Normalize Wave Function for Particle in a Box

Normalize the wave function $\psi(x) = x(L-x)$ for a particle in a box of length $L$.

In this problem, we are tasked with normalizing a wave function for a particle confined in a one-dimensional box. Normalization is a key concept in quantum mechanics because it ensures that the total probability of finding a particle within a given space is equal to one. For this reason, it is essential to understand how to apply the normalization condition to wave functions, especially in systems with defined boundaries such as the infinite square well potential or particle in a box scenarios. The wave function given is a polynomial function of the position variable, x, and includes the boundary defined by the length L of the box. Normalizing involves integrating the absolute square of the wave function over the entire space in which the particle can be found and setting this integral equal to one. In this particular problem, the integral will be evaluated from 0 to L, as these are the limits of the particle's confinement. Understanding the normalization process for wave functions not only reinforces mathematical techniques involving integration but also deepens comprehension of how quantum states are described probabilistically. This problem also serves as a fundamental example of how boundary conditions influence the solutions and behaviors of wave functions in quantum systems.