Okay, in this video I'm going to continue on my tutorials on the quantum lab. This video at number 5 I'm going to talk about converting sign-in-call sign to complex exponentials. I'd like to draw your attention to my website at universityphysicstoreth.com. The previous video today was at number 4, where I talked about solving the Schrodinger equation. And just to say once more, that was a mix of a lot of other videos I've done or the conclusion of a lot of other videos I've done. So there was definitely a lot of information in there because it's just supposed to the conclusion of a lot of other videos. So if it was a bit too much, feel free to go back and look at my playlist on solving second order linear differential equations with constant coefficients. So why are we doing this? Well we're doing this because in that particular video number 4, I showed you that the general solution to, would say, our wave function, which is having solved the Schrodinger equation, is going to be something. E to the alpha x times a cos beta x plus b times the sign of beta x. Now just to remind you, we said that our characteristic equation had roots at lambda 1 and lambda 2. But in each of those, let's say just look at lambda 1, it can be written as alpha plus beta, where alpha was the real component and beta was the imaginary component. So for each of our solutions to our characteristic equation, we had a real and an imaginary solution. And therefore the general solution to our wave function turns out to be this, e to the alpha x, a cos beta x plus b times the sign of beta x, where of course I'm assuming that it's a function of x. Alright, now I also said to you that if there are no imaginary solutions, so beta is equal to zero, the sign of naught is not that disappears. The cos of naught is 1, so that basically disappears and you just get these real exponential solutions. If however you have no real solutions, then each of the alpha x becomes not and you just get these trigonometric cos and sine solutions. And that's usually what happens. You can of course get a mix between them, but it's usually either there are no imaginary or there are no real solutions. So we like working with exponentials because they're really easy. So for that reason, when we get completely imaginary solutions which turn out to be these trigonometric functions, we want to be able to turn them into exponentials. And I'm telling you that trigonometric functions cos and sine become complex exponentials. The hyperbolic functions, cos and sine actually become real exponentials and that's however it's just an aside. So this all comes down to Euler's equation and I've put a link to the proof of Euler's equation on my website. You can have a look at that under the playlist of quantum wells. So this is Euler's formula and it's strange. I'm not going to do the proof now, but let's just accept it that you can go from a complex exponential to this real cosine, but this imaginary sine. And you can go for this negative complex exponential to a real cosine and the negative of this complex sine. So for now on, I'm going to rewrite this as follows. Each the i-theta is going to be cos plus i sine. Let's put it right in that way and e to the minus i-theta is equal to cos minus i sine. All right. And going from these trigonometric functions, cos and sine and complex exponentials is actually very easy. All right. What happens if I add e to the i-theta plus e to the minus i-theta? Simply that's going to be cos plus i- sine plus cos minus i- sine is equal to twice cos. Or what we can say is we can rewrite it and say that cos of theta is equal to e to the i-theta plus e to the minus i-theta over 2 done. Next. Subtract them. When I've e to the i-theta minus e to the minus i-theta, that's going to be equal to cos plus i times sine minus cos plus i times sine is equal to twice i times sine. Rearranging, we can get the sine of theta is going to be equal to e to the i-theta minus e to the minus i-theta divided by 2 times i. All right. 2 times i. So that's really it. That's really it. So the point here is this. You can go from real trigonometric functions, cos and sine to complex exponentials using the Euler's formula. The important point is not here by the way, is that sine has this in the quotient, it's got i-out, i-out, it's got minus 1. So that can be a bit of a pain sometimes. So it's something to be very aware of. All right. So by adding the Euler's formula and subtracting it, you can get the formula for cos and for sine. All right. So that's all I've got to say about that. Thanks for watching. Please pass it on to your friends. Subscribe to my channel. And you might also click on universityphysicsstoryals.com. Thank you.