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

[actionintegral]

Time dependent perturbation theory...

second order term...

For some reason they replace

[tex]<E_{n}|H_{0}^2|E_{m}>[/tex]

with

[tex]\Sigma<E_{n}|H_{0}|E_{i}><E_{i}|H_{0}|E_{m}>[/tex]

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

 

[Meir Achuz]

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.

 

[denian]

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.