Question

Can a particle in a 3 dimensional potential well experience quantum tunneling? Why or why not? How about in two dimensions?

Answer 1

To answer this, we always have to see tunneling parameter (T) for any particular case .we can know there any particle in system will have tunneling coefficient (even small values) as this works for only well type potential which has barrier of some width (shouldn't be extended to infinty) and of potential height (cases considered for particles having always less total energy inside the well than barrier height i.e potential size ). Intutive explainations of these type problem do not work as these are quantum mecanical regions, only can be explained by solving parameters (observables ,here tunneling coefficient).  

off course,2d or 3d - problem i.e potential well barriers holding particle inside will have tunneling . A case, for instance, alpha particle decay is 3d quantum tunneling problem, By W.K.B principle ,if particle inside potential barrier , the potential has very slow change (i.e almost constant potential barrier or continuous curve of potential) in its slope as compared to wavelength of wavefunction of particle then , transmission coefficient is

here limits a and b are well barrier's inner and outer radius, ,     is independent angles , then problem will be of one dimensional in r radius . T is very small but enough to show there will be transmisson sooner or later .

Similarly 2d case , a rectangular potential having inner sides a and b, and of finite widthalong x and y axis . Solving for three regions; inside , in barrier , outside barrier . Inside and outside of potenial well ,solution will be of perodic nature as it will be free limited to these regions. In barrier or potenial it will be of hypebolic nature, after applying boundary conditions and getting solutions, then transmission coefficient

So as long as T exist , tunneling is possible.