It is not accurate to say that physicists do not have a dynamically complete quantum mechanical model. In fact, quantum mechanics provides a highly successful framework for understanding and describing a wide range of phenomena, including the formation of atoms such as hydrogen.
In the case of the capture of an electron by a proton to form a hydrogen atom, quantum mechanics provides a comprehensive description of the process. The dynamics of this process can be accurately modeled using the Schrödinger equation, which describes the time evolution of quantum systems.
Quantum mechanics predicts the probability distribution for the location of the electron around the proton, as well as its energy levels and other properties. The behavior of electrons and protons, including their interactions, can be described using quantum field theory, which is a framework that extends quantum mechanics to include relativistic effects.
However, it is important to note that while quantum mechanics provides accurate predictions and is supported by extensive experimental evidence, there are still open questions and ongoing research in certain areas. For example, in the case of hydrogen, precise calculations of its energy levels and other properties can involve complex mathematical techniques and approximations due to the inherent mathematical challenges of solving the Schrödinger equation exactly for systems with more than one electron.
Additionally, there are phenomena that are not fully understood within the current framework of quantum mechanics, such as the unification of quantum mechanics with general relativity (the theory of gravity). This is an active area of research, and physicists are working on developing more comprehensive theories, such as quantum gravity, to address these open questions.
Overall, while there are still areas of ongoing research and exploration, quantum mechanics provides a robust and successful framework for understanding the dynamics of atoms and a wide range of other physical systems.