STANDING-WAVE FUNCTIONS FOR A PARTICLE IN A BOX
The number 'n' is called a quantum number. It characterizes the wave function for a particular state and for the energy of that state. In our one-dimensional problem, a quantum number arises from the boundary condition on the wave function that it must be zero at x=o and x=L.
The solution of a classical mechanics problem is typically specified by giving the position of a particle as a function of time. But the wave nature of matter prevents us from doing this for microscopic systems. The most that we can know is the relative probability of measuring a certain value of the position If we measure the position for a large number of identical systems, we get a range of values corresponding to the probability distribution.