00:01
Suppose we have a particle that is tunneling.
00:02
That is, the particle is in a region where its energy e is less than the potential u.
00:07
We want to show that this wave function can describe that particle in this region.
00:14
That is, we want to show that this is a solution to the schrodinger equation under these conditions.
00:20
So first, what we're going to do is write down the schrodinger equation to mind ourselves.
00:26
And what we're going to show is that this is indeed a solution.
00:32
So we're going to plug in this function into the trottinger equation and we'll show that it works.
00:42
So here's our wave function.
00:44
We're going to try it.
00:45
I'm going to rearrange this so that i just have the derivative over here and we'll group everything else on the left, on the right hand side.
00:57
So on this side, we'll have e minus u.
01:14
And we can do a little bit better.
01:16
Let's go ahead and multiply each side by negative 1.
01:21
And then we'll also go ahead and multiply each side by 2m over h bar squared.
01:27
So that i'll have d squared, dx squared of our wave function is equal to u minus e times 2m over h bar squared.
01:42
And so this is important because this is a constant.
01:46
This is a constant.
01:46
This is a is a second derivative.
01:48
And so we can start to see that when we do the derivative of this exponential, we're going to pull down this coefficient next to x, and we're going to pull it down twice.
01:58
And so if we have something on the right -hand side that looks like the square of this coefficient, we're looking good.
02:03
And looking ahead, we can see that it's going to look like that.
02:06
Let's go ahead and take that second derivative for the left -hand side, and write out our equation, plus or minus square root, mu minus...