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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?

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    $\begingroup$ Yes, it can always be split in this way. If $ H = 4 $, Then $H_0 = 4 - t$ and $W(t) = t$ is a valid split. The question is whether such a split would be useful. Ideally, the $W(t)$ is small compared to $H$ so that the resulting perturbation series gives a good result. $\endgroup$
    – QuantumDot
    Feb 13 '19 at 7:48
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    $\begingroup$ When we do split the total hamiltonian, wouldnt we do it such that we have a time independent hamiltonian Ho and a time dependent perturbation W(t)? Your answer has both the terms as time dependent. $\endgroup$
    – prananna
    Feb 15 '19 at 2:46
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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 $$-\mu_B\mathbf{S}\cdot\mathbf{B}.$$
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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.

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  • $\begingroup$ Is it correct if I say the following: To use perturbation theory, we need to have an exact solution to an unperturbed system ( and the unperturbed system can be time dependent or independent. It doesnt matter as long as we are able to solve the unperturbed system exactly). So Ho can be a time dependent as well, in the above equation. $\endgroup$
    – prananna
    Feb 15 '19 at 2:52
  • $\begingroup$ Perturbation theory provides you with a systematic way to obtain an analytic solution to a problem which otherwise might not possess (as long as the coupling is small). In quantum mechanics, we study these as corrections to the wavefunction and energy levels that are introduced by slightly disturbing the system. $\endgroup$
    – Judas503
    Feb 15 '19 at 3:20
  • $\begingroup$ I'm sorry but I'm still not clear as to whether Ho is time dependent or time independent. In "Quantum Mechanics by L E Ballentine, Pg 349, " the author talks about the time dependent perturbation theory. There he explicitly mentions that the time dependence is confined to the perturbation term "λW(t)". My question is, is this just an assumption or is there a formal proof that any perturbed hamiltonian can be split into a time independent and time dependent parts? $\endgroup$
    – prananna
    Feb 15 '19 at 5:18
  • $\begingroup$ The term 'perturbed Hamiltonian' speaks for itself. We have very few exactly solvable systems in quantum mechanics. Those Hamiltonians are your H0. If you then want to study how an external influence changes your system, you include a perturbation. Whether it is time-dependent or not depends on the kind of physics you are interested in. $\endgroup$
    – Judas503
    Feb 15 '19 at 5:34
  • $\begingroup$ As we have a very few exactly solvable systems, do we have a choice on selecting what Ho can be among these few systems? By this I mean, given a general perturbed Hamiltonian, can I split it into Ho+λW(t) with Ho being the Hamiltonian of any of the exactly solvable systems? Or is there any way we associate a given perturbed Hamiltonian with a specific exactly solvable system? Thank you for the above answer. That was exactly what I was looking for. $\endgroup$
    – prananna
    Feb 15 '19 at 7:52

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