berlin, 1932.