Heisenberg's entryway to matrix mechanics

For more context, see Introduction to quantum mechanics.

For complete information on the specific topic in quantum physics, see Matrix mechanics.

Werner Heisenberg contributed to science at a point when the old quantum physics was discovering a field littered with more and more stumbling blocks. He decided that quantum physics had to be re-thought from the ground up. In so doing he excised several items that were grounded in classical physics and its modeling of the macro world. Heisenberg determined to base his quantum mechanics "exclusively upon relationships between quantities that in principle are observable." By so doing he constructed an entryway to matrix mechanics.

He observed that one could not then use any statements about the such things as "the position and period of revolution of the electron." Rather, to make true progress in understanding the radiation of the simplest case, the radiation of excited hydrogen atoms, one had measurements only of the frequencies and the intensities of the hydrogen bright-line spectrum to work with.

In classical physics, the intensity of each frequency of light produced in a radiating system is equal to the square of the amplitude of the radiation at that frequency, so attention next fell on amplitudes. The classical equations that Heisenberg hoped to use to form quantum theoretical equations would first yield the amplitudes, and in classical physics one could compute the intensities simply by squaring the amplitudes. But Heisenberg saw that "the simplest and most natural assumption would be" to follow the lead provided by recent work in computing light dispersion done by Kramers. The work he had done assisting Kramers in the previous year now gave him an important clue about how to model what happened to excited hydrogen gas when it radiated light and what happened when incoming radiation of one frequency excited atoms in a dispersive medium and then the energy delivered by the incoming light was re-radiated — sometimes at the original frequency but often at two lower frequencies the sum of which equalled the original frequency. According to their model, an electron that had been driven to a higher energy state by accepting the energy of an incoming photon might return in one step to its equilibrium position, re-radiating a photon of the same frequency, or it might return in more than one step, radiating one photon for each step in its return to its equilibrium state. Because of the way factors cancel out in deriving the new equation based on these considerations, the result turns out to be relatively simple.

To make a long and rather complicated story short, Werner Heisenberg used the idea that since classical physics is correct when it applies to phenomena in the world of things larger than atoms and molecules, it must stand as a special case of a more inclusive quantum theoretical model. So he hoped that he could modify quantum physics in such a way that when the parameters were on the scale of everyday objects it would look just like classical physics, but when the parameters were pulled down to the atomic scale the discontinuities seen in things like the widely spaced frequencies of the visible hydrogen bright line spectrum would come back into sight.

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