quantum number — an integer value, one of the discrete quantum states of the electron
lower and upper limits — the lowest and highest x values for the position of the electron
The one-dimensional solution to Schrödinger's equation for an electron in an infinite square potential well (normalized to be of width 1) is , where is the quantum number. The square of this wave function is the probability density function for the electron.
The probability of finding the electron somewhere inside the square well is 1. (snapshot 1)
The probability of finding the electron with a quantum number of 3 between 0.2 and 0.8 is approximately 0.54. (snapshot 2)
The probability of finding the electron with a quantum number of 4 between 0.25 and 0.75 is approximately 0.5. (snapshot 3)