"In retrospect, he's glad. "Part of my luck was that I couldn't keep up with them," he said. "They were proving theorems, but they had nothing to do with the essence of the situation." Hofstadter instead decided to test out a more down-to-earth approach. Rather than proving theorems, he was going to crunch some numbers using an HP 9820A desk calculator-a computerlike machine that weighed nearly 40 pounds and could be programmed to perform complex computations."
"When fed certain information about an electron and its environment as inputs, the Schrödinger equation describes how the electron will behave. In particular, its solutions tell you the amount of energy the electron can have. In the case that Hofstadter was interested in, the Schrödinger equation includes a variable called alpha, the product of the magnetic field's strength and the area of one grid square."
A difficult proof resolved the Ten Martini Problem by using number theory to explain quantum fractals in electron spectra. Electrons in a crystal lattice near a magnetic field have energy levels determined by a Schrödinger equation that depends on a parameter alpha equal to magnetic-field strength times unit-cell area. Numerical computation of that equation revealed a fractal pattern of allowed energies, known as the Hofstadter butterfly. The analysis connects arithmetic properties of alpha to the structure of the spectrum and shows the spectrum can form a Cantor set, linking spectral theory and deep number-theoretic techniques.
#ten-martini-problem #hofstadter-butterfly #number-theory #quantum-fractals #almost-mathieu-operator
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