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! -