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In other words, let’s choose the external magnetic field we apply not rational , but only irrational . What can change, you might say. After all, there is only 0.5 difference between alpha=1/2 and alpha=1 . But that is not the case. The very interesting thing is that the butterfly only occurs for rational alpha values! If we plot the solutions of Harper’s equation corresponding to the irrational values of alpha ; we get nothing more than horizontal arrays of dots !
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In quantum mechanics, classically forbidden phenomena are likely to occur. For any particle to pass over a potential barrier , it needs energy . If the energy of a particle is lower than the potential barrier , the particle is trapped there. In other w...
and naturally too many equations.
Let’s take a closer look to see one of the beautiful details of theHofstadter Butterfly . You can clearly see the band separations in the left graph above. It is the number in the denominator of the alpha value that determines how many tim...
A world lived by atoms was constructed using quantum mechanics. In condensed matter physics , crystal mediums are represented as lattice models in the presence of magnetic and electric fields (or both) . In this model, when the set of equations (Harper’s Equation)
A one of the most interesting phenomena (Hall voltage) in physics was first observed by Edwin Hall in 1879. Let’s imagine that we have a horizontal conductive plate(Hall element)and with the help of any power source we supply current to the plate in a particular directi...
The version of the Hall effect in the subatomic world is called the quantum Hall effect. We should expect to observe the classic Hall effect at low magnetic field, room temperature and at very high magnetic fields (about 10 Tesla) , very low temperatures (a...
In the quantum mechanics, every particle is also a wave . As we know from daily life, light waves (photon particles) scatter as they pass through raindrops and formed a rainbow. We assume that an electron does not undergo such scattering due to its movement in the crystal lattice medium.
Just as different types of butterflies exist in nature, the Hofstadter Butterfly is not unique in physics. Naturally, fractal patterns will be different in systems that exhibit different behaviors under different types of features. Above, we only talked about two-dimensional square latti...
In the context of this model, the mathematical equation describing the motion of a single electron living on a two-dimensional square lattice is called the Harper equation . It was studied by Rudolf Peierls andR. G. Harper in the 1950s. There is something that makes the...
Potential difference is required for the existence of voltage in a particular direction. In our experiment, electrons and protons stacked on opposite sides to create a potential difference perpendicular to the current. This voltage is called the Hall voltage . Also, this phenomena is called the c...
The main reason why we don’t think of teleportation as a spontaneous natural event is that it never happens in daily life (by the laws of classical mechanics). Because the wavelength of the wave (or particle) much less than the width of the barrier . This also decreases the probability . That’s w...
All that remains in solving the problem is to solve the finite set of Harper equations using numerical solution (computational) techniques. In other words determining the energy values in each of the equations in this finite set. It was solved byDougles R. Hofstadte...
(energy as a function of alpha) are plotted, we will see the Hofstadter Butterfly , which similar to butterfly wings.
Note that the graph is symmetrical with respect to the zero of the energy and alpha = 1/2 value. The symmetry presents us an ima...
Since the ratio will still give the magnetic field unit, the alpha value is the external magnetic field itself. The results will differ depending on whether the value of the external magnetic field is rational or irrational .
Here, the flux can be thought of as magnetic field lines pass thr...
If we look at the graph below, we see visually sense pointless results for irrational alpha values . But, as we saw in the previous graph,Hofstadter Butterflies lives in the white regions (range of rational values) of the graph below.
Multidimensional lattice, a mathematical structure for molecules, was modeled in condensed matter physics (solid state physics) which is a sub-area of physics to solve the problem. In this model, we can imagine the lattice (can be of any geometry) with atoms in point form at eac...
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