# Understanding Time-Dependent Perturbation Theory's 2nd Order Term

• actionintegral
In summary, time dependent perturbation theory involves replacing the second order term of <E_{n}|H_{0}^2|E_{m}> with \Sigma<E_{n}|H_{0}|E_{i}><E_{i}|H_{0}|E_{m}> and this is allowed because of the completeness property of the states |E_i>. This becomes useful when the |E_i> are eigenstates of H_0, while the |E_m> are not.
actionintegral
Time dependent perturbation theory...

second order term...

For some reason they replace

$$<E_{n}|H_{0}^2|E_{m}>$$

with

$$\Sigma<E_{n}|H_{0}|E_{i}><E_{i}|H_{0}|E_{m}>$$

I know why they are allowed to do this, what I don't understand is how it makes my life better?

The fact that \sum|E_i><E_i|=1 is called the completeness property of the states E_i>. It means that you have enough states to expand a function in.
Look at the Fourier sin series. There sin(2pi x/L) form a complete set.
It becomes useful when the |E_i> are eigenstates of H_0, while the |E_m> are not.

Time-dependent perturbation theory is a powerful tool in quantum mechanics that allows us to study the behavior of a system when it is subjected to a time-dependent perturbation. The second order term in this theory is an important aspect as it takes into account the effects of the perturbation on the energy levels of the system.

The expression <E_{n}|H_{0}^2|E_{m}>, where H_{0} is the unperturbed Hamiltonian, represents the second order term in time-dependent perturbation theory. To simplify this expression, we can use the identity <A^2> = <A><A>, which allows us to rewrite the second order term as <E_{n}|H_{0}|E_{i}><E_{i}|H_{0}|E_{m}>, where E_{i} is an intermediate state.

This replacement may seem like a small change, but it actually makes our calculations much more manageable. By breaking down the second order term into two parts, we can treat each term separately and then sum them together. This approach is known as the "sum over intermediate states" method and it greatly simplifies the calculations involved in time-dependent perturbation theory.

In addition, this replacement also allows us to gain a deeper understanding of the system's energy levels and how they are affected by the perturbation. By considering all possible intermediate states, we can see how the perturbation affects the system's energy levels at different points in time.

Overall, the replacement of the second order term in time-dependent perturbation theory with <E_{n}|H_{0}|E_{i}><E_{i}|H_{0}|E_{m}> is a powerful tool that simplifies calculations and provides a deeper understanding of the system's behavior under a time-dependent perturbation. It allows us to make more accurate predictions and better understand the dynamics of quantum systems.

## What is time-dependent perturbation theory and why is it important?

Time-dependent perturbation theory is a mathematical tool used to study the behavior of a quantum system when it is subjected to an external perturbation, or disturbance. It allows us to calculate the changes in the system's energy levels and wave functions due to the perturbation, and is important for understanding the dynamics of quantum systems and predicting their behavior.

## What is the 2nd order term in time-dependent perturbation theory and how does it contribute to the overall solution?

The 2nd order term in time-dependent perturbation theory is the second term in the perturbation series expansion. It takes into account the second-order corrections to the system's energy levels and wave functions due to the perturbation. It contributes to the overall solution by refining and improving the accuracy of the predictions made by the first-order term.

## How is the 2nd order term calculated in time-dependent perturbation theory?

The 2nd order term is calculated by taking into account the second-order corrections to the energy levels and wave functions, which are calculated using the time-dependent Schrödinger equation. This involves solving a set of differential equations and integrating over time to obtain the final result.

## What are the limitations of the 2nd order term in time-dependent perturbation theory?

The 2nd order term is limited in its applicability to weak perturbations and small changes in the system's energy levels and wave functions. It also assumes that the perturbation acts continuously on the system, which may not always be the case. Additionally, it does not take into account higher-order corrections, which may be significant in certain systems.

## How is time-dependent perturbation theory applied in real-world scenarios?

Time-dependent perturbation theory is widely used in many areas of physics, including quantum mechanics, atomic and molecular physics, and condensed matter physics. It has practical applications in fields such as quantum computing, spectroscopy, and material science. It is also used in studying the dynamics of chemical reactions and the behavior of particles in accelerators.

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