Particle tunnelling between two boxes joined with a long thin tube

The ground state of a particle confined to two two-dimensional square boxes connected with a narrow channel is calculated. It is shown that the presence of the channel results in the splitting of the otherwise degenerate ground state. When the width of the channel is small compared to the wavelength of the particle, the wavefunction in the channel is exponentially suppressed, very much like in the case of tunnelling through a potential barrier. Moreover, in the limit of long and narrow channels, any small asymmetry in the shape of the boxes results in the localization of the particle.