The boundary conditions in the diagram above force the particle to be located in a region of space between 0 and L. The potential energy is indfinite outside of these limits and V = 0 inside the limits. Threfore, the wave equation is the same as for the free particle, but with the difference that the boundary conditions define the wave function. They define both the phase of the wave function and the wave vector, k. The reason for this is that the wave function must vanish at the boundaries. Another way to say this is that the probability of finding the particle outside the box is zero and therefore the probability at both boundaries is also zero.
The solutions are quantized. First the energy increases proportional to the square of the quantum number n. Secondly, the wave functions for each solution are orthogonal to one another. This means that if we multiply any two wave functions and integrate over all space we will obtain a result of zero, unless we multiply a wave function times itself. In the case that we square the wave function we will not get zero. In that case, we will be able to use the wave function to calculate probability. However, for that purpose we will ned to normalize the wave function. We will consider normalization in the next section.
