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.


Equation of the Day #8: Absolute Zero and Negative Temperatures

If you’re reading this indoors, the room you are currently sitting in is probably around 20°C, or 68°F (within reasonable error, since different people like their rooms warmer or colder or have no control over the temperature of the room they’re reading this entry in). But what does it mean to be at a certain temperature? Well, we often define temperature as an average of the movement of an ensemble of constituent particles – usually atoms or molecules. For instance, the temperature of a gas in a room is given as a relation to the gas’ rms molecular speed:

v_{\rm rms} = \sqrt{\langle v^2\rangle} = \sqrt{\dfrac{3kT}{m}},

where T is the absolute temperature (e.g. Kelvin scale), m is the mass per particle making up the gas, and k is the Boltzmann constant, and the angular brackets mean “take the average of the enclosed quantity.” For reference, room temperature nitrogen (which makes up 78% of the atmosphere) has an rms speed of half a kilometer (one third of a mile) per second. But this definition is a specific case. In general, we need a more encompassing definition. There is a quantity that arises in thermodynamics known as entropy, which basically quantifies the disorder of a system. It is related to the number of ways to arrange the elements of a system without changing the energy.

For instance, there are a lot of ways of having a messy room. You can have clothes on the floor, you can track mud into it, you can leave dishes and food everywhere. But there are very few ways to have an immaculately clean room, where everything is tidy and put in its proper place. Thus, the messy room has a larger entropy, while the clean room has very low entropy. It is this quantity that helps to define temperature generally. Denoting entropy as S, the more robust definition is

T \equiv \left( \dfrac{\partial E}{\partial S} \right)_V = \left( \dfrac{\partial H}{\partial S} \right)_P,

or, in words, temperature is defined as the change in energy divided by the corresponding change in entropy of something with fixed volume, which is equivalent to the change in enthalpy (heat content) divided by the change in entropy at a fixed pressure. Thus, if you increase the energy of an object and find that it becomes more disordered, the temperature is positive. This is what we are used to. When you heat up air, it becomes more disorderly because the particles making it up are moving faster and more randomly, so it makes sense that the temperature must be positive. If you cool air, the particles making it up slow down and it tends to become more orderly, so the temperature is still positive, but decreasing. What happens when you can’t pull any more energy out of the air? Well, that means that the temperature has gone to zero, and movement has stopped. Since the movement has stopped, the gas must be in a very ordered state, and the entropy isn’t changing. When the speed of the gas particles is zero, we call its temperature absolute zero, when all motion has stopped.

It is impossible to reach absolute zero temperature, but it isn’t intuitive as to why at first. The main reason is due to quantum mechanics. If all atomic motion of an object stopped, its momentum would be known exactly, and this violates the Uncertainty Principle. But there is also another reason. In thermodynamics, there is a quantity related to temperature that is defined as

\beta = \dfrac{1}{kT}.

Since k is just a constant, β can be thought of as inverse temperature. This sends absolute zero to β being infinity! Now, this makes much more sense as to why achieving absolute zero is impossible – it means we have to make a quantity go to infinity! It turns out that β is the more fundamental quantity to deal with in thermodynamics for this reason (among others).

Now, you’re probably thinking, “Well, that’s all well and good, but, are you saying that this means that you can get to infinite temperature?” In actuality, you can, but you need a special system to be able to do it. To get temperature to infinity, you need β to go to zero. How do we do that? Well, once you cross zero, you end up with a negative quantity, so if we could somehow get a negative temperature, then we would have to cross β equals zero. But how do we get a negative temperature, and what would that be like? Well, we would need entropy to decrease when energy is added to our system.

It turns out that an ensemble of magnets in an external magnetic field would do the trick. See, when a compass is placed in a magnetic field, it wants to align with the field (call that direction north). But if I put some energy into the system (i.e. I push the needle), I can get the needle of the compass to point in the opposite direction (south). When less than half of the compasses are pointing opposite the external field, each time I flip a compass needle I’m increasing entropy (since the perfect order of all the compasses pointing north has been tampered with). But once more than half of those compasses are pointing south, I am decreasing the disorder of the system when I flip another magnet south! This means that the temperature must be negative! In practice, the compasses are actually molecules with an electric dipole moment or electrons with a certain spin (which act like magnets), but the same principles apply. So, β equals zero is when exactly half of the compasses are pointing north and the other half are pointing south, and β equals zero is when T is infinite, and it is at this infinity that the sign on T swaps.

Lasers are a more realistic physical system that employs negative temperatures. For lasers to work, atoms or electrons are excited to a higher energy state. When the higher energy state is populated by more than half of the atoms or electrons in the system, a population inversion occurs, which puts the system at a negative temperature in the same way as the compass needles described above.

It’s interesting to note that negative temperatures are actually hotter than any positive temperature, since you have to add energy to get to negative temperature. One could define a quantity as –β, so that plotting it on a line would be a more intuitive way to see that the smaller the quantity, the colder the object is, while preserving the infinities of absolute zero and “absolute hot.”