We can imagine that the complicated array of moving things which constitute "the World" is something like a chess game being played by the Gods, and we are observers of the game. We do not know what the rules of the game are; all we are allowed to do is to watch the playing. Of course if we watch long enough, we may catch on to a few of the rules.The rules of the game are what we mean by fundamental physics. Even if we know every rule, however...what we really can explain in terms of these rules are very limited, because almost all situations are so enormously complicated that we can not flow the playsof the game using the rules, much less tell what is going to happen next. We must , therefore limit ourselves to the more basic questions of the rules. If we know the rules, we consider that we 'understand' the world.
-Richard Feynman, The making of the Atomic Bomb (1980)
For most problems in Quantum Mechanics, it is extremely difficult to obtain exact solutions of Schrodinger equations and one has to resort to approximate methods. The three most important ones are
- The perturbation method
- The variational method
- The JWKB approximation
The approximations must be using mathematical tools and one of the three methods can be chosen for the specific range of complexities.
There the physical picture of the problem comes in.
For example, Variational methods can give good results for ground state and for an excited state the perturbation can be a better substitute and for smoothly varying potentials the JWKB gives a good result.
What I wish to underline is that the physical picture of the problem, the nature of interactions and the choice of perturbing potentials, do give a physical insight, of course with limitations, and it is not the 'mathematical tool's play' only.
The Quantum Mechanics as such are sometimes seen by students as "Mathematical Physics" but they forget that solutions of those partial differential equations can be a large set but only a few limited by the physical boundary conditions are to be taken in as 'real solutions'.