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 ($U > \hbar^2 / (m a^2)$), 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 $U < \hbar^2 / (m a^2)$, 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:

$E_{2D} \approx -U e^{-(\hbar^2/(ma^2))/U}$.

For a weak potential, $E_{2D}$ 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.

1. April 11, 2009 9:39 am

Thanks for writing this blog. Because I am mathematically challenged, I am often frustrated that I can only follow quantum mechanics to a certain point. Your use of simple analogy (spring of a Bic pen) and drawings make ideas very clear and understandable. I appreciate that you take the time to share your expertise!

Melody

April 13, 2009 11:04 am

Thanks Melody! I’m not sure that I would call it “expertise”, but I’m glad it helped you.
Please remember that most of the explanations given here aren’t standard, unimpeachable treatments. Physics is usually presented in terms of very precise statements (for a comparison, try reading the Wikipedia article on the finite potential well!). You won’t find many people who describe quantum wells in terms of squeezing a spring. But that’s the way I have come to think about how they work, and as far as I know, it is completely consistent with the facts. Just don’t be surprised if someone looks at you strangely when you say putting a particle in a quantum well is like squeezing a bic pen spring between your fingers!

June 15, 2012 4:38 pm

There is an error in this article. Any 2-D potential well supports at least one bound state. A proof is given in this article,