Assume the potential U(x) in the time-independent Schrodinger equation to be zero inside a one-dimensional box of length L and infinite outside the box. For a particle inside the box a free particle wavefunction is appropriate, but since the probability of finding the particle outside the box is zero, the wavefunction must go to zero at the walls. This constrains the form of the solution to
For the particle in a box with infinite walls, the probability must be equal to one for finding it within the box. The condition for normalization is then
For the finite potential well, the solution to the Schrodinger equation gives a wavefunction with an exponentially decaying penetration into the classicallly forbidden region.
Confining a particle to a smaller space requires a larger confinement energy. Since the wavefunction penetration effectively "enlarges the box", the finite well energy levels are lower than those for the infinite well.
For a potential which is zero over a length L and has a finite value for other values of x, the solution to the Schrodinger equation has the form of the free-particle wavefunction for -L/2 < x < L/2 and elsewhere must satisfy the equation With the substitution this may be written in the form:
The energy levels for an electron in a potential well of depth 64 eV and width 0.39 nm are shown in comparison with the energy levels of an infinite well of the same size.