include the centered box (xc = 0) and the shifted box (xc = L/2) (pictured). Position wave function In quantum mechanics, the wavefunction gives the most fundamental description of the behavior of a particle; the measurable properties of the particle (such as its position, momentum and energy) may all be derived from the wavefunction. The wavefunction can be found by solving the Schrödinger equation for the system where is the reduced Planck constant, is the mass of the particle, is the imaginary unit and is time. Inside the box, no forces act upon the particle, which means that the part of the wavefunction inside the box oscillates through space and time with the same form as a free particle: where and are arbitrary complex numbers. The frequency of the oscillations through space and time is given by the wavenumber and the angular frequency respectively. These are both related to the total energy of the particle by the expression which is known as the dispersion relation for a free particle. Here one must notice that now, since the particle is not entirely free but under the influence of a potential (the potential V described above), the energy of the particle given above is not the same thing as where p is the momentum of the particle, and thus the wavenumber k above actually describes the energy states of the particle, not the momentum states (i.e. it turns out that the momentum of the particle is not given by ). In this sense, it is quite dangerous to call the number k a wavenumber, since it is not related to momentum like "wavenumber" usually is. The rationale for calling k the wavenumber is that it enumerates the number of crests that the wavefunction has inside the box, and in this sense it is a wavenumber. This discrepancy can be seen more clearly below, when we find out that the energy spectrum of the particle is discrete (only discrete values of energy are allowed) but the momentum spectrum is continuous (momentum can vary continuously) and in particular, the relation for the energy and momentum of the particle does not hold. As said above, the reason this relation between energy and momentum does not hold is that the particle is not free, but there is a potential V in the system, and the energy of the particle is , where T is the kinetic