Hofstadter's Quantum-Mechanical Butterfly

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This Demonstration shows the quantum‐mechanical energy spectrum of an electron in a two-dimensional periodic potential with a perpendicular magnetic field. The fractal nature of this system was discovered by Douglas Hofstadter in 1976.

Contributed by: Enrique Zeleny (July 2012)
Open content licensed under CC BY-NC-SA



The Schrödinger equation for this system takes the form

where is a pseudodifferential operator, is the potential, and is the energy eigenvalue, with given by the ansatz

for integers .

The problem reduces to the solution of the recursive equation



where is the energy, , with being the denominator of nonrepeating rational numbers . Finally, the eigenvalue condition for the energy spectrum is


The plot has energy on the axis and the parameter on the axis.


[1] D. Hofstadter, "Energy Levels and Wave Functions of Bloch Electrons in Rational and Irrational Magnetic Fields," Physical Review B, 14, 1976 pp. 2239–2249. doi:10.1103/PhysRevB.14.2239.

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