Keep reading for FREE
(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 image similar to the patterns on the wings of a butterfly in nature. Additionally, the larger the set of equations being studied, the butterfly pattern (solutions) will become sharper .
As can be seen from the graphs above, the most important reason why Harper’s equation is to be worked only numerically is that it contains too many points (solutions)
16
15 reads
MORE IDEAS ON THIS
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...
17
47 reads
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...
16
12 reads
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)
18
209 reads
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...
16
94 reads
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...
16
53 reads
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.
O...
16
36 reads
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...
16
22 reads
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...
16
28 reads
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...
16
89 reads
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...
16
39 reads
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...
16
24 reads
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...
16
26 reads
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.
16
24 reads
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...
16
41 reads
16
10 reads
Ready for the next level?
Read Like a Pro
Explore the World’s
Best Ideas
Save ideas for later reading, for personalized stashes, or for remembering it later.
Start
31 IDEAS
Start
44 IDEAS
# Personal Growth
Take Your Ideas
Anywhere
Just press play and we take care of the words.
No Internet access? No problem. Within the mobile app, all your ideas are available, even when offline.
Ideas for your next work project? Quotes that inspire you? Put them in the right place so you never lose them.
Start
47 IDEAS
Start
75 IDEAS
My Stashes
Join
2 Million Stashers
4.8
Stars
5,740 Reviews
App Store
4.7
Stars
72,690 Reviews
Google Play
Shankul Varada
Best app ever! You heard it right. This app has helped me get back on my quest to get things done while equipping myself with knowledge everyday.
“
samz905
Don’t look further if you love learning new things. A refreshing concept that provides quick ideas for busy thought leaders.
“
Sean Green
Great interesting short snippets of informative articles. Highly recommended to anyone who loves information and lacks patience.
“
Ashley Anthony
This app is LOADED with RELEVANT, HELPFUL, AND EDUCATIONAL material. It is creatively intellectual, yet minimal enough to not overstimulate and create a learning block. I am exceptionally impressed with this app!
“
Laetitia Berton
I have only been using it for a few days now, but I have found answers to questions I had never consciously formulated, or to problems I face everyday at work or at home. I wish I had found this earlier, highly recommended!
“
Jamyson Haug
Great for quick bits of information and interesting ideas around whatever topics you are interested in. Visually, it looks great as well.
“
Ghazala Begum
Even five minutes a day will improve your thinking. I've come across new ideas and learnt to improve existing ways to become more motivated, confident and happier.
“
Giovanna Scalzone
Brilliant. It feels fresh and encouraging. So many interesting pieces of information that are just enough to absorb and apply. So happy I found this.
“
Read & Learn
20x Faster
without
deepstash
with
deepstash
with
deepstash
Access to 200,000+ ideas
—
Access to the mobile app
—
Unlimited idea saving & library
—
—
Unlimited history
—
—
Unlimited listening to ideas
—
—
Downloading & offline access
—
—
Personalized recommendations
—
—
FAQ
Claim Your Limited Offer
Get Deepstash Pro