and V the potential energy. The size (or amplitude) of the wavefunction at a given position is related to the probability of finding a particle there by . The wavefunction must therefore vanish everywhere beyond the edges of the box. Also, the amplitude of the wavefunction may not "jump" abruptly from one point to the next. These two conditions are only satisfied by wavefunctions with the form where and where n is a positive integer (1, 2, 3, 4, ...). For a shifted box (xc = L/2), the solution is particularly simple. The simplest solutions, or both yield the trivial wavefunction , which describes a particle that does not exist anywhere in the system. Negative values of are neglected, since they give wavefunctions identical to the positive solutions except for a physically unimportant sign change. Here one sees that only a discrete set of energy values and wavenumbers k are allowed for the particle. Usually in quantum mechanics it is also demanded that the derivative of the wavefunction in addition to the wavefunction itself be continuous; here this demand would lead to the only solution being the constant zero function, which is not what we desire, so we give up this demand (as this system with infinite potential can be regarded as a nonphysical abstract limiting case, we can treat it as such and "bend the rules"). Note that giving up this demand means that the wavefunction is not a differentiable function at the boundary of the box, and thus it can be said that the wavefunction does not solve the Schrödinger equation at the boundary points and (but does solve it everywhere else). Finally, the unknown constant may be found by normalizing the wavefunction so that the total probability density of finding the particle in the system is 1. Mathematically, (The particle must be somewhere) It follows that Thus, A may be any complex number with absolute value ; these different values of A yield the same physical state, so A = can be selected to simplify. It is expected that the eigenvalues, i.e., the energy of the box should be the same regardless of its position in space, but changes. Notice that represents a phase shift in the wave function. This phase shift has no effect when solving the Schrödinger equation, and therefore does not affect the eigenvalue. If we set the origin of coordinates to the center