Broadly speaking, the scientific method in physics is
1) observation – 2) theory – 3) observation etc.
1) In the case of quantum mechanics the observation of spectral lines had to be explained.
2) The Schrödinger equation, embedded in the theoretical framework of quantum mechanics, explains the discreteness of the spectral lines and facilitates computing their corresponding frequencies for the most simple atoms like Hydrogen or Helium.
3) Further subtle observations of spectral lines detected the fine structure.
4) The introduction of electron spin allows to explain the fine structure.
Quantum mechanics did not change this established scientific method.
Also Popper’s emphasis on the principle of falsification is not restricted to quantum mechanics. It is an idealized description of the relation between experiment and theory, which applies to all natural sciences. Falsification is independent from Popper’s opinion concerning the Copenhagen interpretation.
A paradigm for the revolutionary step of quantum mechanics is the Heisenberg uncertainty relation. It shows that the classical concept with physical observables, having always a definite value, has to be abandoned in the domain of microphysics.
Hence from the viewpoint of philosophy of nature quantum mechanics has created a lot of deep open questions, in the domain of ontology as well as in the domain of epistemology. Does the concept of individuality pertain to microphysics, do the values of certain observables originate in the act of observation, how to speak about microcosmos with our concepts from mesocosmos? Some of these questions are examined by Heisenberg in his lecture Physics and Philosophy.
The coincidence of quantum mechanics and of the successors like quantum electrodynamics with the observed values from observation is striking. But the interpretation of the mathematical framework is still discussed. The main challenge is to explain the transfer during the act of measuremen from the microcosmic world of possibilities to the mesocosmic world of definite results. The best explanation at hand is the mechanism of decoherence.
Hence quantum mechanics provides us with an example of a theory which makes very precise predictions. But simultaneously, quantum mechanics provides a challenge for interpretation for each generation since its development at about 1925.