Perturbation Theory - Writing the perturbed Hamiltonian as the sum of unperturbed and the perturbation

Question

In the time dependent perturbation theory of quantum mechanics, we start off with the assumption that the Hamiltonian of the perturbed system can be written as the sum of the Hamiltonian of the unperturbed system and the perturbation itself. i.e., $$ H = H_0 + W(t) $$ where W(t) is the time dependent perturbation. My query is when is this possible? Is it always true that the Hamiltonian of the perturbed system can be split in this way? If not, can you provide me with an example where splitting the total Hamiltonian is not possible?

The reason I'm asking this question is because, as far as I understand, the perturbations always reveal themselves in the form of time varying potential energy function, in which the time dependency is woven into the potential energy function. How are we separating out the time dependent part out here?

Answer 1

What is perturbation theory? Perturbation theory is an approximation technique, based on splitting the Hamiltonian into an (exactly) solvable part, $H_0$ and a perturbation, $W$, and then building the solution as an expansion in the strength of the perturbation.

When is it applicable? This technique is not always applicable: in some problems such separation is impossible, and even when it is possible, the perturbation series may not converge, or may not contain some important solutions (if additional approximations are involved). On the other hand both parts of the Hamiltonian may be time-dependent or time-independent. Quantum mechanics textbooks usually consider only the cases with time-independent $H_0$, but with perturbation independent and dependent on time. This is an important technique that provides solution to many problems, and therefore is a must know for anyone learning quantum mechanics.

Is perturbation always a potential? Quantum mechanics treats only electromagnetic interactions (excluding gravity, strong and weak forces, considered by more advanced disciplines). Thus, all the interactions in QM are through the electric and magnetic fields or potentials. Specifically:

• Charge coupling to an electric field via a potential term
• Coupling to a magnetic field via a vector pitential, in the form $$H=\frac{1}{2m}\left(\mathbf{p}-\frac{e}{c}\mathbf{A}\right)^2,$$
• Coupling of spin to magnetic field via Zeeman interaction $$-\mu_B\mathbf{S}\cdot\mathbf{B}.$$

Answer 2

By my understanding, perturbation theory deals with solving a quantum (or classical) system systematically around a parameter $\lambda$. We say we are using perturbation theory when we start out with a system which we can analytically solve and introduce a small perturbation such that $$ H(t)=H_{0}+\lambda W(t) \;\;\;\;\;\;\; \lambda<<1$$ We are not splitting a Hamiltonian into two parts. Rather, we are studying different mechanisms of influences on a known system and studying them order by order in terms of $\lambda$. Also, the perturbations don't always have to be time dependent. Popular examples are the Stark and Zeeman effect of the Hydrogen atom.