Wavefunctions in first order degenerate pertubation theory

[klpskp]

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 :)

 

[eljose79]

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.