Green's function for resonant level

In summary, the Green's function g(l,ikn) is the Fourier transform of the retarded Green's function g(vv',τ) and is defined as g(l,ikn) = ∫∞−∞ dτ g(vv',τ)eiknτ. To calculate it, one can use the formula for the Fourier transform and simplify it using the delta function, resulting in g(l,ikn) = ∑v ∫∞−∞ dτ g(v,τ)eiknτ.
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I would really like some help for exercise 2 in the attached pdf. I know it's a lot asking you to read through all the pages but maybe you can skim them and catch the main points leading to exercise 2.
What I don't understand is pretty basic. What is meant by the Green's function g(l,ikn)? In all of the examples we are working with green's function on the form g(vv',τ) so what does the other notation mean? Does it mean: g(ll,ikn)?
In any case isn't it simply found by identifying g-1(l,τ) as stated in (2.23) and Fourier transforming? That's what I though but I don't think that yields the correct result with the sum over v-states.
 

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The Green's function g(l,ikn) is the Fourier transform of the retarded Green's function g(vv',τ) (defined in Equation (2.23)). It is defined as:g(l,ikn) = ∫∞−∞ dτ g(vv',τ)eiknτ To calculate it, one would need to use the formula for the Fourier transform of a function, which is given by:F[f(t)] = ∫∞−∞ dt f(t)e−iknt By substituting the definition of g(vv',τ) into the above equation, one would get:g(l,ikn) = ∫∞−∞ dτ ∑v,v' δ(v-v')g(v,τ)eiknτ The delta function can be used to simplify the above expression and we get:g(l,ikn) = ∑v ∫∞−∞ dτ g(v,τ)eiknτ Finally, this yields the desired result for the Green's function g(l,ikn).
 

FAQ: Green's function for resonant level

1. What is Green's function for resonant level?

Green's function for resonant level is a mathematical tool used in quantum mechanics to describe the behavior of a resonant level, which is a localized electronic state that can trap electrons at a specific energy level. It is a complex function that relates the input and output states of a resonant level system.

2. How is Green's function for resonant level used in research?

Green's function for resonant level is used in various research fields, including condensed matter physics, quantum computing, and nanotechnology. It is particularly useful in studying the transport properties of quantum systems, such as quantum dots and quantum wells, and in understanding the behavior of electrons in materials with impurities.

3. What is the significance of the resonant level in Green's function?

The resonant level plays a crucial role in Green's function as it represents the localized electronic state that interacts with the rest of the system. It is defined by its energy, which is typically close to the Fermi level, and its coupling strength to the rest of the system. This allows researchers to study the behavior of electrons at a specific energy level within a larger quantum system.

4. How is Green's function for resonant level calculated?

Green's function for resonant level is typically calculated using advanced mathematical techniques, such as perturbation theory and the Keldysh formalism. These methods involve solving complex equations and integrating over the energy spectrum of the system. It is also possible to numerically calculate Green's function using computer simulations.

5. What are some applications of Green's function for resonant level?

Green's function for resonant level has many practical applications, such as in the development of quantum computing devices and nanoelectronic devices. It is also used in theoretical studies of electronic transport, such as in the study of quantum interference effects. Additionally, it has potential applications in developing new materials with desired electronic properties.

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