In quantum mechanics, the atom is described using the Schrödinger equation, which allows us to calculate the behavior of electrons in the presence of a positively charged nucleus (usually a proton or a collection of protons). According to this model, the electron does not exist as a classical particle with a well-defined trajectory or position. Instead, it is described by a wave function that represents the probability distribution of finding the electron at different locations around the nucleus.
In the context of the Schrödinger equation, there is no concept of a "gap" between the electron and the proton. The electron's wave function extends throughout the entire space around the nucleus, including regions very close to the nucleus itself. However, it is important to note that the electron's probability of being found at a particular distance from the nucleus decreases as the distance decreases.
Furthermore, quantum mechanics tells us that there is a minimum energy associated with the electron in an atom, known as the ground state energy. This energy arises from the attractive electrostatic force between the negatively charged electron and the positively charged nucleus. The electron's wave function is typically peaked at a certain average distance from the nucleus, which is determined by the energy level it occupies.
In summary, while there is no classical gap between the electron and the proton in an atom, the electron's wave function extends throughout the atom, and the probability of finding the electron decreases as you get closer to the nucleus.