No, two electrons cannot have the same value for the principal quantum number (n), at least within a given atom. The principal quantum number specifies the energy level or shell that an electron occupies in an atom. According to the Pauli exclusion principle, no two electrons in an atom can have the exact same set of quantum numbers.
In addition to the principal quantum number (n), there are three other quantum numbers that define the unique quantum state of an electron: the azimuthal quantum number (l), the magnetic quantum number (m), and the spin quantum number (s). The combination of these quantum numbers determines the specific electron state.
For a given atom, the principal quantum number (n) can take on integer values starting from 1 (representing the first energy level) and increasing upwards. Each energy level can accommodate a specific number of electrons, depending on the subshells within that level. The azimuthal quantum number (l) further divides the energy level into subshells, and each subshell can hold a certain number of electrons with different magnetic quantum numbers (m).
Therefore, while multiple electrons within an atom can have the same value for the principal quantum number (n), they will have different values for the other quantum numbers, ensuring that no two electrons occupy the exact same quantum state. This principle is fundamental to the electron configuration and the behavior of electrons in atoms.