In the current formulation of quantum mechanics, known as the standard quantum mechanics or quantum theory, there isn't a single missing equation that is required to complete the theory. Quantum mechanics is based on a mathematical framework that includes several fundamental equations, such as Schrödinger's equation, Heisenberg's uncertainty principle, and various mathematical formalisms for describing quantum systems and their evolution.
Schrödinger's equation, which I mentioned earlier, describes how the wavefunction of a quantum system evolves over time. It provides a mathematical representation of the behavior of particles at the quantum level. Heisenberg's uncertainty principle, on the other hand, states that there is a fundamental limit to how precisely certain pairs of physical properties, such as position and momentum, can be simultaneously known.
While the existing equations of quantum mechanics are highly successful in describing a wide range of phenomena and have been extensively tested and verified through experiments, there are still open questions and areas of ongoing research in quantum physics. Some of these questions include the nature of wave-particle duality, the measurement problem, the interpretation of the wavefunction, and the potential compatibility of quantum mechanics with other fundamental theories, such as general relativity.
In the field of quantum gravity, which seeks to unify quantum mechanics with general relativity, there is ongoing research and the exploration of potential new equations or modifications to existing equations to describe the behavior of particles in the framework of a theory of quantum gravity. However, such efforts are still in progress, and a complete and universally accepted theory of quantum gravity has not yet been established.
In summary, while the current formulation of quantum mechanics is mathematically complete and has been successful in explaining a wide range of phenomena, there are still open questions and ongoing research in certain areas of quantum physics. These questions may require further advancements and potentially new equations to provide a more comprehensive understanding of the quantum world.