Condensed matter problem with Feynman integral

In summary, the conversation is about deriving the final expression for a particle in a potential coupled to a "bath" of harmonic oscillators. The person is unsure about how to arrive at the final expression and has questions about assuming q is periodic and using the Fourier series expansion of q. They also mention the Matsubara formalism and suggest looking at Fetter/Valecka for more information.
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Im working on deriving the final expression on the attached picture. The problem is that of a particle in a potential coupled to a "bath" of harmonic oscillators but I'm not sure how you arrive at the final expression. First of all, why are you allowed to assume that q is periodic on the interval [0,T-1]? And secondly how should I use this? Should I plug in the Fourier series expansion of q? And in that case what should I do with the integral in dt' (which I suppose has been transformed to an integral in τ by t'->-iτ)
 

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  • #2
Did you have a look on how the Matsubara formalism works, e.g. in Fetter/ Valecka?
 

FAQ: Condensed matter problem with Feynman integral

1. What is the condensed matter problem with Feynman integral?

The condensed matter problem with Feynman integral is a mathematical problem in the field of condensed matter physics. It involves the use of Feynman integrals, which are complex mathematical expressions used to calculate the probability of a particle's path in quantum mechanics. In condensed matter physics, these integrals are used to study the behavior of large numbers of particles, such as in solid materials or liquids.

2. What makes the condensed matter problem with Feynman integral challenging?

The condensed matter problem with Feynman integral is challenging because it involves solving complex integrals with many variables. These integrals may have no analytical solutions, meaning they cannot be solved using traditional mathematical methods. Instead, numerical methods and approximations must be used to solve these integrals, making the problem computationally intensive and time-consuming.

3. How is the condensed matter problem with Feynman integral relevant to real-world applications?

The condensed matter problem with Feynman integral has many practical applications in fields such as material science, nanotechnology, and electronics. By studying the behavior of particles in condensed matter systems, we can better understand and predict the properties and behavior of materials. This knowledge is essential for developing new technologies and improving existing ones.

4. What are some techniques used to solve the condensed matter problem with Feynman integral?

Some techniques used to solve the condensed matter problem with Feynman integral include Monte Carlo simulations, perturbation theory, and variational methods. These methods involve breaking down the integral into smaller, more manageable parts and using numerical or analytical approaches to solve them. Additionally, computer algorithms and high-performance computing are often used to speed up the calculation process.

5. Are there any current advancements or developments in solving the condensed matter problem with Feynman integral?

Yes, there are ongoing research and advancements in solving the condensed matter problem with Feynman integral. Some recent developments include the use of machine learning and artificial intelligence to solve these integrals more efficiently. Additionally, new numerical and analytical techniques are being developed to solve more complex integrals and improve the accuracy of results. These advancements are crucial in furthering our understanding of condensed matter systems and their applications.

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