So, I was studying quantum mechanics and reached the point where perturbation theory is discussed. It is my first time in this topic, and something called my attention: it was said that we need perturbation theory when there is this hamiltonian that we cannot solve and them we can write:

$$H = H_0 + \lambda V$$

where $H_0$ is a hamiltonian that we can solve - or better put, is a hamiltonian that we KNOW how to solve -, and $V$ is another one that we can solve or not, that we do know how to solve or we do not, the question is that their sum, $H$, is the one that we cannot solve.

Ok, $H$ may be written in something way that is really hard to solve, or impossible, so we perturb the system and solve an easier problem. My question is: if we know how to solve $H_0$ and $V$, why we cannot solve $H$ = $H_0 + \lambda V$? The Schrödinger equation is linear.

What am I missing?

  • $\begingroup$ Are you asserting that if we know the eigenfunctions of $H_0$ and $V$, then we can simply add them together to get the eigenfunctions of $H_0 + \lambda V$? $\endgroup$ – J. Murray Aug 29 at 20:51

When people say "The Schrödinger equation is linear." they mean that for a particular $\hat{H}$, given two (or more solutions) $\Psi_i$ to $$ \hat{H}\Psi = i \hbar \frac{\mathrm{d}}{\mathrm{d}t} \Psi \,$$ then $$\Psi_\text{combination} = \sum_i c_i \Psi_i $$ with $c_i \in \mathbb{C}$ for all $i$ is also a solution.

They don't mean that given two different Hamiltonians $\hat{H}_i$ whose solutions are know you can trivially find solutions for $\hat{H}_\text{combination} = \sum_i a_i \hat{H}_i $.


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