# Time-Dependent Perturbation Theory & Completeness

• Runei
In summary, the conversation discusses the use of the unperturbed system's eigenfunctions as a basis for the Hilbert space in order to find solutions to the perturbed system. The argument states that the solution to the perturbed system can still be written as a sum of the eigenfunctions of the unperturbed system, even though the full hamiltonian now depends on time. The expansion coefficients will also depend on time, but the main point is that the solutions can still be found using the eigenfunctions of the unperturbed system. The conversation concludes by confirming that this method is correct and that the unperturbed eigenfunctions form a complete basis for the Hilbert space.
Runei
Hello!

I just want to make sure that I have understood the following argument the correct way:

For a given quantum system we take the hamiltonian to be a time-independent (and soluble) part, and a time-dependent part.

## \hat{H} = \hat{H_0} + H'(t) ##

Now, the solutions to the unperturbed system are given by

## \Psi_n(x,t) = \psi_n(x) e^{-iE_nt/\hbar} ##

And any solution to the system can be written as

## \Psi(x,t) = \sum\limits_n c_n \psi_n(x) e^{-iE_nt/\hbar} ##

Argument:
When we go ahead and introduce the perturbed system, we can still write the solution to the system as a sum of the eigenfunctions of the unperturbed system, since these eigenfunctions represent a basis to Hilbert-space. And the solution to the perturbed system must also belong to Hilbert-space. Therefore:

## \Psi'(x,t) = \sum\limits_n c_n(t) \psi_n(x) e^{-iE_nt/\hbar} ##

The expansion coefficients in this case will depend on time, since the full hamiltonian now also depends on time, but the central argument is that we can still write the solution we are looking for, as an sum of the solutions to the unperturbed system.

Is this correct or am I missing something essential? :)

Runei said:
The expansion coefficients in this case will depend on time, since the full hamiltonian now also depends on time, but the central argument is that we can still write the solution we are looking for, as an sum of the solutions to the unperturbed system.

Yes, you can expand any vector using the eigenvectors of the unperturbed hamiltonian since they (are supposed at least) form complete basis for the Hilbert space.

Runei
Thanks a lot, mate! :)

## 1. How does time-dependent perturbation theory work?

Time-dependent perturbation theory is a method used to solve the Schrodinger equation for a quantum system that is subject to a time-dependent perturbation. It involves expanding the wave function in terms of the unperturbed Hamiltonian and using perturbation theory to calculate the corrections to the energy levels and wave functions of the system.

## 2. What is the importance of completeness in time-dependent perturbation theory?

Completeness is a fundamental concept in quantum mechanics that ensures that the wave function contains all the information about a system. In time-dependent perturbation theory, completeness allows us to express the perturbed wave function in terms of the unperturbed wave function, making the calculations more manageable.

## 3. How is the completeness of the wave function achieved in time-dependent perturbation theory?

The completeness of the wave function in time-dependent perturbation theory is achieved by expanding the perturbed wave function in a series of the unperturbed wave function and its derivatives. This expansion allows us to express the perturbed wave function as a linear combination of the unperturbed wave function, making it complete.

## 4. What are the limitations of time-dependent perturbation theory?

Time-dependent perturbation theory is only applicable to small perturbations and cannot be used for systems with strong interactions. It also assumes that the perturbation is turned on and off gradually, which may not always be the case in real-life situations.

## 5. How is time-dependent perturbation theory used in practical applications?

Time-dependent perturbation theory has numerous applications in various fields of physics, such as atomic and molecular physics, solid-state physics, and nuclear physics. It is used to study the dynamics of quantum systems subject to time-dependent external fields and to calculate transition probabilities between different energy levels.

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