## Equation of the Day #11: The Uncertainty Principle

**Akano**, in Math/Physics Apr 11 2014 · 268 views

**You've probably heard of the Uncertainty Principle before. In words, it says "you cannot simultaneously measure the position and the momentum of a particle to arbitrary accuracy." In equation form, it looks like this:**

**What this says is that the product of the uncertainty of a measurement of a particle's position multiplied by the uncertainty of a measurement of a particle's momentum has to be greater than a constant (given by the reduced Planck constant,**

*h*

**over**

*τ*= 2

*π*

**). This has nothing to do with the tools with which we measure particle;**

*this is a fundamental statement about the way our universe behaves*. Fortunately, this uncertainty product is very small, since*ħ*

**is around 1.05457 × 10**

^{-34}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, wavelength determines momentum, so here we see a localized wave doesn't really have a well-defined wavelength, 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. Because waves exhibit this phenomenon – and quantum particles behave like waves – quantum particles 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 particle (how long the particle exists before it decays), the less you can know about its energy. Since mass and energy are equivalent via Einstein's

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 particle (how long the particle exists before it decays), the less you can know about its energy. Since mass and energy are equivalent via Einstein's

*E = mc*

^{2}

**, this means that particles that "live" 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 lab, we use this so-called "energy-time" uncertainty to determine whether certain configurations of the hydrogen molecule, H_{2}, are long-lived or short lived; the longer-lived states have thinner spectral lines, and the short-lived states have wider spectral lines.**So while we can't simultaneously measure the position and momentum of a particle to arbitrary certainty, we can definitely still use it to glean information about the world of the very, very small.**