2.3. SOLUTION OF THE PROBLEM - Deepstash

How to set new goals

How to take action towards a new life

How to create a plan for change

Discover 35 similar ideas in

It takes just

# 2.3. SOLUTION OF THE PROBLEM

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. Hofstadter in 1976.

Now that we have the solutions, let’s look at their dependence on external magnetic field (alpha values) . Let’s consider alpha values a rational number. In other words, the magnetic field that we apply to the system should only be in rational values. In this case, if the energy solutions corresponding to each alpha values

17

MORE IDEAS ON THIS

### 2. QUANTUM HALL EFFECT

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...

17

### We will look at how discovered the Hoftstadter Butterfly fractal pattern in quantum physics.

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)

19

### Let’s take a closer look to see one of the beautiful details of

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...

17

### ...

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...

17

### 2.1. QUANTUM TUNELLING

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...

18

### 1. HALL EFFECT

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...

18

### What do we mean by scatter ?

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...

17

### 2.4. ALTERNATIVE BUTTERFLIES

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...

17

### ...

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...

17

, we will examine the Hall Effect , which is the initial discovery of the main problem (the motion of the electron in a crystal medium) and the Hofstadter Butterfly , which is the solution of the same effect in the subatomic world.

17

### ...

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...

17

### ...

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...

17

### ...

(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...

17

### ...i

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.

17

### 2.2. HARPER EQUATION

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...

17

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 fo...

17

CURATED FROM

CURATED BY

卐 || एकं सत विप्रा बहुधा वदन्ति || Enthusiast || Collection Of Some Best Reads || Decentralizing...

Related Collection

## MORE LIKE THIS

### Einstein and the shape of spacetime

Albert Einstein held that space and time together made up the universe's fabric that could stretch and compress. Einstein suggested that the shape of spacetime is what gives rise to the force of gravity. A concentration of mass, like the earth or sun, ben...

## Explore the World’sBest Ideas

### 200,000+ ideas on pretty much any topic. Created by the smartest people around & well-organized so you can explore at will.

#### An Idea for Everything

Explore the biggest library of insights. And we've infused it with powerful filtering tools so you can easily find what you need.

Knowledge Library

#### Powerful Saving & Organizational Tools

Save ideas for later reading, for personalized stashes, or for remembering it later.

# Personal Growth

### Organize your ideas & listen on the go. And with Pro, there are no limits.

#### Listen on the go

Just press play and we take care of the words.

#### Never worry about spotty connections

No Internet access? No problem. Within the mobile app, all your ideas are available, even when offline.

#### Get Organized with Stashes

Ideas for your next work project? Quotes that inspire you? Put them in the right place so you never lose them.

My Stashes

Join

2 Million Stashers

4.8

5,740 Reviews

App Store

4.7

72,690 Reviews

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!

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.

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.

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.

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.

20x Faster

without
deepstash

with
deepstash

with

deepstash

Unlimited idea saving & library

Unlimited history

Unlimited listening to ideas

Personalized recommendations

FAQ