Confused about perturbation theory with path integrals

In summary, the conversation discusses using path integrals to calculate the ground state energy shift in a harmonic oscillator. The equations and attempts at a solution are also mentioned, with a question about how to apply the method to other energy levels. The person eventually finds the answer in a set of notes.
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Hazz
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Homework Statement


Hi, I am just trying to wrap my head around using path integrals and there are a few things that are confusing me. Specifically, I have seen examples in which you can use it to calculate the ground state shift in energy levels of a harmonic oscillator but I don't see how you can find the shift for any other energy level. I am aware you don't need to use a path integral for this but I am hoping to get a better understanding of them by doing this.

Homework Equations


I know that you can calculate the energy shift in the ground state using the partition function by using

[itex] E_0=-lim_{\beta\rightarrow\infty}\frac{1}{\beta} log K_E[J] |_{J=0}[/itex]

I know if I am using the Hamiltonian
[itex]\mathcal{H}=\frac{\hat{p}^2}{2m}+\frac{1}{2}m\omega^2\hat{q}^2+\gamma\hat{q}^4[/itex]

then to first order you can just write ## K_E[J] ## as

[itex] K_E[J]=K_E^0[J]-\gamma \int d\tau (\frac{\delta}{\delta J(\tau)})^4K^0_E[J]|_{J=0}[/itex]

where

[itex] K_E^0[J]=\frac{C}{2}\int d\tau d\tau'J(\tau)G(\tau,\tau')J(\tau')[/itex]

and G is the green's function.

The Attempt at a Solution


[/B]
So you can evaluate this all and work it out for the ground state which I don't have a problem with, but what confuses me is how this works for anything that isn't the ground state. Clearly the ##\beta\rightarrow\infty## limit is always going to put you there.

I understand that you probably need to use

[itex] \langle 0 |\hat{a}e^{-\beta H} \hat{a}^\dagger |0\rangle [/itex]

In some capacity but I don't really see where to proceed from there. Do you have to worry about commuting ##\hat{a}^\dagger## through the exponential?

I don't really know where to look for guidance with this sort of thing so any help would be appreciate. Thank you very much!
 

Related to Confused about perturbation theory with path integrals

What is perturbation theory with path integrals?

Perturbation theory with path integrals is a mathematical framework used in quantum mechanics to approximate the behavior of a system when it is subjected to small external influences or perturbations. It combines the concepts of perturbation theory, which deals with small changes in the Hamiltonian of a system, and path integrals, which represent the probability of a particle traveling between two points in space.

How does perturbation theory with path integrals work?

In perturbation theory with path integrals, the Hamiltonian of a system is expanded into a series of terms, with the first term representing the unperturbed system and the subsequent terms representing the effects of the perturbation. The path integral is then calculated for each term, and these terms are summed together to approximate the behavior of the system under the influence of the perturbation.

What are the advantages of using perturbation theory with path integrals?

Perturbation theory with path integrals allows for the calculation of approximate solutions for complex quantum systems that cannot be solved exactly. It also provides a systematic way to incorporate small perturbations into the description of a system, making it a powerful tool for studying the behavior of particles in a wide range of physical systems.

What are the limitations of perturbation theory with path integrals?

One of the main limitations of perturbation theory with path integrals is that it only works for small perturbations. If the perturbation is too large, the series expansion may not converge, and the results obtained may be inaccurate. Additionally, this method can become computationally intensive for systems with many degrees of freedom.

What are some applications of perturbation theory with path integrals?

Perturbation theory with path integrals has many applications in quantum mechanics, including the study of the behavior of atoms, molecules, and condensed matter systems. It is also used in quantum field theory to analyze the behavior of particles and fields in high energy physics. Additionally, it has applications in other fields such as statistical mechanics, nuclear physics, and cosmology.

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