Original subtitles

Why come to this lecture about the Schrodinger equation, and before I start, let me say that this

is one of the most important lectures of the whole course, so please try your best to follow and to

understand how we end up with the Schrodinger equation.

So as a disclaimer, this is no derivation of the Schrodinger equation because this is in fact not possible.

The Schrodinger equation can only be postulated, and here we want to understand how Schrodinger came

up with the idea of writing down his famous equation.

So we want to find a motivation for this equation.

So as we have learned in the previous section, electrons and basically every piece of matter is a particle

and a wave at the same time.

So we can write down these two different terms for the energy.

And we can say that the wave function of our object is characterized by suchan plane wave, which is

an E function with an imaginary argument.

So here we have a dependence on the position and we have a dependence on the time.

Now, what we have here is the frequency omega, which basically tells us that we have a wave because

something is fluctuating.

And this plane wave, as we have seen previously, is the solution of the classical wave equation.

We will see that this wave equation holds very well for, let's say, classical waves.

But there are actually some downsides in terms of quantum mechanics that this allow that this wave equation

can actually be true for our wave function.

So what we have to find is an alternative differential equation, but we want that the solution still

looks like this because we know that this is true.

And so we take a whiteboard and write down the following derivative of this wave function.

So at the moment, it is not clear to you why I'm doing this, and that cannot be clear.

But in the end, I hope you will realize what I'm doing here.

So I'm just calculating the time derivative of this wave function and multiply it by these two prefectures.

And as you can see, we get these the energy in front of it.

And we have these two prefectures from here that cancel out with these two prefectures.

So we will just have the energy times to function PSI itself.

Now, as a second derivative that I'm calculating is that now I calculate the second derivative with

respect to the coordinate X, so this time what we get is the momentum squared, actually, and we have

this prefecture and now this age bar cancels with this age bar.

And what we have now is the momentum of a two M times our wavefunction itself.

Now, we know from this energy relation here that the energy of our particle is equal to P Square,

over to M.

So this means that also these two parts have to be equal.

So let's just write it down.

And what we have done here is we have written down the Schrodinger equation.

So basically, it means it's a differential equation that gives plane waves as the solution, and that

includes particle and wavelike properties for this wave.

Now, in one dimensions for free particles, this is exactly the equation we have just written down,

and in general, we can also treat it as a three dimensional system and also additionally account for

some finite potential, which is this V and now this is the total Schrodinger equation as you find it

in the literature.

So we have here this partial derivative and we have, again, this lipless operator that we also had

in the wave equation.

So this is the sum of the second derivatives with respect to X, Y and C.

So that's really, really cool and actually at this point, our cost could be done because if we would

be super smart or if we would have computers with enormous capabilities, we could calculate now the

wave function of the whole universe, given the fact that we could formulate this potential of all interacting

particles that take part in the universe.

However, of course, we cannot discuss the whole earth or the whole universe.

So we have to make some assumptions, some simplifications and treat some real examples, which we will

do in the following.

So now let's grab this board here and let's make notes what is really important in this whole section.

So from this lecture, we have the Schrodinger equation that is written down here.

So please keep it in mind.

We will need it many, many times throughout this course and you will see it very often.

So now this is nice.

We have derived to Schrodinger equation or more precisely, we have motivated the Schrodinger equation.

In the next lecture, we will discuss it and interpret it.

And I will tell you what all the quantities mean exactly.