# Perturbation theory in second quantization

I am dealing with electron/phonon interaction in QM. In particular, given the Hamiltonian of a solid,

$$H=H_\text{el}+H_\text{ion}+H_\text{el-ion}$$

we have that the el-phonon Hamiltonian is treated perturbatively with respect to $H_\text{el}+H_\text{ion}$

and, neglecting Umklapp processes we have

$$H_\text{el-ion}=\Sigma_{\vec{q}} \;\nu(\vec{q})\cdot (a^\dagger(-\vec{q})+ a(\vec{q}))\cdot c^\dagger_{\vec{k}+\vec{q}}c_{\vec{k}}$$

According to this Hamiltonian we can see that only two first order processes are admitted (emission of a phonon of momentum $-\vec{q}$ and absorption of a phonon of momentum $\vec{q}$).

Then, supposing to know all the states of the unperturbed Hamiltonian $H_\text{el}+H_\text{ion}$, denoting them with $|\psi_n^{(0)}\rangle$ we compute the correction to the ground state of this Hamiltonian using perturbation theory, obtaining

$$E_{GS}^1=\langle\psi_0^{(0)}|H_\text{el-ion}|\psi_0^{(0)}\rangle=0$$

meaning that 1st order processes (absorption/emission) do not change energy levels, while

$$E_{GS}^2=\Sigma_{n>0}\frac{|\langle\psi_{n}^{(0)}|H_\text{el-ion}|\psi_0^{(0)}\rangle|^2}{E_0^{(0)}-E_n^{(0)}}\neq 0$$

meaning that the 2nd order process (el/el effective attractive coupling due to an exchange of a phonon) changes energy.

It seems me that there is a relation between order of correction in perturbation theory and order of the process that caused that correction (and physically this is intuitive because in 1st order correction calculations involve only one wavefunction while in second order correction we have 2 different wavefunctions involved).

Is what I am saying correct? If yes, what is the formal way to say this? In other words, is there a relationship between the order of a process and order in perturbation theory correction?

• Yes you are correct and it becomes more obvious in the path integral formulation. I'm not proficient enough to demonstrate this on my own but you should look up things on path integral and Feynman diagrams. – Shane P Kelly Jul 31 '15 at 16:03
• I'm a little puzzled by what you're asking. You say "It seems me that there is a relation between order of correction in perturbation theory and order of the process that caused that correction" but that's trivially true, in a sense by definition. The "order of a correction in perturbation theory" and the "order of a process" are literally the same thing. Are you trying to ask why, physically, energy levels are affected by 2nd order processes? – DanielSank Feb 23 '16 at 3:09