To use the wave function for a particle in a box to calculate the probability that the particle is found in various regions within the box. The quantum mechanical probability that a particle described by the (normalized) wave function ψ(x) is found in the region between x=a and x=b is P=∫ba|ψ(x)|2dx. The specific example of a particle trapped in an infinitely deep potential well, sometimes called a particle in a box, serves as good practice for calculating these probabilities, because the wave functions for this situation are easy to write down. If the ends of the box are at x=0 and x=L, then the allowed wave functions are ψ(x)={2L−−√sin(nπxL),0, for 0≤x≤L, for all other x, where n=1 is the ground-state wave function, n=2 is the first excited state, etc. Here are a few integrals that may prove useful: ∫sin(kx)dx=−1kcos(kx)+C, ∫cos(kx)dx=1ksin(kx)+C, ∫sin2(kx)dx=x2−14ksin(2kx)+C, and ∫cos2(kx)dx=x2+14ksin(2kx)+