Quantum Viruses?

The Twitter conversation that prompted yesterday's post about composite objects was apparently prompted by a comment somebody made about how a virus left alone would see its quantum wavefunction spread out on a time scale of minutes. This led to wondering about whether a virus could really be considered a particle that would move as a single quantum unit, and then to whether that estimate is reasonable. So, let's look at that specific question a little more closely. This is going to be one of those swashbuckling physicist estimation deals, in which I'm going to attempt to come up with numbers without even the benefit of an envelope to scribble on.

So, how would we make an estimate of this? Well, the first order of business is to take on the question of whether it's reasonable to consider a virus a particle. I think the answer is pretty clearly yes, for the reasons described yesterday: as long as the energy scale is such that you don't have to worry about the composite object falling apart, you can consider it a single particle for estimation purposes. A virus is a fairly hardy thing-- if you're trying to sterilize something, you generally heat it to temperatures of a hundred-odd C and hold it there for some time. And that's probably not doing real damage to the physical integrity of the virus as a whole, just scrambling some bits of DNA. so I'd say it's safe to call it a particle.

Given that, how would you estimate the time to spread out from some initial position? Well, you could come up with a detailed model of the initial distribution, and work out what the evolution of the wavefunction should look like, but we're going for quick and dirty here, which means this is the time to break out one of the two calculations you can actually do using the equation from which this blog derives its name:

$latex \Delta x \Delta p \geq \frac{\hbar}{2} $

That is, of course, the Heisenberg Uncertainty Principle, relating the uncertainty in position to the uncertainty in momentum. We can safely assume that relativity doesn't enter into this, meaning that momentum is just mass times velocity, and re-arrange this to find:

$latex \Delta v \geq \frac{\hbar}{2 m \Delta x} $

This gives you an uncertainty in velocity in terms of the mass of the virus and the initial uncertainty in position. And that's all we really need to get a sense of the situation we're interested in.

Note that this isn't a specific velocity-- that would be impossible in quantum physics. Instead it's a range of velocities. The actual velocity of any one virus would be somewhere within that range, but is impossible to predict. But if you ran the experiment many times-- a million, say-- you would find that the distribution of measured velocities traced out some overall distribution characterized by a spread given by the formula above: Planck's constant divided by the mass and the initial position uncertainty.

And over time, that spread of velocities would lead to a spread in position, as the different viruses flew off in all different directions at different speeds. This is described mathematically by a spreading-out of the wavefunction, which is the thing we're trying to describe. So the spread in velocity really is the thing we're after.

So, we've got an equation, now we need some numbers. So, what's the scale of a virus? Well, a cell is kind of micron-sized, give or take a bit, and viruses are smaller than that. So let's say that if we've got a single virus, and we've pinned down its position reasonably well, we know it to within plus or minus 100nm. That's probably a reasonable order of magnitude for $latex \Delta x $.

What's the mass, then? I could probably pick a type of virus and find this out, provided I was willing to spend a bunch of time on unit conversions, which I'm not. Instead, let's say that it has around the same density as water, and fits in a cube 100nm on a side-- that's a volume of (10^-7m)³ or 10^-21m³, and water has a density of 1000 kg per cubic meter, so a mass of an attogram: 10^-18 kg.

Then we just plug in to get the velocity, keeping only orders of magnitude. Planck's constant is 10^-34 kg m²/s, so we have:

$latex \Delta v \geq \frac{10^{-34}kgm^2/s}{(2)(10^{-18}kg)(10^{-7}m)} = \frac{1}{2} 10^{-9}$ m/s.

So the wavefunction would spread out by half a nanometer per second, which means it would take a bit more than three minutes to double the initial position spread of 100nm.

So, yeah, assuming you can call a virus a single quantum particle, it should spread out due to quantum effects on a time scale of minutes.

Now, I've made some simplifying assumptions there, and putting more exact numbers in would probably tend to make the velocity spread a bit bigger-- the initial position spread and mass are probably both overestimated. But that's not going to be off by many orders of magnitude, and roughly validates the original comment that kicked all this off. The characteristic time scale for the wavefunction of a single virus to spread out due to quantum-mechanical effects would be tens of seconds.

But is this something you'd actually be able to see? I suppose you might, if you could float a virus in a vacuum and observe its position somehow. A more realistic situation would involve a virus suspended in some sort of fluid-- air or water-- and in that case, the Brownian motion due to collisions with molecules from the fluid would probably swamp anything you might see from quantum effects. But in the spherical frictionless world of low-energy quantum mechanics, it's kind of a fun little problem.

(Featured false-color image of an influenza virus from Wikipedia.)