# Diagram-like perturbation theory in quantum mechanics

There seems to be a formalism of quantum mechanics perturbation that involve something like Feynman diagrams. The advantage is that contrary to the complicated formulas in standard texts, this formalism is intuitive and takes almost zero effort to remember (to arbitrary orders).

For example, consider a two level atom $\{|g\rangle, |e\rangle\}$ coupled to an external ac electric field of frequency $\omega$. Denote the perturbation by $\hat V$, with nonzero matrix element $\langle e|\hat V |g\rangle$.

Then the second order energy correction reads $$E^{(2)} = \langle e|\hat V |g\rangle\frac{1}{\omega_g - \omega_e +\omega} \langle g|\hat V |e\rangle + \langle e|\hat V |g\rangle\frac{1}{\omega_g - \omega_e -\omega} \langle g|\hat V |e\rangle$$ where the first term corresponds to the process absorb a photon then emit a photon while the second process is emit a photon then absorb a photon.

Does anybody know the name of this formalism? And why it is equivalent to the formalism found in standard texts?

## 2 Answers

There is an exposition of a diagrammatic representation of the terms in the quantum mechanical perturbation expansion here. Basically the diagrams are just used to represent the combinatorial properties resulting from eigenstate degeneracy.

(Just for fun there is also a diagrammatic approach to perturbation expansions in classical mechanics here).

• Thank you! But the first reference is sophisticated and too mathematical. The formalism I'm looking for is quite intuitive and physical. – Machine Oct 28 '12 at 5:59

Try Shankar Intro to Quantum Mechanincs page 489, he discusses the mathematical connection and the relation to feynman diagrams, or rather all the possible paths of interaction: If you find something better, I'd be curious to know too.