# Modeling perturbations of a quantum mechanical system

I'm wondering how to properly perturb a quantum mechanical system.

I'm looking for a way to express the return to equilibrium of such a perturbed state, where the perturbation is in the Hamiltonian. The unperturbed Hamiltonian is $H_{\text{u}}$ and the perturbed Hamiltonian is $H_{\text{p}}$.

I can compute the eigensystem for both $H$ and thus find the relevant energy spectra and eigenfunctions, and can add the time-dependence factor $e^{-iEt}$, but I do not know how to generate the final system from its component, isolated systems. I would guess an inner product to be of use here, but...

What theory/mathematical methods do I need to learn to perform such simulations?

Something I want to model is a molecule in a polarized state, that is, with an asymmetric electron distribution.

I don't understand exactly what is your question:

If you start in an eigenstate of $H_p$, say $\varphi_j$ and then your Hamiltonian changes like $H(t) = H_u + f(t) (H_p - H_u)$ where $f(0) = 1$ and $f(T) = 0$ -$T$ being the final time of the perturbation- then in order to follow the evolution of $\psi(t)$ all you need to do is to solve the time dependent Schroedinger equation $$i\hbar \partial \psi(t) / \partial t = H(t) \psi(t)$$ where $\psi(0) = \varphi_j$. There are many numerical procedures to do so. For instance if you are interested, as it seems, in following the dynamics in the basis of the eigenstates of $H_u$, then you expand $\psi(t) = \sum_k a_k(t) \phi_k$, where $\phi_k$ are eigenstates of $H_u$, and you solve a set of ordinary differential equations.

If $f(t)$ is very fast ($T << (\Delta E)^{-1}$, where $\Delta E$ is a characteristic energy of the system, say the energy difference between adjacent eigenstates) then you can use the sudden approximation; if $f(t)$ is very slow you can use the adiabatic approximation, if $H_p - H_u$ is small you can use first order perturbation theory, etc.

In any case it is important to realise that in Quantum Mechanics there is no relaxation unless you couple your initial state to a continuum and there is no return to equilibrium unless you do the same (couple to continuum) and work with the proper formalism that allows you to define the (thermal) equilibrium, which are the density matrices, unless by equilibrium you are referring to the initial state. Again, in ordinary Quantum Mechanics $\psi(t)$ will oscillate and will not relax to the initial state $\varphi_j$ (or $\phi_0$); at most, it will return to the initial state as part of an oscillation, called revival.

Provided that I understood your question correctly, the keyword you're looking for is time-dependent perturbation theory. Formally, it is very similar to the more common time-independent perturbation theory, which is used in particular to express solutions for a (hard to solve) Hamiltonian using as a basis the solutions for a similar but easier to solve Hamiltonian, as you noted in your question. An time-dependent pertubation ($H_p-H_u$) indeed includes the case you mention, where perturbation is turned on (or off) at a given time (usually taken as reference, $t=0$). The behavior of the system at $t>0$ can then be expressed.

You can find good introductions to this topic in many quantum physics books, but I wouldn't know which one in particular to recommend… The Wikipedia page on the topic might also help you get a good start.

• I'll re-read the wikipedia entry, but I didn't find it particularly inspiring. I think you understood my question correctly, yes. I'm just looking for a way to set a perturbed system at t=0 and watch it decay. – CHM Aug 26 '12 at 21:45
• @CHM I re-read it also, and the so-called “method of variation of constants” is what I was taught in my first QM class… it should be a good fit for your problem! – F'x Aug 26 '12 at 21:59