# Another 2 questions in perturbation theory.

• MathematicalPhysicist
In summary, the conversation discusses perturbation theory and finding corrections to the energy and eigenstates for a particle in a square well potential and a two-dimensional harmonic oscillator with perturbations. The conversation includes a question about understanding the computation of matrix elements and the eigenfunctions of the problem.
MathematicalPhysicist
Gold Member

## Homework Statement

1. A particle of mass M is in a square well, subject to the potential:
$$V(x)= V0\theta(x-a/2)$$ for x in (0,a) and diverges elsewhere, where theta is heaviside step function.
In perturbation theory, find O(V0^2) correction to the energy and O(V0)to the eigenstate.

2. A particle is moving in a two dimensional harmonic oscillator with perturbation $$\delta V=\lambda x^2 y^2$$ For the ground state and the first excited state, find the first order correction to.

## The Attempt at a Solution

1. As I see it the hamiltonian has a perturbation in x in [a/2,a), which is V0.
, but from the equation for the second order correction to the energy which is:
$$\sum_{m doesn't equal n} \frac{|<m|V0|n>|^2}{E_n-E_m}$$, where $$E_k=\frac{(\hbar k)^2}{2M}+V_0$$, so that means the second order correction is zero cause <m|n> for m not equal n equlas zero, doesn't make sense to me, what am I missing?

For 2, for the first excited state I get that it's degenrate, cause $$E_{1,0}=E_{0,1}$$, not sure how to find for the first excited state its first order correction.

P.S
I sent an email to the TA with my questions, so far he hasn't replied, this is why I am asking here.

For #1, you can't just assume that $\left\vert \langle m \vert V_0 \vert n\rangle\right\vert = 0$ whenever $m \neq n$. Remember how you compute a matrix element:
$$\langle m \vert V_0 \vert n\rangle = \int\psi^{*}_m(x) V_0(x) \psi_n(x)\,\mathrm{d}x$$

Ok thanks.
Can someone help me with question 2?

bump, question 2?

diazona said:
For #1, you can't just assume that $\left\vert \langle m \vert V_0 \vert n\rangle\right\vert = 0$ whenever $m \neq n$. Remember how you compute a matrix element:
$$\langle m \vert V_0 \vert n\rangle = \int\psi^{*}_m(x) V_0(x) \psi_n(x)\,\mathrm{d}x$$

I still get that it's zero.
What are eigenfunctions of this problem?

I think that they are:
$$\psi_n(x)=\sqrt(\frac{2}{a})sin(nx\pi/a)$$ but if it's so then <n|V|m>=0.

## 1. What is perturbation theory?

Perturbation theory is a mathematical method used to solve problems that involve small changes or disturbances to a known system or process. It allows us to approximate the behavior of a complex system by breaking it down into smaller, more manageable components.

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

Perturbation theory is used in many scientific fields, such as physics, chemistry, and engineering. It is particularly useful in situations where the full system cannot be solved exactly, but small changes to a known system can be analyzed and understood. It is also used to make predictions about the behavior of a system under different conditions.

## 3. What is the difference between first-order and higher-order perturbation theory?

In first-order perturbation theory, we only consider the effect of a small change on the first iteration of the system. In higher-order perturbation theory, we take into account multiple iterations of the system, resulting in a more accurate approximation. The higher the order of perturbation theory used, the closer the approximation will be to the exact solution.

## 4. Can perturbation theory be used for any system?

No, perturbation theory is most effective when the changes to a system are small compared to the overall behavior of the system. It is also limited to systems that are linear, meaning that the response to a small change is proportional to the size of the change. Non-linear systems may require other methods for analysis.

## 5. Are there any limitations to perturbation theory?

Yes, perturbation theory may not always provide an accurate approximation, especially for systems with strong non-linearities or large changes. It also relies on the assumption that the small changes made to the system are independent of each other, which may not always be the case. Additionally, higher-order perturbation theory can become computationally intensive and may not be practical for all systems.

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