# Litle help with perturbation theory

• baranas
In summary: Then,after the perturbation is applied, one can show that the wave-function of thedegenerate unperturbed state is the same as the wave-function of theunperturbed state with the additional label.
baranas
Why when we analyse time dependant perturbation theory, we take that the diagonal elements of matrix <i|W(t)|j> are equal to zero?
Why in degenerate perturbation theory we assume that perturbed wavefunctions of degenerate states can be expressed in the base of unperturbed wavefunctions of degenerate states and deny other functions?

P.S. Sorry for my English

baranas said:
Why when we analyse time dependant perturbation theory, we take that the diagonal elements of matrix <i|W(t)|j> are equal to zero?
Why in degenerate perturbation theory we assume that perturbed wavefunctions of degenerate states can be expressed in the base of unperturbed wavefunctions of degenerate states and deny other functions?

I'm not sure I understand your question, but I think the answer is:
"because we're assuming that the set of all eigenstates of the
unperturbed Hamiltonian is a basis for the Hilbert space".

(If that doesn't make sense, then I guess I didn't understand
the question properly.)

strangerep said:
I'm not sure I understand your question, but I think the answer is:
"because we're assuming that the set of all eigenstates of the
unperturbed Hamiltonian is a basis for the Hilbert space".

No, the problem is that in degenerate perturbation theory we take the basis for our perturbed state function only the wave-functions of degenerate states from unperturbed system and skip others, which belong to nendegenerate states.

H0$\phi$0n=E0n$\phi$0n

Assume that we have r degenerate states E0f$\rightarrow$$\phi$0fa, with a=1,2,3...r. The question is why do we assume, that a wave-function after perturbation will be in the form
$\Psi$na=$\sum$cb$\phi$0fb and skip the part $\sum$cn$\phi$0n of nondegenerate functions

baranas said:
No, the problem is that in degenerate perturbation theory we take the basis for our perturbed state function only the wave-functions of degenerate states from unperturbed system and skip others, which belong to nondegenerate states.

In Ballentine, "QM - A Modern Development", sect 10.5, pp284-285, one starts with a full set of (unperturbed) states, with an additional label to represent any degeneracies. Then one derives an equation involving only the degenerate unperturbed states belonging to a
particular unperturbed energy. [See Ballentine's eq(10.92) and discussion leading to it.]

So we haven't "skipped" the others. They're still there, but we derived an extra
condition that allows the perturbation procedure to go ahead.

## 1. What is perturbation theory?

Perturbation theory is a method used in physics and mathematics to approximate solutions to problems that cannot be solved exactly. It involves breaking down a complex system into a simpler one that can be solved, and then adding small corrections to improve the accuracy of the solution.

## 2. How is perturbation theory applied in science?

Perturbation theory is widely used in physics, chemistry, and engineering to solve problems involving small changes or disturbances in a system. It is particularly useful in analyzing systems that are difficult to solve using other methods, such as quantum mechanics and fluid dynamics.

## 3. What are the limitations of perturbation theory?

Perturbation theory is most effective for small perturbations and may not provide accurate results for larger perturbations. It also assumes that the system is linear, which may not be the case for all systems. In addition, perturbation theory can be time-consuming and may not always provide exact solutions.

## 4. What are the benefits of using perturbation theory?

Perturbation theory allows scientists to approximate solutions to problems that would otherwise be impossible to solve. It also provides insights into the behavior of complex systems and can help identify important parameters and relationships within these systems. Additionally, perturbation theory can be used to improve the accuracy of numerical simulations and experimental measurements.

## 5. Can perturbation theory be used in real-world applications?

Yes, perturbation theory has been successfully applied in various fields such as quantum mechanics, celestial mechanics, and fluid dynamics. It has also been used in engineering applications, such as designing aircraft and spacecraft, and in economic models. Its versatility and effectiveness make it a valuable tool for solving practical problems in the real world.

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