Perturbation theory

Perturbation theory is the study of quantum systems which are in some sense near some solvable systems, in such a way that we may be able to approximate via such solutions.

Consider a quantum system where our operator can be decomposed in two parts,

\begin{equation} \hat{A} = \hat{A}_0 + \hat{B} \end{equation}

Here we know the spectrum of $\hat{A}_0$. In other words,

We will assume here that we're dealing with systems obeying the canonical commutation relations

\begin{equation} [\hat{x}, \hat{p}] = i\hbar \hat{I} \end{equation}

So that whatever the case may be, we can suppose that we are working on the Hilbert space $L^2(\mathbb{R}^n, d\mu)$

What we will want to do is to express $\hat{H}$ as a function of some parameter $\lambda$, so that we can consider a small variation around our known Hamiltonian $\hat{H}_0$ :

\begin{equation} \hat{H}_\lambda = \hat{H}_0 + \lambda \hat{V} \end{equation}

Thanks to our assumptions, $\hat{H}_\lambda$ is an operator on the same Hilbert space for all values of $\lambda$. For $\lambda = 0$, we also have the spectrum

Last updated : 2019-10-28 09:26:57
Tags : physics , quantum-mechanics