Particle in A Box - Higher-dimensional Boxes

Higher-dimensional Boxes

If a particle is trapped in a two-dimensional box, it may freely move in the and -directions, between barriers separated by lengths and respectively. Using a similar approach to that of the one-dimensional box, it can be shown that the wavefunctions and energies are given respectively by

,

,

where the two-dimensional wavevector is given by

.

For a three dimensional box, the solutions are

,

,

where the three-dimensional wavevector is given by

.

In general for an n-dimensional box, the solutions are

An interesting feature of the above solutions is that when two or more of the lengths are the same (e.g. ), there are multiple wavefunctions corresponding to the same total energy. For example the wavefunction with has the same energy as the wavefunction with . This situation is called degeneracy and for the case where exactly two degenerate wavefunctions have the same energy that energy level is said to be doubly degenerate. Degeneracy results from symmetry in the system. For the above case two of the lengths are equal so the system is symmetric with respect to a 90° rotation.