…says the following: you don’t have to
know what’s going on anywhere outside of a little ball; if you want to know what the
potential is here, you tell me what it is on the surface of any ball, no matter how
small—you don’t have to look outside, you just tell me what it is in the neighborhood—and
how much mass there is in the ball. The rule is this… The man you just saw speaking is physicist
Richard Feynman, giving a lecture at Cornell in 1964. I remember watching this in my dorm
room my freshman year of undergrad, and I was bewildered by Feynman’s “average on
a ball” concept, whatever that meant. In this video, I want to really dive into what
Feynman is talking about, and in the process, we’ll develop a deep intuitive understanding
of what the second derivative really does in math and physics, and why it shows up in
the schrodinger equation, E&M, and elsewhere. As a heads up, I will assume you have some
familiarity with Taylor series, in so far as you know that we can expand a function
as such. Now before we dive in, really quick I want
to let you all know of a really cool opportunity from a few Harvard physics PhD students working
with the Harvard Quantum Initiative. In celebration of World Quantum Day, they are hosting a Quantum
Shorts Contest over on their HQI Blog. This is a contest open to absolutely anyone, regardless
of your background or experience in physics. Basically, they invite you to create a short
video on a quantum topic of your choosing, using your creativity to explain some aspect
of quantum physics. After submitting your entry, you have the chance to win Harvard
merchandise and even a trip to Harvard to explore their quantum research facilities
and meet the scientists that push quantum research forward. This is a really neat opportunity
hosted by some really passionate people, so check out their blog and contest if you’re
interested, I’ll have it linked in the description. The deadline is May 14, 2024, so good luck
if you choose to submit anything! Now, to begin our journey on the second derivative,
I think we should quickly review our intuition of the first derivative. Essentially, say
we have some variable x, and a point x0. Likewise, let’s say we have some function f(x), where
I’ve indicated where f of x0 lands on this number line. If we move x a tiny bit away
from x_0, we will correspondingly move f(x) a tiny bit away from f(x0). If then we take
the change in f(x) and divide by the change in x, you intuitively get the first derivative
at the point x0. Formally you’ve got all the limits and whatnot, but this the intuition.
So the first derivative intuitively tells us how much f changes when we change x by
a tiny amount. So what about the second derivative? What
does it tell us? Well, usually we’re taught that it tells us how the first derivative
changes when we change the input by a tiny amount. But…this understanding kind of sucks.
I don’t wanna know what the second derivative tells me about the first derivative, I wanna
know what it tells me about the function itself! So…how do we go about trying to intuitively
understand the second derivative? Well, here is where we are going to follow Feynman’s
lead – so let’s dig into this “average on a ball” concept he was talking about,
first in one dimension. Let’s say we have some function, and let’s
look at a particular point x_0. What I want to do is look at the points right next to
x_0, both a distance dx away. Note that this is what a “ball” is in one dimension – it’s
all the points of radius dx away from x0. Now, again trusting Feynman for a moment,
I want to know if the value of f at the points next to x_0 are on average greater than or
less than the value of f at x_0. Here we see they are both greater, but how do we quantify
this? One way to measure this is by calculating
the average of f for the points around x0 (where I’ve used this fancy double bracket
to indicate the average), then subtract the value of f at x_0. This should tell just how
much higher or lower, on average, the points around x_0 are. Take a second to make sure
you understand what this quantity represents. This might seem like a random expression,
but let’s follow through with it for a moment. First, let’s calculate the value of this
average term. The average of the two values around x0 is
calculated exactly as we’d expect: by adding then dividing by two. Now, remember that dx
is supposed to be pretty small, so that should inspire us to taylor expand both of these
quantities about the point x_0. The taylor series of the point to the right
of x_0 can be written as follows, while the series of the point to the left of x_0 can
be written similarly. Now if we add the two, note that the terms with an odd power of dx
will cancel out, leaving only the terms with an even power of dx. So, we get the following.
Now note that using a first order expansion for both terms wouldn’t have worked here.
Usually that does the trick, but notice that the first order approximation canceled out!
– we’ll have more on this in a moment, but keep this in mind. So, dividing by 2,
we get that the average of the points around x_0 can be written as follows. Now, we can subtract the point at the center,
f(x0), from both sides. To proceed, I’m going to divide both sides by dx^2. Now, let’s
take the limit as dx goes to zero on both sides. Note that all the terms with dx^2 and
higher on the right hand side will go to zero. If we then move the ½ in front of the second
derivative to the left hand side, we are left with the following. This is a really neat
result: we have found that the second derivative at a point is related to the average of the
values around that point, minus the value of the function. And if we take a moment to
think about this result, this should make a lot of sense. Remember that we usually use the second derivative
to study the curvature of functions. If a function is concave up, then at any point,
the values of the function around that point are on average higher, so the limit we derived
a few moments ago would be positive, giving us a positive second derivative. And if a
function is concave down, then at any point, the values of the function around that point
are on average lower, so the limit is negative, and we get a negative second derivative. So
this whole “average on a ball” business is really just a way to quantitatively capture
the curvature of a function, which happens to be related to the second derivative. Now, what is the curvature of a straight line?
