SKEDSOFT

Introduction and Explanation:

We know that the probability density of the particle is given by the square of the wavefunction; we also know that the particle must be somewhere in the box. Therefore, if we add up the total probability density in the box we must get the answer 1, i.e. there is 100% probability of finding the particle somewhere in the box.

Mathematical representation:

Mathematically, adding up this probability density in the box corresponds to integrating the square of the wavefunction in the range 0 to L:

where to go to the last line we have recalled that as n is an integer, sin(2nπ ) is zero. This integral of [ψ(x)]² gives the total probability, and so should have the value 1; this is plainly not the case. However, since we have shown that ψ(x) = A sin(kx) solves the SE for any value of the constant A, we can choose the value of A so that the integral is equal to 1:

With this choice of A the wavefunction is said to be normalized, meaning that the total probability it predicts is indeed equal to one. A is called the normalization constant or normalization factor.