00:01
Okay, so given this wave function, the square root of a divided by pi times x squared plus a squared, we're going to determine the following.
00:10
We're going to get a certain probability.
00:12
That's what we're going to do.
00:16
So we have to first specify what probability that we're getting.
00:22
And normally you would calculate a probability over the entire number line from minus infinity to positive infinity.
00:31
But you can also specify probabilities over finite intervals.
00:37
So in this case, we're going to calculate from minus a to positive a along the x axis.
00:47
So that's what i mean by the number line is just a particular axis.
00:51
So we usually, for one -dimensional problems in quantum mechanics, refer to either the x -axis or the r -axis if we're working in polar coordinates.
01:00
So anyway, here we're just working with rectangular coordinates.
01:06
So when we do a probability calculation, we have to take the integral of the probability density.
01:12
So we set this up as an integral from negative a to positive a, and the probability density is the absolute square of the wave function.
01:23
And when i say absolute square, i just mean that if this had been a complex function, we would have to multiply, this function by its complex conjugate, and that's how we would obtain a square in that case.
01:37
But here, it'll turn out to be only the square of the function, since this is a real valued function.
01:43
So proceeding, we'll go ahead and simply multiply the function by itself.
01:54
So as we can see, the square root will go away, and we now have a over pi times x squared plus a squared, dx.
02:12
And then taking this integral, we can pull a and pi outside, and we're simply integrating 1 over x squared plus a squared.
02:29
This is a trigonometric integral, and when you take this integral, you get the following, you just simply get the inverse tangent of x over a...