Archive for April, 2012

Ballparking

Posted in Philosophy of learning, practical math on April 28, 2012 by Alex

One of my favorite mental exercises is something I like to call “ballparking.” The idea is to make a quick, rough estimate for some quantity using prior knowledge and some mental arithmetic. Here is an example I often use with my students:

Finding the radius of the Earth

It’s about 3000 miles from the west coast of the United States to the east coast.

There are 4 time zones in the continental U.S.

There are 24 time zones in the whole world, so the circumference of the Earth is about 18000 miles.

To get the radius, we just divide by 2\pi, or about 6, and we get about 3000 miles for the radius.

If you prefer the metric system, then there are about 2 km in a mile, so the radius of the Earth is about 6000 km. According to Wikipedia, the radius of the Earth is 6378 km, so my rough estimate was good to one significant figure.

You may have noticed I used the word “about” an awful lot in that calculation. This was by design. I do not know the exact distance from the west coast to the east coast, or the exact fraction of the 4 time zones taken up by that distance. I did not take into account the fact that we are not located at the Equator, but a little more than 30 degrees above it. I divided by 6 instead of 2\pi, mostly because it required less effort, and I was a bit off in my conversion from miles to kilometers. Despite all these imprecisions, I was still able to make an estimate that was “in the ballpark” and then get on with my life. If, for whatever reason, I needed a more accurate value for the radius of the Earth, I could go back and correct all these estimates, and get it. I could also use the ballpark estimate as a reference in case I make a mistake in the more laborious calculations needed to find the exact value.

One physicist who was famous for this type of calculation was Enrico Fermi. He was able to estimate the blast power of the atomic bomb Trinity by eyeballing how far it blew some pieces of paper miles from the blast site. Another classic question answered using this technique was “how many piano tuners work in the city of Chicago?” He was so good at this type of estimate, that most people refer to this technique as a “Fermi Calculation.” It was a method that helped scientists determine what a reasonable value for something would be before anyone had figured out what kind of math to use in order to determine the exact value.

Things for my readers to do:

What are some good problems for ballparking? I would like to use some of your suggestions for a class I am teaching this summer.

The Alternative to Many Worlds

Posted in Science! with tags , on April 18, 2012 by Alex

This could be my engineering training talking, but the “Many Worlds” interpretation of quantum mechanics has never rung true for me. For those who are unfamiliar with this, here’s a brief summary:
Very small particles appear to behave probabilistically, satisfying something called the “Born Rule.” This means that it is impossible to determine the exact position (or momentum, or energy) of a particle, it is only possible to determine a set of possible positions (or momenta, or energies) and the probability of the particle having each one.
Ok, so far we are still in the land of verified facts, with no mention of parallel worlds containing (possibly) evil goatee sporting versions of ourselves.

The Many Worlds interpretation of QM is an attempt to grapple with the question: Why does nature choose this particular path for the particle rather than another one? Many Worlders would answer this question with “it doesn’t! The particle takes every possible path, each one creating a universe of its own, and we happen to live in the universe where it took this one.”

This all sounds very cool and science fiction-y, but it does have some problems.

1. We cannot ever observe these other universes, so this interpretation will always be speculative.

2. Unless I’m mistaken (which is always possible), the creation of a kajillion universes at every moment in time would grossly violate conservation of energy.

3. It doesn’t explain why the paths of particles obey the particular statistics that they do, the Born Rule, the thing that makes particles look like waves when there are a lot of them together. Granted, some very smart people (actual experts in this sort of thing) have come up with solutions to this third problem. See here.

So what’s the alternative?
Let’s look at the facts:

Particles do take actual paths through space.

For some reason, the set of possible paths a particle can take behaves like a wave.

The probability that a particle takes a particular path is the square of the amplitude of that wave.

When the possible paths of one particle collide with the possible paths of another, the waves interfere with one another and the particles become “entangled.”

If enough particles become entangled, the set of possible paths loses its wavy nature and things start to behave like classical objects such as baseballs and W-2 forms.

The mysterious question is: what exactly is that wave? And what determines the particular path of a particle? I should note that any interpretations beyond this point do not change any of the above facts (Many Worlds included.) They are all attempts to answer a question for which nobody has an answer.

The alternative to Many Worlds is something called “Hidden Variable Theory,” or sometimes “Bohmian Mechanics.” The idea is that there is some as yet undiscovered (possibly undiscoverable in principle) mechanism that tells the particle which of the possible paths to take. For the purposes of making observations and predictions, this interpretation of QM is exactly the same as Many Worlds. It also has the burden of possibly being unverifiable. It doesn’t violate conservation of energy, however. And it sidesteps the issue of why the set of possible paths acts like a wave. In the end, it is an alternate speculation, but at least it doesn’t make my head explode by trying to think about an extremely large number of universes being created at every moment in time.

One universe is mysterious enough.