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The Heisenberg uncertainty principle

People are familiar with measuring things in the macroscopic world around them. Someone pulls out a tape measure and determines the length of a table. A state trooper aims his radar gun at a car and knows what direction the car is traveling, as well as how fast. They get the information they want and don’t worry whether the measurement itself has changed what they were measuring. After all, what would be the sense in determining that a table is 80 cm long if the very act of measuring it changed its length!

At the atomic scale of quantum mechanics, however, measurement becomes a very delicate process. Let’s say you want to find out where an electron is and where it is going (that trooper has a feeling that any electron he catches will be going faster than the local speed limit). How would you do it? Get a super high powered magnifier and look for it? The very act of looking depends upon light, which is made of photons, and these photons could have enough momentum that once they hit the electron they would change its course!

It’s like rolling the cue ball across a billiard table and trying to discover where it is going by bouncing the 8-ball off of it; by making the measurement with the 8-ball you have certainly altered the course of the cue ball. You may have discovered where the cue ball was, but now have no idea of where it is going (because you were measuring with the 8-ball instead of actually looking at the table). Werner Heisenberg was the first to realize that certain pairs of measurements have an intrinsic uncertainty associated with them.

For instance, if you have a very good idea of where something is located, then, to a certain degree, you must have a poor idea of how fast it is moving or in what direction. We don’t notice this in everyday life because any inherent uncertainty from Heisenberg’s principle is well within the acceptable accuracy we desire. For example, you may see a parked car and think you know exactly where it is and exactly how fast it is moving. But would you really know those things exactly?

If you were to measure the position of the car to an accuracy of a billionth of a billionth of a centimeter, you would be trying to measure the positions of the individual atoms which make up the car, and those atoms would be jiggling around just because the temperature of the car was above absolute zero! Heisenberg’s uncertainty principle completely flies in the face of classical physics. After all, the very foundation of science is the ability to measure things accurately, and now quantum mechanics is saying that it’s impossible to get those measurements exact!

But the Heisenberg uncertainty principle is a fact of nature, and it would be impossible to build a measuring device which could get around it. In 1922 Otto Stern and Walther Gerlach performed an experiment whose results could not be explained by classical physics. Their experiment indicated that atomic particles possess an intrinsic angular momentum, or spin, and that this spin is quantized (that is, it can only have certain discrete values). Spin is a completely quantum mechanical property of a particle and cannot be explained in any way by classical physics.

It is important to realize that the spin of an atomic particle is not a measure of how it is spinning! In fact, it is impossible to tell whether something as small as an electron is spinning at all! The word ‘spin’ is just a convenient way of talking about the intrinsic angular momentum of a particle. Magnetic resonance imaging (MRI) uses the fact that under certain conditions the spin of hydrogen nuclei can be ‘flipped’ from one state to another. By measuring the location of these flips, a picture can be formed of where the hydrogen atoms (mainly as a part of water) are in a body.

Since tumors tend to have a different water concentration from the surrounding tissue, they would stand out in such a picture. Every quantum particle is characterized by a wave function. In 1925 Erwin Schrdinger developed the differential equation which describes the evolution of those wave functions. By using Schrdinger’s equation scientists can find the wave function which solves a particular problem in quantum mechanics. Unfortunately, it is usually impossible to find an exact solution to the equation, so certain assumptions are used in order to obtain an approximate answer for the particular problem.

As mentioned earlier, the Schrdinger equation for a particular problem cannot always be solved exactly. However, when there is no force acting upon a particle its potential energy is zero and the Schrdinger equation for the particle can be exactly solved. The solution to this ‘free’ particle is something known as a wave packet (which initially looks just like a Gaussian bell curve). Wave packets, therefore, can provide a useful way to find approximate solutions to problems which otherwise could not be easily solved.

First, a wave packet is assumed to initially describe the particle under study. Then, when the particle encounters a force (so its potential energy is no longer zero), that force modifies the wave packet. The trick, of course, is to find accurate (and quick! ) ways to ‘propagate’ the wave packet so that it still represents the particle at a later point in time. Finding such propagation techniques, and applying them to useful problems, is the topic of my current research.

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