According to the mathematical framework of quantum mechanics, there is a non-zero probability for an object, such as a vase on your table, to appear at a different location through a phenomenon known as quantum tunneling. However, it is crucial to understand the context and the scale at which quantum tunneling operates.
Quantum tunneling is a quantum phenomenon that allows a particle to penetrate a classically forbidden energy barrier. It arises from the wave-like nature of particles and the inherent uncertainty associated with their position and momentum. In theory, there is a finite probability for a particle to tunnel through an energy barrier even if it does not have enough classical energy to overcome it.
However, the probability of macroscopic objects, such as a vase, undergoing quantum tunneling to appear on the Moon is extremely vanishingly small. The reason is that the probability of quantum tunneling decreases exponentially with the mass and complexity of the object involved. For a macroscopic object like a vase, the probability of tunneling over such vast distances and overcoming the enormous energy barriers is effectively negligible.
Quantum tunneling is more relevant and observable at the atomic and subatomic scales. It plays a crucial role in phenomena like radioactive decay, scanning tunneling microscopy, and various semiconductor devices. But for everyday objects in our macroscopic world, the effects of quantum tunneling are incredibly tiny and virtually impossible to observe.
So, while quantum mechanics allows for the possibility of quantum tunneling, the likelihood of a vase on your table spontaneously appearing on the Moon due to quantum tunneling is infinitesimally small and can be safely considered negligible.