Epilogue: quantum wells in 2 dimensions
In the previous two posts, I discussed the difference between 1D and 3D potential wells. Conspicuously absent, though, was any talk about a 2D potential well. The reason is that the 2D case is more difficult to deal with and less generally applicable; even my graduate quantum mechanics book doesn’t bother going into it, even though it discusses the 1D and 3D cases in some detail.
Nonetheless, the 2D potential well is important (superconductivity, for example, cannot be understood without it), so I’ll give the general results here.
If you try to repeat the arguments of the previous post for two dimensions, you get a funny result: the potential well can trap a particle as long as the depth of the well is greater than the confinement energy (), and the size of the probability cloud doesn’t matter at all. Usually, if you’re trying to solve for some quantity x and you find that it drops out of the problem alltogether, that’s a sign that you need to do things more carefully. And that is the case for the two-dimensional potential well problem: our rough “probability cloud of size x” estimation is too crude to give the right answer.
Actually, the answer stated in the previous paragraph is almost correct. It is true that whenever the depth of the well is greater than the confinement energy, you can have a bound state. When , however, you can still trap a particle. It’s just that in this latter case the bound state is extremely weak. Mathematically speaking, it is exponentially weak:
For a weak potential, can be extremely close to zero. That means that just about any little disturbance will knock the particle out of the trap.
So usually people refer to the two dimensional potential well as a “marginal case”. It is a little like the one-dimensional result: it is technically possible to trap a particle in any two-dimensional well. But it is also like the three-dimensional result: when the depth of the well is much smaller than the confinement energy, it is effectively impossible to trap the particle in a 2D potential well.