No, a hydrogen atom cannot absorb a photon whose energy is greater than its binding energy (13.6 eV) and remain as a bound state. According to the principles of quantum mechanics, the binding energy of an electron in an atom corresponds to the energy difference between the bound state and the free state of the electron.
When a photon interacts with an atom, it can be absorbed by an electron, causing the electron to transition to a higher energy level or be completely ejected from the atom. However, if the energy of the photon is greater than the binding energy of the electron, the excess energy cannot be absorbed by the electron and the excess energy will not contribute to the binding of the electron to the nucleus.
In the case of a hydrogen atom, if a photon with energy greater than 13.6 eV interacts with the atom, the excess energy will be imparted to the electron as kinetic energy, causing the electron to be completely ejected from the atom. This process is known as ionization, and the resulting hydrogen atom will be left in an ionized state.
In summary, a hydrogen atom cannot absorb a photon with energy greater than its binding energy and remain as a bound state. The excess energy will cause ionization of the atom, resulting in a free electron and an ionized hydrogen atom.