# Simple explaination of pertibation theory

• venomxx
In summary, perturbation theory is a tool used to predict small corrections to the Hamiltonian in order to obtain a more accurate result. These corrections can be crucial in understanding the behavior of quantum states, especially when introducing new physics such as spin effects. While it may not always provide exact analytical solutions, perturbation theory is a valuable tool in quantum mechanics for predicting energy gaps between shifted states. However, for more complex systems like helium, the variational method is often preferred over perturbation 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!

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)

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

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!

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.

## 1. What is perturbation theory?

Perturbation theory is a mathematical tool used in science and engineering to approximate solutions for complex problems by breaking them down into simpler, more manageable parts. It is often used when the exact solution cannot be easily determined.

## 2. How does perturbation theory work?

Perturbation theory works by introducing a small parameter, called the perturbation parameter, into a system of equations. This parameter allows us to approximate the solution by considering the effects of the perturbations on the system's behavior.

## 3. What are the applications of perturbation theory?

Perturbation theory has many applications in physics, chemistry, and engineering. It is used to study the behavior of systems in quantum mechanics, celestial mechanics, fluid dynamics, and many other fields where complex systems can be broken down into simpler components.

## 4. What are the limitations of perturbation theory?

Perturbation theory is most effective when the perturbations are small and the system is close to equilibrium. If the perturbations are too large or the system is far from equilibrium, the approximation may not accurately reflect the true behavior of the system.

## 5. How is perturbation theory related to other mathematical methods?

Perturbation theory is closely related to other mathematical methods such as Taylor series, power series, and asymptotic expansions. These methods are often used to approximate solutions for problems that cannot be solved exactly.

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