Consider a particle in a one-dimensional infinite square well potential of width $L$. Solving the Time Independent Schrödinger equation, and considering the boundary conditions, gives wavefunction solutions with wavenumbers: $k = \frac{n\pi}{L}$ For $n=1,2,3,...$
Solving a system of a periodic potential with infinite potential walls gives wavefunction solutions with wavenumbers: $k=\frac{2\pi n}{L}$ for $n= \pm1, \pm2, \pm3, ...$
This means that the energy eigenvalues for a periodic potential are four times larger than the ones for a box potential.
- Why are only positive wavenumbers allowed for the wavefunction of a particle in a box?
- Intuitively, why do particles in periodic potentials have higher wavenumbers and energies?