00:01
All right, so let's look at the one -dimensional momentum operator in quantum mechanics.
00:06
So this looks at, this is like negative i times h -bar, times the spatial derivative operator in h -bar is the ordinary planks constant divided by 2 pi.
00:18
So we want to know what does the kinetic energy look like for this operator.
00:23
So our kinetic energy term is going to be classically p squared over 2m.
00:27
So if we do that here, that's going to be negative, bar squared over 2m times the second derivative with respect to position.
00:36
So this is our kinetic energy operated now.
00:40
And so for part b, let's say we have a wave function that looks like a times the sign of 2 pi x over l.
00:51
I believe, or sorry, a is the squared of 2 over l.
00:58
So we have this.
00:59
And we want to find the expectation value, the kinetic energy of this particle.
01:06
So the average value of k is going to be the integral of psi star, which si is real in this case, times k, times acting on site, integrated over x.
01:20
All right.
01:21
So if we plug in the values that we're going to have, we'll have 2 over l, and then this integral is going from like presumably zero, to l.
01:33
And so we'll have the sign of 2 pi x over l times our kinetic energy operator, which will be negative h bar squared over 2m times our second derivative.
01:49
And then this is acting on the sign of 2 pi x over l dx.
01:56
So what we can do, let's bring the constants out front.
01:59
So we'll have negative h bar squared over ml.
02:09
Yeah, so that's what we'll have.
02:11
And then the integral from 0 to l of sine of 2 pi x over l.
02:17
And then we'll just have the second derivative of the sign of 2 pi x over l...