The concept that electrons in an atom can only occupy certain specific energy levels is a fundamental principle of quantum mechanics. This principle is known as the quantization of energy. The idea was first proposed by Niels Bohr in 1913 and later developed further by other physicists.
The proof for the quantization of energy in an atom comes from solving the Schrödinger equation, which is the fundamental equation of quantum mechanics. The Schrödinger equation describes the behavior of quantum particles, including electrons, in terms of wave functions. These wave functions provide information about the probability distribution of finding an electron at a particular energy level.
When the Schrödinger equation is solved for the hydrogen atom, which has only one electron, it yields a set of discrete energy levels. These energy levels correspond to specific orbits or shells, represented by quantum numbers (n=1, 2, 3, ...). The lowest energy level (n=1) is the closest to the nucleus, and the energy increases as the value of n increases.
The proof of quantization arises from the requirement that the wave function of an electron must be single-valued and continuous. The wave function also needs to satisfy boundary conditions, such as vanishing at infinite distances from the nucleus. By solving the Schrödinger equation under these constraints, the only solutions that satisfy these conditions are the ones corresponding to the discrete energy levels.
Experimental evidence also supports the existence of discrete energy levels. Spectroscopy, for example, reveals distinct lines in the electromagnetic spectrum emitted or absorbed by atoms. These spectral lines correspond to transitions between energy levels, and the fact that they are discrete and well-defined further confirms the quantized nature of electron energies.
Overall, the quantization of electron energies in an atom is a consequence of the underlying mathematical framework of quantum mechanics and has been supported by experimental observations.