###### By Don Lincoln, Fermilab

## Measuring the mass of subatomic particles rested on the principles of quantum mechanics, specifically on the Heisenberg Uncertainty principle. Invented by German physicist, Werner Heisenberg, it states that it is impossible to simultaneously precisely measure the position and momentum of an object.

### The Heisenberg Uncertainty Principle

The Heisenberg Uncertainty principle itself was formally written down a few months later by American physicist, Earle Kennard, who was on sabbatical in Germany at the time. It states that if *x* is position and *p* is momentum, then the equation can be written as *Delta-x times Delta-p is greater than or equal to h-bar divided by 2*. Delta just means uncertainty and *h*-bar is called the reduced Planck constant.

Now that’s the most familiar version of the Heisenberg uncertainty principle. However, there’s another one that is basically the same. This one says that energy and time experience a similar relationship. This time, using the symbol *E* for energy and *t* for time, then this second form of Heisenberg’s principle says that *Delta-E times Delta-t is greater than or equal to h-bar over 2*.

Now that greater than sign can be confusing, so let’s just talk about the smallest possible numbers that Delta-E and Delta-t can be. The smallest happens when the greater than or equal sign is just an equal sign. If it’s greater, then the numbers can be as big as we choose, and we want to talk about the smallest case. So, let’s just replace the greater than or equals sign with a simple equals sign. And we’ll just remember that *h*-bar divided by 2 is a super tiny number. Thus, focusing on the most important components of the equation*, it says that* Delta-E times Delta-t equals a constant small number. What does that mean really?

### Meaning of the Energy and Time Version

Well, if we multiply two numbers together and the product is a constant, then they have an inverse relationship. If one goes up, the other goes down. For instance, if *X* times *Y* equals 1, then if *X* is 10, then *Y* is 0.1. If *X* is 100, then *Y* is 0.01. And the opposite is true. If *Y* gets big, *X* gets small. It’s like a seesaw from a children’s playground. If one side goes up, the other goes down.

This interestingly assumes significance when physicist calculate a particle’s lifetime as it is connected to the basic meaning of the energy and time version of the Heisenberg uncertainty principle. At its deepest level, the principle says that energy doesn’t have to be conserved, as long as the amount of time it’s not conserved isn’t very long. Furthermore, the more that the energy isn’t what it should be, the shorter amount of time it can be different than it should be.

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### The Lifetime of a Particle

So, how can we work out the lifetime of a particle knowing this information?

The first thing we need to remember is Einstein’s equation *E = mc ^{2}*. It’s a special case specifically for particles that a) have mass and b) are stationary. So, Einstein’s simplest equation applies, once we measure the rest mass, which is a physicists-code for stationary. We can then simply use Einstein’s equation to calculate this rest energy and make plots of either the mass or the energy of the particle. But, what should we expect to see?

This is where Heisenberg’s principle comes in. If a particle lives for only a short amount of time before it decays, then Delta-T is very small. If we put that into the Heisenberg uncertainty principle, we have *Delta-E equals h-bar over 2, divided by Delta-T*. A small Delta-*T* means a large Delta-*E* and, by *E* equals *m c*-squared, a large Delta-*M*. Similarly, the Heisenberg principle says that a particle that lives for a very long time has a very large Delta-*T*. If it has a large Delta-*T*, then it has a very small Delta-*E* or *M*. This is a clear prediction from a simple equation.

### The Lifetime of Electrons and Quarks

And that’s exactly what we see. If we identify and measure the mass of various particles, we find that some of them exist in a very small mass range. The electron, for example, lives forever because it is stable. If we measure the mass of any electron, we get a single value. In fact, if we use the very best measurement currently available of the mass of an electron, we find that it is 0.510 998 946 1 plus or minus the tiny uncertainty of 0.000 000 003 1 MeV.

This tiny uncertainty in the mass of an electron reflects the fact that it appears to have an infinite lifetime. In fact, that tiny uncertainty is really just a measurement thing. If the equipment were perfect, we’d expect a particle with an infinite lifetime would have a mass range of zero.

### The Top Quark

Let’s contrast this with the top quark, which is the heaviest known and shortest lived subatomic particle. The lifetime of the top quark is reported to be about 10 to the minus-25 seconds. But, of course, we can’t measure anything that fast. What we can do is measure the mass of the top quark and the best current measurement says it’s about 173.3 billion electron volts.

Now, we measure the mass of the top quark to be 173.3 billion electron volts, but not all top quarks have that value. Most of them are in the range of 171.3 to 175.3 billion electron volts. That means the uncertainty is plus or minus 2 billion electron volts.

### Heisenberg’s Equation

We can take that range and put it in Heisenberg’s equation, this time putting in a value for *h*-bar, which is 6.6 times 10 to the minus-25 billion electron volts. Divide by the 2 in the equation and then use the top quark’s Delta-*E* equals 2 billion electron volts and you get a Delta-T of 1.6 times 10 to the minus-25 seconds.

The bottom line is that this is the method used to figure out the lifetime of short-lived subatomic particles. We measure the range of observed masses and use Heisenberg’s equation to work out the lifetime. We never measure the lifetime directly, except for particles that live for a very long time.

### Common Questions about Using the Heisenberg Uncertainty Principle to Measure the Lifetime of Subatomic Particles

**Q: According to Heisenberg’s principle**, when is Delta-T very small?According to Heisenberg’s principle, if a particle lives for only a short amount of time before it decays, then Delta-T is very small.

**Q: What does the tiny uncertainty in the mass of an electron reflect**?The tiny uncertainty in the mass of an electron reflects the fact that it appears to have an infinite lifetime.

**Q: When do we measure the lifetime directly**?We never measure the lifetime directly, except for particles that live for a very long time.