Heisenberg's Hallway

An Analogy to Quantum Mechanical Probabilities

copyright 1996 by Dave Van Domelen

Based on a paper presented at the Ohio Section of the AAPT Meeting, April 13, 1996

One of the more thorny concepts to try and wrap your brain around in physics is that of the position and velocity uncertainty of quantum-scale particles. As something without an obvious and direct classical analog, the matter can take years to get a handle on...certainly not a good timeframe for a quarter-long course on Quantum Mechanics, and certainly not for an overview of Modern Physics at the Freshman level.

The following analogy is presented as a partial solution to the problem of finding a classical version of the Uncertainty Principle.

I hit upon the basic idea while leaving a movie theater as other people were trying to get in. I gave wide berth to an old woman with a cane, but all but tried to ram a group of highschoolers that were in my way. Then I stopped just outside the crowded area and realized I'd just mentally computed probability distributions...at a Classical level! I'd made sure the chances of running into the old woman were very low, but was willing to accept fairly high odds of hitting a 9th grader. And thus I had an analogy.

Few students have had direct experience with quantum effects, but they've all had the experience of trying to navigate a hallway or theater entrance full of people. Person A needs to guess where Person B will be in a few seconds in order to avoid a collision. Person A will almost automatically guess how likely any given location is for occupation and set limits on how close he's willing to cut it. Probably at that point, pretty definitely within that general area, 100% positively within that larger area. The person has just set up a probability map without even thinking about it. Helping students actually start to think about what they're doing in this sort of situation can go a long way towards helping them develop an understanding of quantum probabilities.

To further illustrate this point, two special cases will be considered, and analogies drawn to orbitals in a Hydrogen atom. After this has been done, mass and time will be discussed as a means to extend the everyday experiences down to the quantum level.

Example: "S-ORBITAL"

Consider a woman standing in a foyer. At the moment she's not moving, merely looking around. You can say you know her position and velocity with a high degree of certainty.

However, one second from now she may no longer be standing there. She could have started moving in just about any direction, or shifted around slightly to look at something. You can assign a high probability of her being in the same place one second from now, but not 100%. Given a relaxed walking pace of about 1 meter per second, you can draw a circle 1 meter in radius around her current position and say with even higher certainty that she'll be inside that circle one second from now. Perhaps she'll start moving at a more urgent speed, around 2 meters per second, giving the circle of 2 meters radius a slightly higher probability of containing her. And so forth, until you have to invoke wildly improbable events to get her farther out in one second...but not totally impossible events. A graph of her distance from the starting point at one second from now might look something like the following graph.

As you can see, the probability for a given radius decreases as the radius itself increases, but the larger a circle you draw around her current position, the greater total chance she'll be in that circle a second from now.

Now, remove the knowledge of where she is now and keep only the graph of probabilities. You now have a map of possible locations for a classical object at a fixed time, much like the probability plot of a ground state hydrogen electron (although not exactly like it). No particular direction is preferred, and you can't be totally sure the person or electron is at the mean radius or location.

Example: "P-ORBITAL"


(This section was not included in the talk for reasons of time and focus, and probably could use some more work if it is to be presented to students.)

Consider the woman now moving down the hall at a fairly steady pace. If she keeps going the same speed and direction, you can say where she should be in one second. However, it's possible that she'll speed up, slow down, stop entirely, turn or even do an about-face. A contour graph of her likely positions will resemble a bowling pin contour map, with two oblong regions joined together. The more likely and larger region will be centered on where she would be if she stayed on course, the second oval would center on her current position, assuming a stop and then random motion from there. A graph of her velocity one second from now would have a similar double hump, one around her current velocity and one around zero.

Again, remove her current position and velocity and you have a quantum-like graph of position and velocity at a given time. This one might resemble a p-orbital instead of an s-orbital. While not exactly like a p-orbital, it's easy to see the resemblance between the two probability distributions.

Obviously, the metaphor starts to seriously break down once you reach the f-orbital and higher angular momenta, but hopefully s-orbital and p-orbital analogies should be enough to help students grasp the idea.

Introduction of Mass:

Consider now what would happen if the woman in the previous two examples was replaced by a man twice her weight. All other things being equal, the man wouldn't be able to change speed or direction as quickly or easily as the woman, owing to inertia. An observer might be slightly more certain of the man's location in one second from now for that reason, graphs would become a little more sharply peaked. Now double the man's mass again, to about 200kg. He's definitely not going to be as fast on his feet as either of the previous two people, and his graphs will be even further sharply peaked. The chance of not moving at all in the "S-orbital" situation would increase, and the maximum reasonable distance would decrease. Continuing to double the mass would lead to greater and greater chances of remaining stationary in the "S-orbital" case, until you could state with effectively total certainty that the person was going to stay put, like a fixed roof support would.

Now look at the other direction. Replace the woman with an elementary school child half her mass. The child is less likely to stay in one place, and more likely to run about at random. Knowledge of either position or velocity for the child now will tell you very little about those quantities one second from now. Graphs will flatten out, it will be harder to predict anything about future activity. Extend this further, all the way down to a butterfly, which is very light and very unpredictable when in motion, moving sometimes almost at random. The lighter an object gets, the harder it is to say anything with certainty.

Now, while the analogy gets a bit weak as you move outside the range of human masses (a snail's small, but pretty certain, for example), the following relationships should be fairly acceptable to students:

Position Uncertainty is LESS for More Massive Objects

Position Uncertainty is MORE for Less Massive Objects

Introduction of Time

Changing how far you look into the future will change the amount of uncertainty seen in a fairly easy to see manner. If you only give someone half a second, they're not going to get as far as they do in a second. And if you ask about someone's position tomorrow based on her position now, any estimates will be VERY vague. So we get the next set of relationships:

Looking Less Far into the Future gives LESS Uncertainty

Looking Farther into the Future gives MORE Uncertainty

We can combine the relationships on mass and time to get the following statements:

To keep the SAME uncertainty in position:

Increase Mass And How Far Ahead You Look


Decrease Mass And How Far Ahead You Look

Extension to Quantum Scale:

From here it's only a short hop to the Quantum level effects.

Atoms are really small, so even at the current time they have some position uncertainty left.

People are pretty big, so at the current time their uncertainty is so small we can't see it...but look a second into the future and it appears.

Something to think about next time you're navigating a crowded hallway.
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