## Adapting de Broglie waves

De Broglie's ideas applied only to free electrons, and Schrödinger was faced with the problem of adapting the wave picture to electrons which are subject to forces, and in particular to electrons in an atom such as hydrogen. The effect of forces was represented by 'potential wells' within which the wave function had to be accommodated. An even more important and basic problem was to express the laws of physics in the form of a mathematical relation which the wave functions have to satisfy. Using clues provided by classical mechanics, and in particular the classical methods developed by William Rowan Hamilton (1805-1865), Schrödinger was able to construct an equation which described the laws of Nature in terms of the wave representation.

Over a period of just a few months Schrödinger had applied his theory to explain the spectral wavelengths of the hydrogen atom, although initially he had some difficulties in predicting the observed spectrum, which we now know has a fine structure caused by the spin of the electron. In about a year he had developed practically the whole of what is now known as non-relativistic wave mechanics.

It is difficult to find in the history of physics two theories which appeared to differ more radically than Heisenberg's matrix mechanics, and Schrödinger's wave mechanics, and yet both correctly interpreted the same experimental results. This could hardly be a coincidence; in fact Schrödinger soon discovered that they were just two different mathematical ways of looking at the same thing. In Spring 1926 he published a paper in Annalen der Physic entitled 'On the Relationship of the Quantum Mechanics of Heisenberg-Born-Jordan to mine', which proved the formal mathematical identity of the two representations.

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