Equation of the Day #9: The Uncertainty Principle

The Uncertainty Principle is one of the trickiest concepts for people learning quantum physics to wrap their heads around. In words, the Uncertainty Principle says “you cannot simultaneously measure the position and the momentum of a particle to arbitrary precision.” In equation form, it looks like this:

\Delta x \Delta p \ge \dfrac{\hbar}{2},

where \Delta x is the uncertainty of a measurement of a particle’s position, \Delta p is the uncertainty associated with its measured momentum, and ħ is the reduced Planck constant. What this equation says is that the product of these two uncertainties has to be greater than some constant. This has nothing to do with the tools with which we measure particles; this is a fundamental statement about the way our universe behaves. Fortunately, this uncertainty product is very small, since

\hbar \approx 0.000000000000000000000000000000000105457 \text{ J s}.

The real question to ask is, “Why do particles have this uncertainty associated with them in the first place? Where does it come from?” Interestingly, it comes from wave theory.

Take the two waves above. The one on top is very localized, meaning its position is well-defined. But what is its wavelength? For photons and other quantum objects, wavelength (λ) determines momentum,

p = \dfrac{h}{\lambda},

so here we see a localized wave doesn’t really have a well-defined wavelength, and thus an ill-defined momentum. In fact, the wavelength of this pulse is smeared over a continuous spectrum of momenta (much like how the “color” of white light is smeared over the colors of the rainbow). The second wave has a pretty well-defined wavelength, but where is it? It’s not really localized, so you could say it lies smeared over a set of points, but it isn’t really in one place. This is the heart of the uncertainty principle.

So why does this apply to particles? After all, particles aren’t waves. However, at the quantum level, objects no longer fit into either category. So particles have wavelike properties and waves have particle-like properties. In fact, from here on I will refer to “particles” as “quantum objects” to rid ourselves of this confusing nomenclature. So, because waves exhibit this phenomenon – and quantum objects have wavelike properties – quantum objects also have an uncertainty principle associated with them.

However, this is arguably not the most bizarre thing about the uncertainty principle. There is another facet of the uncertainty principle that says that the shorter the lifetime of a quantum object (how long the object exists before it decays), the less you can know about its energy. Since mass and energy are equivalent via Einstein’s E = mc2, this means that objects that exist for very short times don’t have a well-defined mass. It also means that, if you pulse a laser over a short enough time, the light that comes out will not have a well-defined energy, which means that it will have a spread of colors (our eyes can’t see this spread, of course, but it means a big deal when you want to use very precise wavelengths of light in your experiment and short pulses at the same time). In my graduate research, we used this so-called “energy-time” uncertainty to determine whether certain configurations of the hydrogen molecule, H2, are long-lived or short lived; the longer-lived states exhibit sharper spectral lines, indicating a more well-defined energy, and the short-lived states exhibit wider spectral lines, a less defined energy.

So while we can’t simultaneously measure the position and momentum of an object to arbitrary certainty, we can definitely still use it to glean information about the world of the very, very small.

-A

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