Zero! Which explains why the first order terms in the taylor expansions canceled out! Those
terms represent the linear part of the function, which contribute nothing to the curvature. So we see that the second derivative for a
single variable function has a really neat geometric interpretation in terms of the average
value around a point, minus the value at the point itself. Now, say we wanted to extend
this idea to three dimensions, how would we do that? Necessarily, this becomes a problem
in multivariable calculus, but we can guess what the solution would be in this case. In three dimensions, to find the average value
of a function around a point, we would look at a tiny sphere of radius dx around that
point, and take the average value of all the points on that sphere. Then, we would just
subtract that average by the value in the middle of the sphere. Our claim is that this
is related to some three dimensional version of the second derivative. It turns out, this
is one hundred percent correct! In three dimensions, the corresponding expression is as follows,
where the second derivative in three dimensions is written using this symbol here, called
the Laplacian. The only difference is that the 2 has become a 6 (and in fact, that number
is always 2 times the number of dimensions you’re in). For those of you who have taken a multivariable
calculus course, you will recognize the laplacian and know how to calculate it, but for anyone
who hasn’t, let’s just take this to mean a second derivative in 3 dimensions, which
it really is. Now, for those of you interested in a derivation of this fact, which you are
much entitled to, I’ve included a link to a clean derivation in the description. This
is one of those expressions that is much better suited to being derived on paper where you
can see each calculational step – as opposed to me flashing a hundred equations on a screen
for 20 minutes. That being said, the intuition for this equation is exactly the same as the
one dimensional case. Now, taking this expression as a fact, we
can actually already use it to intuitively derive some of the most important equations
in modern physics, without doing any tedious math. For example, say we have some electrical charge
distribution defined by a function rho(x). How would we come up with an equation for
the potential energy function generated by this charge distribution? Although this seems
like a tough problem, we can use our newfound intuition to give it a shot. Let’s say we have a region of negative charge.
If we take a positive charge and pull it away from the region of negative charge, the attractive
force means we have to put in work to do so, so it gains electrical potential energy as
we pull it away from this region. Similar to how lifting something up gives it more
gravitational potential energy. Likewise, if we instead had a region of positive
charge, and we pulled our particle away from this positive region, the positive charge
is being repelled, so it loses potential energy. So, let’s use this intuition to form a differential
equation! Say we have our potential energy function defined in all space, and let’s
look at some point x0. We can then examine the average potential energy on a tiny sphere
around x0. If the potential energy is bigger at points away from x0, then our intuition
tells us that there should be some negative charge here, because this means that our particle
gains potential energy by moving away from x0. Likewise, if the potential is smaller at points
away from x0, then our intuition instead states that there is some positive charge here. So, summarizing all this, we can use our physicists
intuition to guess that maybe an equation of the following form is right: the second
derivative of the potential (which tells us the average of how much greater the potential
is around any given point) should be proportional to the negative of the charge density at that
point x0. Take a second and digest this, and you’ll see that this exactly describes the
conclusions we made a moment ago. And it turns out, this is exactly right, this
is in fact one of Maxwell’s equations, with A equal to one over the permittivity of free
space. So we have intuitively derived an equation without needing any fancy electromagnetic
theory. Although we got somewhat lucky that the relationship on rho wasn’t something
more complex, you’d be surprised how often nature seems to choose the simplest expression. Now, given that this channel has been dedicated
to quantum mechanics in the past, we can also use our newfound understanding to develop
further intuition into quantum phenomena. Specifically, I want to look at the kinetic
energy operator in quantum mechanics, which in the position basis can be written as the
second derivative. Now, even if you’ve never seen this before or never even taken a course
in quantum mechanics, we can still understand why this quantity should represent kinetic
energy. To recap some quantum physics, the wavefunction
is a function that tells us the probability amplitude of where our particle is. With just
this, we can use our second derivative intuition to derive some quantum facts. First, let’s
assume the position wavefunction of our particle looks like a simple gaussian. For now, let’s
assume it’s a fairly tight gaussian, meaning our particle is well localized around some
point in space. So, how do we interpret the kinetic energy
operator on this wavefunction? Well, if we take this relation as fact for the moment,
then at the peak of our wavefunction, all the points around it are on average much much
smaller, so, using our newly developed intuition, we expect the second derivative to be a big
negative number, which when multiplied by the negative, means that this quantity is
a relatively large positive number. I say relatively because hbar is tiny, but the smaller
we localize the gaussian, the bigger we make this number. Now, this should be a somewhat shocking result.
Although it’s a bit wishy washy to interpret an operator at a single point, this statement
approximately holds true in the region where our particle is localized. So, why should
this be surprising? Well, all we did was define where our particle was localized in space,
and nowhere did we input how it was moving or in what direction. So…why and where is
our particle getting this magnitude of kinetic energy? What we are discovering here is in fact a
vestige of the heisenberg uncertainty principle. Note that our particle is really tightly confined
in space, so we have a really low uncertainty in its position. The uncertainty principle
then dictates that we must be wildly uncertain in our particle’s momentum, and therefore
it can take on large magnitudes, increasing our particle’s kinetic energy. And in fact,
if we time evolve this initial quantum state, the solution we would get be a gaussian that
spreads out through time, since that extra kinetic energy from momentum uncertainty pushes
our particle outwards. So, now we can see that the kinetic energy
operator in quantum mechanics not only measures how fast our particle is moving, but it also
carries information on how the uncertainty principle affects its motion: the more localized
our wavefunction is, and therefore the smaller the average values around it are, the more
its motion and energy will be warped by momentum uncertainty. I think this is absolutely fascinating,
and it gives us some intuition into how the uncertainty principle is baked into the schrodinger
equation through the second derivative. Now before wrapping up the video, I encourage
you to think of how you can use this understanding of the second derivative to build new intuitions.
For example, think about what the differential heat equation is actually saying. Likewise,
in quantum physics, higher energy wavefunctions tend to have smaller and smaller wavelengths–
how can we now understand this? I will leave this up to you to think about!
As physicists and mathematicians, hopefully I’ve given you another tool that you can
use to understand our world, much in the same way the Feynman did for me many years ago.
As always, if you have any questions, feel free to leave a comment and I’ll do my best
to answer it. Hope you all had a good quantum day!
