Hofstadter's Quantum-Mechanical Butterfly

​
denominator
14
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.

Details

The Schrödinger equation for this system takes the form
[ϵ(p)+V(r)]
Φ
E
(r)=
EΦ
E
(r)
,
where
ϵ(p)
is a pseudodifferential operator,
V(r)
is the potential, and
E
is the energy eigenvalue, with
Φ
E
(r)
given by the ansatz
Φ
E
(ma,na)=
-iνn
e
g(m)
for integers
m,n
.
The problem reduces to the solution of the recursive equation
g(m+1)
g(m)
=A(ϵ,m,α,ν)
g(m)
g(m-1)
with
A(ϵ,m,α,ν)=
ϵ-2cos(2πmα-ν)
-1
1
0

,
where
ϵ
is the energy,
ν=2π/q
, with
q
being the denominator of nonrepeating rational numbers
α
. Finally, the eigenvalue condition for the energy spectrum is
Tr
q-1
∏
k=0
Aϵ,k,α,
π
2q
≤4
.
The plot has energy on the
x
axis and the parameter
α
on the
y
axis.

References

[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.

External Links

Magnetic Field (ScienceWorld)
Electron (ScienceWorld)
Schrödinger Equation (ScienceWorld)

Permanent Citation

Enrique Zeleny
​
​"Hofstadter's Quantum-Mechanical Butterfly"​
​http://demonstrations.wolfram.com/HofstadtersQuantumMechanicalButterfly/​
​Wolfram Demonstrations Project​
​Published: July 3, 2012