<Dd> E = ħ ω = ħ 2 k 2 2 m, (\ displaystyle E = \ hbar \ omega = (\ frac (\ hbar ^ (2) k ^ (2)) (2m)),) </Dd> <P> 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 p 2 2 m (\ displaystyle (\ frac (p ^ (2)) (2m))) 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 p = ħ k (\ displaystyle p = \ hbar k)). 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 E = p 2 2 m (\ displaystyle E = (\ frac (p ^ (2)) (2m))) 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 E = T + V (\ displaystyle E = T + V), where T is the kinetic and V the potential energy . </P> <P> The size (or amplitude) of the wavefunction at a given position is related to the probability of finding a particle there by P (x, t) = ψ (x, t) 2 (\ displaystyle P (x, t) = \ psi (x, t) ^ (2)). 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 </P> <Dl> <Dd> ψ n (x, t) = (A sin ⁡ (k n (x − x c + L 2)) e − i ω n t, x c − L 2 <x <x c + L 2, 0, otherwise, (\ displaystyle \ psi _ (n) (x, t) = (\ begin (cases) A \ sin \ left (k_ (n) \ left (x-x_ (c) + (\ tfrac (L) (2)) \ right) \ right) \ mathrm (e) ^ (- i \ omega _ (n) t), &x_ (c) - (\ tfrac (L) (2)) <x <x_ (c) + (\ tfrac (L) (2)), \ \ 0, & (\ text (otherwise,)) \ end (cases))) </Dd> </Dl>

The total energy eigenfunctions of the infinitely deep box are