Particle in a Box: Probability Density and Positional Variance
The square of the absolute value of the Schrödinger wavefunction equals the probability density of finding a particle at any point in a box. It is relatively straightforward to show that for all values of , the average or expectation value of the position of the particle will be at the exact midpoint of the box (, where is the length of the box) . The variance, however, increases with increasing quantum number, to a limiting value of . This Demonstration plots the probability density (in blue) and a normal distribution (in yellow) with mean and variance to represent the change in the variance with increasing .
The variance is , where the expectation values and are obtained from and , respectively, where is the length of the box. The (normalized) probability function for the one-dimensional particle in a box is given by :
 D. A. McQuarrie and J. D. Simon, Physical Chemistry: A Molecular Approach, Sausalito, CA: University Science Books, 1997.