# How Long Can You Balance A (Quantum) Pencil

Sorry for the delay between posts. Here in Virtuosi-land, we’ve all begun our summer research projects and I think we’ve just become a little bogged down in the rush that is starting a summer research project. You feel as though you have no idea what the heck is going on, and just try desperately to keep your head up as you hit the ground running, but thats a topic for another post. In this post I’d like to address a fun physics problem.

How long can you balance a pencil on its tip? I mean in a perfect world, how long?

No really. Think about it a second. Try and come up with an answer before your proceed. What this question will become by the end of this post is something like the following:

Given that Quantum Mechanics exists, what is the longest time you could conceivably balance a pencil, even in principle?

I will walk you through my approach to answering this question. I think it is a good problem to illustrate how to solve non-trivial physics problems. I will try and go into some detail about how I arrived at my solution. For most of you this will probably be quite boring, so feel free to skip ahead to the last section for some numbers and plots.

### Finding an Equation of Motion

The first thing we need to do is find an equation of motion to describe
our system. Lets consider the angle theta that the pencil makes with
respect to the vertical. Lets treat this as a torque problem. Dealing
with rotating systems is almost identical to dealing with free particles
in Newtonian mechanics. Instead of Netwon’s first law, relating forces
to acceleration *model* the system. I need to break up the real world, rather
complicated idea of a pencil, and turn it into an approximation that
retains all of the important bits but enables me to actually proceed.
So, I will model a pencil as a rod, a uniform rod with a constant mass
density. In doing so, I can proceed. The moment of inertia of a rod
about its end is rather easy to calculate. If you are not familiar with
the result I recommend you try the integral yourself.

### Looking at the equation of motion

Now that we’ve found the equation of motion, lets look at it a bit.
First off, what does an equation of motion tell us? Well, it tells us
all of the physics of our system of interest. That little equation
contains all of the information about how our little model pencil can
move. (Notice that while I haven’t yet been explicit about it, in my
model of the pencil, I also don’t allow the tip to move at all, the
pencil is only able to pivot about its tip). Great. A useful thing to do
when confronting a new equation of motion is to try and find its fixed
points. I.e. try and find states in which your system can be which do
not evolve in time. How can I do that? Sounds complicated. In fact, I’ll
sort of work backwards. I want to know the values that do not evolve in
time, meaning of course that if I were to find such a solution, all of
the terms that depend on time would be zero. So, if such a solution
exists, for that solution the derivative term will vanish. So the
solutions have to be solutions to the much simpler equation

### When your approximations fail

So what went wrong again? It seems like I’ve gotten an answer, namely in my model you could, at least in principle balance your pencil forever. But you and I both know you can’t. Something is amiss. Hopefully, the first thought that occurs to you is something along the lines of the following.

Of course you dummy! You could

in principlebalance a pencil forever, but in the real world, you can’t set the pencil up standing perfectly straight. Even if its tilted just a little bit, its going to fall. This is exactly the problem with your physicists, you don’t live in the real world!

Whoa, lets not be so harsh there. I made some rather crude
approximations in order to get such a simple equation. You are allowed
to make approximations provided (1) they are still right to as many
digits as you care about, and (2) you keep in mind the approximations
you made, and think a bit about how they could go wrong. So, before we
do anything too drastic, lets do with your gut. I agree, it seems like
if the pencil would be at any small angle, it ought to fall. Lets double
check that our equation does that. So for the moment imagine theta being
some small number. In fact, I will use the usual notation and call it
epsilon. What does our equation say then?

### A short side comment on Taylor Series

Imagine a function. Any function. Picture the graph of the function.
I.e. imagine a line on a graph. No matter what function you imagine, if
you zoom in far enough, at any point that function ought to look like a
line. Seriously. Zoom out a little bit and it will look like a line plus
a parabola. Zoom out a little more and it will look like a cubic
polynomial. You can make these statements precise, and thats the Taylor
Expansion. But the idea
isn’t much more complicated than what I’ve described. Taylor expanding
the sine, we obtain *really* small angles, sin(x) looks just like x. Whats really small?
Well as long as x^3 is too small for you to care about. For me, for the
rest of the problem that will be for angles that are less than about 0.1
radians, for which that second term is about 0.00017 radians or 0.01
degrees, which is too small for me to care.

### Coming back to the approximation bit

Anywho, for really small angles, our equation of motion is approximately

### A little preliminaries

Before proceeding any further, lets actually solve the equation of
motion we just got for the smallest angles. To remind you, the equation
I got for my model of a pencil in the limit of the smallest angles was

### Abandoning Realism

At this point of considering the question, I turned down a different
route. I don’t really care about balancing pencils on my desk. You see,
I know a curious fact. I know that in quantum mechanics there is an
uncertainty principle which says that you cannot precisely know both the
position and momentum of an object. This of course means that *even in
principle*, since our world is dominated by quantum mechanics, I could
never actually balance even my model pencil forever, because I could
never prepare it with perfect initial conditions. The uncertainly
principle tells us
that the best possible resolution I could have in the position and
momentum of an object are set by Planck’s constant:

Assuming a completly rigid pencil which you place in a vaccum and cool down to a few millikelvin so that it is in its ground state. Roughly how long will it take this pencil to fall?

### Do it to it

So lets do it. This is going to be a bit quick, mostly because its
getting late and I want to go to bed. But the procedure is kind of
straight forward now. I need to choose initial conditions subject to the
above constraint, figure out how long a pencil with those initial
conditions takes to get to theta = pi/2 (i.e. fall over), and then do it
over and over again for different values of the initial conditions. So,
the first thing to do is figure out how to pick initial conditions that
satisfy the constraint. I’ll do this systematically by parameterizing
the problem in terms of the ratio of the initial conditions. I.e. lets
define

And second, zooming into the interesting region.

So, what is the best time you could balance a quantum mechanical pencil, i.e. what is the absolute longest time you could hope to balance a pencil in our universe? About 3.5 seconds. Seriously. Think about that for a second. Usually you hear about the uncertainty principle, and it seems like a neat parlor trick, but something that couldn’t influence your day to day lives, and here is a remarkable problem where even in the best case, the uncertainty principle puts a hard limit on championship pencil balancing which seems tantalizingly close. And there you have a graduate student working through a somewhat non trivial problem. I probably went into way too much details with the basics, but we are still trying to feel out who our audience is. Please leave comments and let me know if I either could have left things out, or should have went into more details at parts.

### EDIT

As per request, here is how the max fall time scales with the length of the pencil assuming a pencil with uniform density.

Plotted on a log-log plot, that is a pretty darn good line. The power
law dependence is

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