Editing Particle in a box (section)

== Higher-dimensional boxes ==

=== (Hyper-)rectangular walls ===

[[Image:Particle2D.svg|thumb|320px|right|The wavefunction of a 2D well with n<sub>x</sub>=4 and n<sub>y</sub>=4]]

If a particle is trapped in a two-dimensional box, it may freely move in the <math>x</math> and <math>y</math>-directions, between barriers separated by lengths <math>L_x</math> and <math>L_y</math> respectively.  For a centered box, the position wave function may be written including the length of the box as <math>\psi_n(x,t,L)</math>. Using a similar approach to that of the one-dimensional box, it can be shown that the wave functions and energies for a centered box are given respectively by
<math display="block">\psi_{n_x,n_y} = \psi_{n_x}(x,t,L_x)\psi_{n_y}(y,t,L_y),</math>
<math display="block">E_{n_x,n_y} = \frac{\hbar^2 k_{n_x,n_y}^2}{2m},</math>
where the two-dimensional [[wavevector]] is given by
<math display="block">\mathbf{k}_{n_x,n_y} = k_{n_x}\mathbf{\hat{x}} + k_{n_y}\mathbf{\hat{y}} = \frac{n_x \pi }{L_x} \mathbf{\hat{x}} + \frac{n_y \pi }{L_y} \mathbf{\hat{y}}.</math>

For a three dimensional box, the solutions are
<math display="block">\psi_{n_x,n_y,n_z} = \psi_{n_x}(x,t,L_x)\psi_{n_y}(y,t,L_y) \psi_{n_z}(z,t,L_z),</math>
<math display="block">E_{n_x,n_y,n_z} = \frac{\hbar^2 k_{n_x,n_y,n_z}^2}{2m}, </math>
where the three-dimensional wavevector is given by:
<math display="block">\mathbf{k}_{n_x,n_y,n_z} = k_{n_x}\mathbf{\hat{x}} + k_{n_y}\mathbf{\hat{y}} + k_{n_z}\mathbf{\hat{z}} = \frac{n_x \pi }{L_x} \mathbf{\hat{x}} + \frac{n_y \pi }{L_y} \mathbf{\hat{y}} + \frac{n_z \pi }{L_z} \mathbf{\hat{z}} .</math>

In general for an ''n''-dimensional box, the solutions are
<math display="block"> \psi =\prod_{i} \psi_{n_i}(x_i,t,L_i)</math>

The ''n''-dimensional momentum wave functions may likewise be represented by <math>\phi_n(x, t, L_x)</math> and the momentum wave function for an ''n''-dimensional centered box is then:
<math display="block"> \phi = \prod_{i} \phi_{n_i}(k_i,t,L_i)</math>

An interesting feature of the above solutions is that when two or more of the lengths are the same (e.g. <math>L_x = L_y</math>), there are multiple wave functions corresponding to the same total energy. For example, the wave function with <math>n_x = 2, n_y = 1</math> has the same energy as the wave function with <math>n_x = 1, n_y = 2</math>.  This situation is called ''[[Degenerate energy level|degeneracy]]'' and for the case where exactly two degenerate wave functions 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.

=== More complicated wall shapes ===

The wave function for a quantum-mechanical particle in a box whose walls have arbitrary shape is given by the [[Helmholtz equation]] subject to the boundary condition that the wave function vanishes at the walls. These systems are studied in the field of [[quantum chaos]] for wall shapes whose corresponding [[dynamical billiards|dynamical billiard tables]] are non-integrable.