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I don't seem to understand what perturbation theory really is, and what it is needed for. Can someone please provide an explanation for what it is, and why it is needed in QM?

There are two main reasons.

The first is practical, and it is that QM is hard. Very few systems are solvable, particularly in the presence of interactions. There's a wide variety of situations where perturbation theory is the only available way to make any headway at all.

The second one is conceptual and it is basically that it allows us to establish a hierarchy in the strengths of the different components of a system, and it allows us to have a clearer picture of how this strengths scale with the parameters of the system. Thus, even if you can solve the full system, if one component is weak, it often makes sense to treat it explicitly as a perturbation, so that you can keep a clearer track of how the different aspects of the solution (energies, eigenfunctions, dipole moments, what have you) scale.

Perturbation Theory in Quantum Mechanics is probably deep rooted with the kind of the mathematical formulation of Quantum Mechanics in terms of Self Adjoint Operators (Von Neumann formulation )and specially the nature of analytical functions expected to describe Physical parameters on quantum physics :The energy bound state levels of a given quantum Hamiltonian is always expected to be at worse a meromorphic function of the "tuning" parameters on the theory (essentially those coupling constants mediating the interaction of the systems interacting to made up a bigger one (the full Hamiltonian is always constructed mathematically rigorously from PERTURBATION KATO-RELICH like self adjoint theorems -The general Atomic Hmiltonian for instance ) .Even if one wishes to handle non perturbative phenomena in Quantum Physics ,in much of the cases , one must reformulate the problem in news variables where now the non perturbative scheme becomes a perturbative scheme .However ,computer oriented calculations are somewhat non perturbative in their mathematical nature , but they remains used in a perturbative scheme .So , concluding :Our Differential Integral calculus and Functional Analysis thinking on Quantum Mechanics and Classical Physics remains almost complete only for sort of "Analytical " functions and their close variants .The reader should compare with the mathematical thinking need in probability theory problems , where "analytical functions thinking-perturbation theory is almost zero !(All sample function in space functions of Measurable sets are at best continuous functions like those made up Wienner trajectories ) .