In the previous post I set up the problem of a particle in a potential well, and explained that any one-dimensional trap is good enough to confine a particle, whereas only sufficiently deep and wide traps are good enough in three dimensions.  In this follow-up post I’ll try to explain why.

In quantum mechanics, every particle is described by a “probability cloud”.  In formal terms, the probability cloud is represented by a strictly mathematical object called a wavefunction, but it can be thought of as a region through which the particle passes from time to time.  There are denser portions of the cloud, which correspond to places where the particle is likely to be, and sparse portions of the cloud which represent places seldom-visited by the particle.

A funny consequence of quantum mechanics is that a particle may extend its probability cloud into regions where it shouldn’t be allowed.  For example, even if the particle does not have enough energy to “break through the walls” of the potential well it sits in, it may still extend its probability cloud outside the well.  That is, a particle may pass fleetingly into a region where it has no business being because it doesn’t have enough energy to get there.  This is the phenomenon of quantum tunneling: a ghostly flickering of a particle’s motion into “classically unallowed” regions.  As a visual example, imagine a marble rolling slowly around the bottom of a large mixing bowl.  If the marble suddenly, without provocation, jumped up the wall to the lip of the bowl only to fall back down and resume its previously slow motion, we would call that an act of quantum tunneling.  The marble “tunneled” to a location it shouldn’t have had enough energy to reach.  Unfortunately (or, perhaps, fortunately), quantum tunneling only occurs on the smallest of scales: usually electron-sized objects in atom-sized traps.  Nonetheless, it’s a fascinating phenomenon that surely deserves its own post later in the future.

For the case of our particle sitting in a square well, we may ask how big its probability cloud is.  That answer is unclear, but we can expect that it depends on the particle’s energy.  The closer the particle is to escaping (total energy E just barely negative) the larger the probability cloud.  If the particle escapes the potential well alltogether ($E > 0$), its probability cloud becomes infinite — the particle could be anywhere at all.  If the particle is indeed trapped, though, its probability cloud must have some particular size x.   Here’s how the situation looks schematically:

A particle and its "probability cloud" inside a square well

If the well has a width a and depth U, how big must x be?  How is the answer different in 1 and 3 dimensions?

The size of the probability cloud affects the energy of the particle in two distinct ways.  First of all, increasing x decreases the “confinement energy” guaranteed by the exclusion principle — if the probability cloud is larger, the particle is less “squeezed” and has smaller kinetic energy.  Mathematically,

$K \approx (\Delta p)^2/(2 m) \approx \hbar^2/(m x^2)$,

where m is the mass of the particle.  This is just an approximation (we are not worried about the exact shape of the cloud), so it isn’t worth keeping track of all the factors of 2.  Our answer won’t be completely accurate, but it will display the qualitative behavior that we need to solve our mystery.

The size of the probability cloud also affects the potential energy.  Even though the well has a constant depth U, the potential energy of the particle is not exactly equal to -U because it doesn’t spend all its time in the well.  Remember, the particle’s probability cloud extends to places that are not in the well.  So roughly, we can estimate the potential energy as

$P \approx -U \times$ (fraction of probability cloud that is inside the well) $+ 0 \times$ (fraction of probability cloud that is outside the well).

In one dimension, the fraction of the cloud that is inside the well is easy to estimate: it’s just the width of the well divided by the width of the cloud.  We’ll call it $f_1 = a/x$, so the potential energy $P_1 \approx -U \times f_1 = -U a/x$.

In three dimensions, we have to remember that the potential well is actually a sphere, and the probability cloud is a larger sphere that envelops the potential well.  So the fraction $f_3$ of the cloud that is inside the well is the ratio of the volumes of these two spheres: $f_3 = (\frac{4}{3} \pi (a/2)^3) / (\frac{4}{3} \pi (x/2)^3 ) = a^3/x^3$.  So here the potential energy is $P_3 \approx -U a^3/x^3$.

What does x have to be in order for the particle to be trapped?  Remember that a trapped particle has total energy $E = K + P < 0$, or in other words, $K < -P$.  For a trapped particle, the potential energy of the trap must beat the kinetic energy.  In one dimension, this requirement becomes

$\hbar^2/(m x^2) < U a/x$.

Solve for x to get the one-dimensional requirement on x:

$x > \hbar^2/(m U a)$.

This result suggests that as long as x isn’t too small, the particle can be trapped.  In other words, a particle can always remain in the potential well by spreading itself over a wider probability cloud.  This is why any one-dimensional potential can trap a particle.  While we haven’t solved exactly for the particle’s total energy, a good estimate is

$E_1 \approx -U a/x \approx -U \cdot \frac{U}{\hbar^2/(m a^2)}$.

It is worth noticing that the expression for energy contains thefollowing fraction: depth of the well U divided by the confinement energy of the well $\hbar^2 / (m a^2)$.  This fraction comes up again and again when describing quantum behavior of potential wells.  It is a pretty universal indicator of how strong the potential well is: how deep is the well compared to its quantum confinement energy?

For a three-dimensional potential, the situation is different.  The requirement $E < 0$ is

$\hbar^2 / (m x^2) < U a^3/x^3$

meaning that a particle is trapped when

$x < U a^3 m/\hbar^2$.

This result is very different than for the one-dimensional well.  It suggests that trapped particles must have small probability clouds.  But there’s no way that the probability cloud can be smaller than the size of the well: the particle isn’t going to spontaneously confine itself to some arbitrary region smaller than the well.  This suggests that

$a \le x < U a^3 m/\hbar^2$.

Or, in order for the potential well to trap the particle, it must satisfy

$U > \hbar^2/(m a^2)$.

So, finally, we have determined that a 3D potential well cannot trap a particle unless its depth U is greater than the confinement energy $\hbar^2/(m a^2)$ associated with its width.

In the end, this post has a lot more equations than I intended, and I hope that they don’t obscure the main idea.  The difference between a 1D and a 3D potential well has everything to do with quantum tunneling and confinement energy.  In one dimension, a particle that extends its probability cloud outside the well can lower its (kinetic) confinement energy while still “feeling” the potential of the well strongly enough to remain trapped.  In three dimensions, a particle that extends its probability cloud weakens the influence of the potential more quickly than it lowers its confinement energy, so only sufficiently deep and wide potentials can hold the particle in place.

As a footnote, I can’t resist pointing out that even though this post involved a fair number of equations, it is nothing compared to the exact solution of the “particle in a square well” problem that is standard in quantum mechanics.  Like most of my fellow graduate students, I have solved this problem multiple times using all the machineries of the Schrodinger equation.  But despite going through the problem in all its detail multiple times, I never really understood the answer until someone showed me the “simple” way that I have described here.  This paradoxical situation is very common in physics: the more exactly you solve a problem, the less you understand what is going on.  My goal for Gravity and Levity is to bring to light as many of the “simple” solutions as possible.