Eugene Wigner wrote famously about the unreasonable effectiveness of mathematics in science. His own ideas illustrate the point as well as any others. In 1956, at a conference on neutron physics in Gatlinburg, Tennessee, Wigner spoke on the energy levels of large, complex nuclei such as uranium, for which data were just becoming available. He expressed the view that a good deal might be learned by making a virtue of theoretical ignorance, and simply assuming random values for the elements of the Hamiltonian matrix, which in quantum theory determines the nuclear energy levels through its eigenvalues.
Wigner showed that this 'simple minded' approach could establish baseline expectations for the spacing of nuclear levels in the absence of any other knowledge. “The question”, he noted, “is simply, what are the distances of the characteristic values of a symmetric matrix with random coefficients?” Wigner's result, worked out in a few lines of algebra, gave a probability distribution of the form p(x) ∼ xexp(−ax2), with x being the energy spacing and a = π/4, thereby pointing to a dearth of levels of similar energy. The result contrasted sharply with what might have been the expected Poisson form, p(x) ∼ exp(−x), for which x = 0 would be a maximum rather than a minimum.
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