# Wavefunctions in first order degenerate pertubation theory

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In summary, first order degenerate perturbation theory is a mathematical method used to calculate the energy levels and wavefunctions of a quantum system with degenerate energy levels. It differs from non-degenerate perturbation theory in that it is used for systems with degenerate energy levels. The purpose of using this method is to make more accurate predictions of a system's behavior. The wavefunctions can be solved using first-order perturbation equations. However, there are limitations to this method, including its validity for only small perturbations and its inability to accurately predict behavior under large perturbations or for highly degenerate energy levels.
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Suppose we want to solve the Hamiltonian ##H=H_0+\lambda V## pertubatively. Let ##E_1,...,E_n## be the eigenvalues of ##H_0## and ##S_1,...,S_n## the eigenspaces that belong to them.

In order to do that, one usually choses an orthonormal Basis ##|\psi_{i,j}>## of each ##S_i## with the property, that ##<\psi_{i,j}|V|\psi_{i,k}>=0## whenever ##j \neq k##. Let ##|\tilde{\psi}_{i,j}>## the first order correction of ##|\psi_{i,j}>##. Then pertubation theory gives

$$<\psi_{i',j'}|\tilde{\psi}_{i,j}>=\frac{<\psi_{i',j'}|V|\psi_{i,j}>}{E_{i}-E_{i'}}$$

when ##i'\neq i##

However, for ##i=i'## one doesn't obtain any information on ##<\psi_{i',j'}|\tilde{\psi}_{i,j}>##. What are the components of ##|\tilde{\psi}_{i,j}>## in ##S_i##?

Thank you :)

Hello,

Thank you for your question. It is important to note that perturbation theory is a powerful tool for solving problems where the Hamiltonian can be written as a sum of a known part (H0) and a small perturbing term (λV). In this case, the eigenvalues and eigenvectors of the unperturbed Hamiltonian (H0) are used as a starting point for finding the eigenvalues and eigenvectors of the perturbed Hamiltonian (H). The perturbative solution is then obtained by expanding the eigenvalues and eigenvectors in powers of λ.

In order to obtain the first-order correction to the eigenvectors, we use the orthonormal basis |ψi,j> of each eigenspace Si. This basis is chosen to ensure that the matrix elements of the perturbing term V are zero when the indices j and k are different, as stated in the post. This allows us to simplify the calculation and focus on the terms where j=k. The first-order correction to the eigenvector |ψi,j> is then given by |ψi,j> = |ψi,j> + λ|ψ~i,j>, where |ψ~i,j> is the first-order correction.

For i≠i', the formula given in the post is correct and can be derived using perturbation theory. However, for i=i', the formula does not provide any information on the components of |ψ~i,j> in Si. To obtain this information, we need to consider higher-order corrections to the eigenvector. In general, the higher-order corrections can be obtained by solving a set of coupled equations, known as the secular equations, which involve the matrix elements of V and the components of the higher-order corrections in the basis of Si. Solving these equations will give us the components of |ψ~i,j> in Si.

In summary, for i≠i', the formula given in the post is correct and can be derived using perturbation theory. However, for i=i', higher-order corrections are needed to obtain the components of |ψ~i,j> in Si. I hope this helps clarify your question.

## 1. What is first order degenerate perturbation theory?

First order degenerate perturbation theory is a mathematical method used to calculate the energy levels and wavefunctions of a quantum system that experiences a small perturbation. Degenerate perturbation theory is used when the original unperturbed system has degenerate energy levels, meaning there are multiple states with the same energy.

## 2. How does first order degenerate perturbation theory differ from non-degenerate perturbation theory?

In non-degenerate perturbation theory, the unperturbed system has non-degenerate energy levels, meaning each energy level has only one corresponding state. First order degenerate perturbation theory, on the other hand, is used for systems with degenerate energy levels, where each energy level has multiple corresponding states.

## 3. What is the purpose of using first order degenerate perturbation theory?

The purpose of first order degenerate perturbation theory is to calculate the energy levels and wavefunctions of a quantum system that is experiencing a small perturbation. This method allows for more accurate predictions of the behavior of a system, especially when the unperturbed system has degenerate energy levels.

## 4. How does one solve for the wavefunctions in first order degenerate perturbation theory?

The wavefunctions in first order degenerate perturbation theory can be solved using the first-order perturbation equations. These equations involve calculating the matrix elements of the perturbation operator and solving for the wavefunctions using linear algebra techniques.

## 5. Are there any limitations to first order degenerate perturbation theory?

Yes, first order degenerate perturbation theory has limitations. It is only valid for small perturbations and cannot accurately predict the behavior of a system under large perturbations. Additionally, it may not give accurate results for systems with highly degenerate energy levels or when the perturbation operator does not commute with the unperturbed Hamiltonian.

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