Built on Facts

We live in a nice, classical world. We can tell where we are and how fast we’re going. We can spin things around at whatever speeds we feel like, and if we throw something through the air we can predict where it’s going to go.

Quantum mechanically, however, things are a lot fuzzier. In theory the trajectory of a thrown object is subject to quantum mechanical uncertainty. But quantum effects almost always are only relevant on very small scales. While the path of an electron might be pretty uncertain in some cases, the trajectory of a baseball won’t noticeably deviate from what Newton’s brilliant formulation of classical physics tells should happen. Just like as in relativity, just because an effect is too small to easily measure doesn’t mean it doesn’t exist. So just how much will a baseball generally be thrown off by quantum effects?

Answer: Not much. The mathematically inclined can read through the details below, but everyone else can skip to the bottom and find out the exact result. Onward!

What we’re trying to do is find the quantum width of a classical trajectory. This suggests a very simple (comparatively speaking) application of Feynman’s path integral formulation. Just find the path integral along the classical trajectory and the path integral along a slightly deflected trajectory. We’ll assume we’re dealing with a free particle so the classical trajectory is just a straight line of length L. Calculate how much deflection is necessary to get a significant deflection, i.e, equal to ?. To make the math easy, we’ll choose the deflected trajectory to be two straight lines: one from the start to a point a distance d to the side of the center, and one from that point to the end. Now find the difference in path lengths. A simple application of the Pythagorean theorem, and then taking the first term in the Taylor series (to make the answer much simpler) tells us the difference in length between the classical and deflected path is

Since the particle traveling along either path must arrive at the same time, we know that the particle traveling along the deflected path has a slightly higher velocity. Taking distance over time, we find that the additional velocity along the deflected path is

This gives us a change in kinetic energy (to second order in d) of

Which means the change in action (energy integrated over time) is simply

Now this is quantum mechanics, and we know that our phases are multiplied by a factor of the reduced Plank’s constant, so we want to solve the above equation for ?S = ?h, where the h is really h-bar.  We get:

Ok, nonmathematical readers, you will have hopefully skipped down to this point without letting the equations scare you off!  We can simply plug in typical numbers for our baseball into the above equation, and we get that the quantum mechanical uncertainty in its path is… about 2% of the size of a proton.  Definitely not something you’d notice in everyday life.  In fact it’s so absurdly far beyond even the most sensitive techniques of modern physics that there’s simply no way to ever measure such a tiny effect.  But we can measure bigger effects.  The smaller and slower the particle, the bigger the effect is.  For things like individual atoms or subatomic particles the effect can be millimeters or more, and is easily observable.