Simple explaination of pertibation theory

[venomxx]

Hi,

I understand pertibation theory is very important in predicting the energy of say, the ground state of helium. My qualitative understanding goes as far as it makes small corrections to the hamiltonian to get a more accurate result...

Can anyone expand on this for me? Or make it clearer? No major math necessary id be happy with just a simple explanation!

 

[venomxx]

Found something interesting...for the record 'Quantum physics of atoms, molecules, solids, nuclei and particles by Eisberg and Resnick covers it nicely in the appendix (J)

 

[Snarky Fellow]

Well, perturbation theory is mainly not only the tool to predict small corrections. The thing is that even small changes in the hamiltonian (which are usually connected with some new physics) can result in dramatically new behaviour. Imagine that we have two quantum states with the same energy. For example, we consider no effects connected with spins. Thus these states are up and down states. Then we want to introduce some spin-effect so that up state can't be considered on equal foot with down-state. (The simplest example is when we apply a magnetic field). Then the energy of one state decreases and, on the contrary, the energy of the other state increases. Therefore perturbation theory may help us to predict the energy gap between these shifted states. In the case of just a magnetic field we can easily solve the problem without any perturbation theory. But often (when perturbation is more sophisticated) the problem may have no exact analytical solution.

Mathematically, you write $$\hat H = \hat H_0 + \hat V, \psi = \psi_0 +\delta \psi$$, Schroedinger equation for $$\hat H_0$$ and $$\psi_0$$, and then for $$\hat H$$ and $$\psi$$ Remember that $$\hat V \ll \hat H_0, \delta \psi \ll \psi$$, so that you can neglect $$V\delta \cdot \psi$$ terms in the first order theory

 

[venomxx]

Cheers Snarky Fellow, that's a great help. I can read on now, i just wish quantum mechanic books had good concise explanations like this!

 

[alxm]

Just a nitpick: While it's a common textbook example, perturbation theory (to the first order) doesn't actually do a great job of helium, or even for atoms/molecules. The variational method is therefore usually used.