Development of matrix mechanics

                        In 1925, Werner Heisenberg, Max Born, and Pascual Jordan formulated the matrix mechanics representation of quantum mechanics.

Epiphany at Helgoland

In 1925 Werner Heisenberg was working in Göttingen on the problem of calculating the spectral lines of hydrogen. By May 1925 he began trying to describe atomic systems by observables only. On June 7, to escape the effects of a bad attack of hay fever, Heisenberg left for the pollen free North Sea island of Helgoland. While there, in between climbing and learning by heart poems from Goethe's West-östlicher Diwan, he continued to ponder the spectral issue and eventually realised that adopting non-commuting observables might solve the problem, and he later wrote 

"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."

The Three Fundamental Papers

After Heisenberg returned to Göttingen, he showed Wolfgang Pauli his calculations, commenting at one point: 

"Everything is still vague and unclear to me, but it seems as if the electrons will no more move on orbits."

On July 9 Heisenberg gave the same paper of his calculations to Max Born, saying, "...he had written a crazy paper and did not dare to send it in for publication, and that Born should read it and advise him on it..." prior to publication. Heisenberg then departed for a while, leaving Born to analyse the paper. 

In the paper, Heisenberg formulated quantum theory without sharp electron orbits. Hendrik Kramers had earlier calculated the relative intensities of spectral lines in the Sommerfeld model by interpreting the Fourier coefficients of the orbits as intensities. But his answer, like all other calculations in the old quantum theory, was only correct for large orbits.

Heisenberg, after a collaboration with Kramers,  began to understand that the transition probabilities were not quite classical quantities, because the only frequencies that appear in the Fourier series should be the ones that are observed in quantum jumps, not the fictional ones that come from Fourier-analyzing sharp classical orbits. He replaced the classical Fourier series with a matrix of coefficients, a fuzzed-out quantum analog of the Fourier series. Classically, the Fourier coefficients give the intensity of the emitted radiation, so in quantum mechanics the magnitude of the matrix elements of the position operator were the intensity of radiation in the bright-line spectrum.

The quantities in Heisenberg's formulation were the classical position and momentum, but now they were no longer sharply defined. Each quantity was represented by a collection of Fourier coefficients with two indices, corresponding to the initial and final states. When Born read the paper, he recognized the formulation as one which could be transcribed and extended to the systematic language of matrices,  which he had learned from his study under Jakob Rosanes  at Breslau University. Born, with the help of his assistant and former student Pascual Jordan, began immediately to make the transcription and extension, and they submitted their results for publication; the paper was received for publication just 60 days after Heisenberg’s paper.  A follow-on paper was submitted for publication before the end of the year by all three authors.  (A brief review of Born’s role in the development of the matrix mechanics formulation of quantum mechanics along with a discussion of the key formula involving the non-commutivity of the probability amplitudes can be found in an article by Jeremy Bernstein.  A detailed historical and technical account can be found in Mehra and Rechenberg’s book The Historical Development of Quantum Theory. Volume 3. The Formulation of Matrix Mechanics and Its Modifications 1925–1926.)