Hi guys! I'm Jade, lovely to meet you. So
the Schrodinger equation is one of those things that pops up a lot in like
quantum science articles and journals and stuff but the journalist doesn't
usually go into what it means, which is fair because it's a fairly complex topic.
So today I just wanted to share with you guys like what it actually means so next
time you read it in an article you can get a better gist of what it's about. So
the short version is the Schrodinger equation tells us everything we can
possibly know about a quantum system. It's basically the F=ma of the
quantum world. If you throw a ball and solve F=ma
you can predict its position and momentum for any moment in time. Once you
have these two you can derive basically everything else you could possibly know
about it, velocity, energy etc. But when we get down to particle land Newton's
equations don't work anymore. If we put a particle in a box and we
want to know where it is F=ma just doesn't cut it. The Heisenberg
uncertainty principle says that we can't know both the exact position and
momentum of quantum objects, but, we can know other things, like the energy levels
and the wave function, which we'll explore in this video. But that information is
all inside the Schrodinger equation and with some hardcore mathematics we can
tease it out. Now the long version! So before I get started I should say that
this is the time independent version meaning that it doesn't involve time. And
if at any point throughout the video you get lost, don't worry, even Schrodinger
didn't know exactly what his equation meant. So let's say we have a quantum
system: an electron in a box. We want to know everything we can about this
electron so we can make predictions, where it might be,
what energy level it might be at. These answers are all buried within the
Schrodinger equation. So first let's start with this guy, the pitchfork thing.
This is the Greek letter psi and it stands for what's called the wave
function. It tells you where the electron is likely to be. But not where it will be.
See quantum objects are sneaky in that you can't predict exactly where they'll
be until you measure them. You can only predict where they'll probably be. Say
there's this kid you know. You've grown up with them your whole life. They're locked
in their room with their homework, a Playstation, and a bed. If you had to
guess where they were you'd say there's about an 80% probability they're on the
PlayStation, 19% probability they're in bed and a 1% probability they're
doing their homework. When you open the door you'll know for sure. But you were
able to make these predictions because you know this guy. What if you had to
guess where this electron is? You don't know this electron. Well that's what the
wave function tells us. It gives us the probabilities of where it's likely to be.
But a big difference is that while the guy is only in one place at a time the
electron is in a superposition of all possible places at the same time. You may
have heard the famous thought experiment Schrodinger's cat who's in a
superposition of being dead and alive until the box is opened and it's forced
to choose a state. It's the same deal here. The act of not knowing where the
electron is allows it's probability distribution to be spread out over a
large space kind of like a wave. Different kinds of waves can represent
different probabilities of where it's likely to be, hence the name wave
function. It's a function that describes the wave shape of probability
distribution of the electron. Oh yeah and when you open the door and
measure where it is this wave probability cloud function collapses and
the electron becomes a particle again. No wonder Schrodinger was confused. So
that's what the pitchfork means, the wave function tells us where our
electron is likely to be. Now let's take a look at this E. It represents the
energies the electron is allowed to have. Now before I go into what that means I
just want to point out that the way this equation is arranged these values are
the ones we're trying to solve for, so it's telling us that hey if you do all
this stuff you can find out the energy levels of the wave functions of the
electron! And if we know these two things we can derive everything else we can
possibly know about the particle, just like the position and momentum of the ball.
But let's backtrack a sec, what do I mean when I say energy levels the electron is
allowed to have? Like it's a grown electron it can have whatever energy
levels it wants right? Well, no. In the regular world we see around is,
energy can go up and down in a smooth continuous way, but this isn't the
case in the quantum world and the reason comes from the wave-like nature of the
probability distributions. Because our particle is inside the box it has a zero
probability of being found in or outside the wall, so this means the wave function
always needs to be zero there otherwise there's some probability that the
electron could be outside the box which we know it's not. That means that the
electron can only have certain frequencies associated with it. This
frequency is allowed as the wave function is zero at both edges, and this
frequency is not, this frequency is allowed and this one is not. So Einstein
discovered that energy is actually proportional to frequency by this
relation E=hf where E is the energy, f is the frequency and this h
here is Planck's constant. Don't worry too much about that h for now, all you
need to know is that it's a constant which means its value doesn't change. So
if only certain frequencies are allowed inside the box and this is a constant
then it follows that only certain energy levels are allowed inside the box too. This
property of discrete or quantized values is where quantum mechanics gets its name.
Things that can take on continuous values in the regular world like energy
levels can only take on certain quantized values at the quantum scale.
Now let's look at the other side of the equation. We know what we're solving for:
the energy levels and the wave functions. But how is all of this going to help us
get there? Well overall energy is made up of kinetic energy and potential energy.
If a skateboarder is on a skate ramp they'll be traveling at some speed and
have some kinetic energy, but when they stop at the top they still have energy,
it's just transformed into a different kind: potential energy. The entire energy
of the system is just the kinetic energy plus the potential energy and while it
can move back and forth between those two states its total value is always
conserved. Sometimes potential energy is written as a V, so this term is the
potential energy of the wave function. So if this is the potential energy that
must mean that this term here is the kinetic energy I know it doesn't really
look like any kind of kinetic energy equation we've seen before right, so
here's the derivation if you don't believe me. So if we can solve for the
potential and kinetic energy of our quantum system
this will tell us the energy levels allowed, and that's everything there is
to know about our little electron! So what would some typical Schrodinger
solutions look like? Well to this particular problem
all the solutions to the wave functions take these two forms and the energy
equation that popped out was this. Great Jade so what the hell does that mean?
Well the first thing to note is that every term in this expression is either
a constant or a whole number: h-bar is a constant, 2 is obviously a constant, m the
mass of the electron is a constant, pi is a constant, and L the length of the box
is a constant, and n stands for the different states of the electron and
they're all whole numbers, 1, 2, 3, etc. So then the energy E can only have certain
values. It's quantized. But what about the wave function? Where is the electron? Well
let's look at this guy when the electron is in its first energy state, when n is
equal to 1. We get this. That's one of the wave functions of the electron and if we
square it we get the probability distribution,
aka where the electron is likely to be. We can see that there's a high
probability that it'll be found in the middle here but a zero probability
it'll be found right at the edges. Here are some more wave functions and
probability densities for other energy states. See how the wave function is
always always 0 right at the edges. This took me an entire semester of a physics
degree to understand and what really helped me was working through a lot of
problems and taking the time to build a strong intuition. Brilliant.org has an
entire course dedicated to quantum mechanics which starts with the
experiments which first discovered quantum behavior and leads up to the
derivation of the Schrodinger equation. It has examples you can work through at
your own pace. I actually just went through it to refresh my memory and
learned some things I didn't know and definitely understood some things better.
They also have heaps of other courses mainly on physics, math and computer
science and they're always adding more. The first 200 people to click the link
below and sign up will get a 20% discount. Just go to brilliant.org/UpandAtom. The link is on screen and in the description. And if you're wondering
why I didn't include the math behind the solutions to the Schrodinger equation
it's because it would have taken me about an hour just to write out
probably another five weeks to explain, but for those of you who are especially
curious I've posted my final exam essay on quantum physics in the description. It
includes the derivation of the Schrodinger equation as well as the
solutions and some other interesting stuff like random questions to my
professor. So guys please be honest, do you understand the Schrodinger equation
a bit better now? I hope you do but if you don't please feel free to give me
any feedback on how I can make my explanations better for future videos.
Quantum physics is so cool and I really don't want anybody to miss out. So yeah
that's all from me. Bye! -
