Quantum Hall Effect: Hamiltonian & Finding Solutions

In summary, A good resource for learning about quantum hall effect is the book edited by Prange and Girvin. The Hamiltonian for the Qunatum hall effect can be found in this book, as well as in other modern condensed matter books. When approaching the Hamiltonian, it is important to look for conserved quantities and try to relate it to a harmonic oscillator Hamiltonian.
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Hi , I need a good book or lecture on quantum hall effect.

my supervisor wants me to find the Hamiltonian for the Qunatum hall effect, he want me to find this from this relation if it looks like a 2nd order differential of harmonic oscillation. can you please guide me doing this?


$$H= \frac {h^2}{2m} ( \delta_x^2+ (\delta_y+ \frac{iq}{h} B_x)^2)^2$$
 
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A great book is this one edited by Prange and Girvin, with a lot of good chapters by experts. There are a few more books/reviews that I can think of, but I think this is a good one for an intro. Many modern condensed matter books will have an intro too.

For the Hamiltonian you posted: do you see any conserved quantities? Can you find a way to relate it to a harmonic oscillator Hamiltonian?
 
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1. What is the quantum Hall effect?

The quantum Hall effect is a phenomenon observed in two-dimensional electron systems at very low temperatures and strong magnetic fields. It describes the quantization of the Hall resistance, which is the ratio of the electric field applied to a sample to the resulting electric current.

2. What is the Hamiltonian in the context of the quantum Hall effect?

The Hamiltonian is a mathematical operator that describes the total energy of a system and its time evolution. In the context of the quantum Hall effect, the Hamiltonian represents the energy of electrons in a magnetic field and can be used to calculate their behavior and properties.

3. How is the Hamiltonian used to find solutions in the quantum Hall effect?

The Hamiltonian is used as part of the Schrödinger equation to describe the quantum states of electrons in a magnetic field. By solving this equation, we can find the energy levels and wave functions of the electrons, which can then be used to understand and predict their behavior.

4. What are some common methods for solving the Hamiltonian in the quantum Hall effect?

One common method is the Landau level representation, which describes the energy levels of electrons in a magnetic field. Another method is the composite fermion theory, which explains the behavior of the electrons in terms of new quasi-particles called composite fermions.

5. How does the quantum Hall effect have practical applications?

The quantum Hall effect has many practical applications, including the precise measurement of electrical resistance, the development of new semiconductor devices, and the potential for creating quantum computers. It also has implications for understanding the behavior of electrons in other systems, such as superconductors and topological insulators.

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