In quantum mechanics, the principal quantum number (denoted by "n") describes the energy levels or shells of an electron in an atom. The higher the value of n, the greater the average distance of the electron from the nucleus and the higher the energy level.
The energy of an electron in an atom is determined by its position and motion, and it can be described by its wave function. The wave function is a mathematical expression that describes the probability distribution of finding the electron in different locations around the nucleus.
When an electron occupies an orbital with a higher principal quantum number (higher energy level), it is in a state that is farther away from the nucleus. This increased distance results in a weaker attraction between the electron and the positively charged nucleus. The electron experiences less electrostatic attraction, which means it has a higher potential energy.
According to the Bohr model of the atom, electrons exist in quantized energy levels or shells. The energy levels are represented by discrete values. Electrons can transition between these energy levels by absorbing or emitting specific amounts of energy in the form of photons.
The energy difference between two energy levels is given by ΔE = E_final - E_initial, where E_final and E_initial represent the energies of the final and initial states, respectively. For electrons in an atom, the energy difference between two energy levels decreases as the principal quantum number increases.
In summary, an electron in a higher principal quantum number has more energy than one with a lower principal quantum number because it occupies an orbital that is farther away from the nucleus, resulting in weaker electrostatic attraction and higher potential energy.