Absolute Zero

Absolute zero is approximately -273.16 °C or -459.67 °F or exactly 0 K (a K or Kelvin is defined as 1/273.16 of the triple point of water) and is defined as the state in which the entire system is in its lowest possible energy state.

• This does NOT mean that there is no energy.
• NOR does it mean that there is no motion.

Let me reiterate: absolute zero is the temperature state at which the system is in the lowest possible energy configuration. That's it. It's actually a quite simple concept, but what follows is the explanation of the two things that absolute zero is not. So, if your curiosity begs to know why, let's look at the quantum harmonic oscillator. This is a very good and commonly used device to introduce molecular mechanics because diatomic molecules act like two masses on opposite ends of a spring; that is, they both exhibit periodic motion. The spring is basically the repulsive and attractive Coulomb forces (Coulomb is a unit of charge) from each atom's eletrons and protons interacting with the other atom's. A molecule may collide with another, and the Coulomb repulsion causes each molecule to compress a little before speeding away from the other. Then, within each molecule, the compression proves to be a little too close for comfort and the protons and electrons of one atom push the other atom's parts away. But then the attractive Coulomb force between one atom's protons and the other's electrons pulls them back together and now the molecule is oscillating like a spring.

Since this is a quantum harmonic oscillator, the energy levels are, of course, quantized, respresented by

E_n = (n + 1/2)ℏω

where n = 0, 1, 2, ..., ℏ is h/(2π) = 1.055*10^-34 J•s where h is Planck's Constant, and ω is angular frequency(radians/sec) which is just 2πf. The frequency is number of cycles per second of the inverse of the period, the time from one amplitude to the next. Think sine waves. Why the n + 1/2, you ask? This is very good question, since letting n = 0, the lowest possible state, results in a nonzero energy E₀ = ℏω/2. This is the zero-point energy. As the name says, this is the smallest quanta of energy the harmonic oscillator can have and it is not zero! The following explanation involves some calculus and some knowledge of physics and QM, resulting in the justification of a nonzero zero-point energy. Continue at your own confusion though I will try to explain with complementing words or skip to the end for the shorter explanation. Well, consider two equations:

• the complete energy expression (1) E = Δp²/(2m) + 1/2*mω²Δx²
• and the lower limit of the Heisenberg Uncertainty Principle (2) ℏ/2 = ΔxΔp,

where m is mass, Δx is the position uncertainty, and Δp is the momentum uncertainty. The first term of the energy expression represents the kinetic energy, while the second term is the potential energy. You'll notice that there's only one mass expressed. This is the reduced mass which is equal to (m₁m₂)/(m₁+m₂). We do this to simplify our calculations. Think of it like this: originally, we have one mass on each end of the spring and the spring is not fixed. As the masses oscillate, they do so around some point on the spring and this is annoying to conceptualize and work with mathematically. So we say, well, if we have this reduced mass on one end of the spring and the other end is fixed, it so happens that the system behaves exactly the same and is much simpler! From now on, all references to mass are referring to the reduced mass. Having two variables in Eq. (1) (Δx and Δp) is no good so we re-express it in terms of Δx by substituting Eq. (2) which yields

(3) E = ℏ²/(8mΔx²) + 1/2*mω²Δx².

There, now we have a simple (relatively) function that relates the oscillator's energy to the uncertainty in the mass's position (the amount of compression in the "spring"). We want to know what the smallest possible energy that the system can have is so taking the derivative of E with respect to Δx to find the minimum Δx gives us

(4) dE/dΔx = -ℏ²/(4mΔx³) + mω²Δx

The derivative says the rate of change of energy is a function of Δx; this basically the slope. If we look more closely at Eq. (3), we see that it is a quartic function (x⁴) and looks sort of like two upward opening parabolas next to each other. Go ahead and graph it on a calculator if you so please, or draw a a smooth 'W'. We want to know where the lowest point on this graph is; that is, where the slope is zero. That's the either one of the valleys in our 'W'! Setting Eq. (4) equal to zero and solving for Δx gets us Δx = (+/-)√(ℏ/(2mω)). There's actually two solutions, a positive and negative one which ends up not mattering since we'll square it away later on. And by later on, I mean now; inserting this into our previous energy expression, Eq. (3), and reducing:

(5) E = ℏω/4 + ℏω/4 = ℏω/2 = E₀

Tada! ℏω/2 is indeed the lowest possible energy state for quantum harmonic oscillators, a nonzero energy! So there is always energy as motion at the atomic level no matter how cold it gets!

As far as molecular motion at absolute zero, we can take a look at Eq. (5). Remember that each term represented kinetic and potential energies, respectively? Well, the kinetic term, upon inspection, is ℏω/4 which means the oscillator is indeed oscillating and will never cease!

But can we ever observe absolute zero? The answer, unfortunately, is no. The Second Law of Thermodynamics (or Third depending if you start numbering from the Zeroeth or First) forbids 100% efficiency. So, if we had a system that we were trying to cool to absolute zero , there are "two" problems. The first is the isolation. There's no way to fully thermally isolate a system without either a perfect insulator or a perfect refrigerator/heat engine, which are impossible to create so says the Second Law. The second, even if we did, measuring the system immediately disrupts the equilibrium of our isolated absolute zero environment. But we have gotten pretty darn low temperatures, in the nanoKelvin range through laser cooling and trapping and then evaporative cooling with a dilution refrigerator last time I checked. Scarves and mittens!

So, basically the uncertainty principle forbids a system with zero energy because knowing the energy definitely violates the energy-time form of the principle (ΔEΔt ≥ ℏ/2), and the laws of thermodynamics forbid us from ever getting to absolute zero because there are no perfect insulators and no way to observe a system at absolute zero without changing the energy of the system.

Sources: My class notes and brain.