00:01
So in this question, we're told that we have a wave function proportional to sign n pi x over l, where x is in the interval from 0 to l, and we're asked to normalize the wave function, so calculate this coefficient, and we're also asked to find the expectation value of the momentum.
00:19
So first of all, let's normalize the wave function.
00:24
So we know that the integral from x equals 0 to l of mod phi squared, dx, has got to be equal to 1 because this is just the total probability of finding the particle, which has got to be 1.
00:40
So we get a n squared, integral from 0 to l of sine squared n pi x over l, the x has got to be equal to 1.
00:55
Well let's have a look at sine squared n pi x over l.
00:59
When x goes to l, sign squared, sign squared looks like this.
01:11
It's another kind of sine wave or a cos wave, but it has period now pi rather than 2 pi.
01:21
And so if we're going to integrate x up to l, we're integrating sine squared up to n pi, which means we're integrating it over an integer number of wavelengths.
01:31
But the average of sine squared over an integer number of wavelengths is a half because it's this periodic function.
01:46
And it has a maximum of 1 and a minimum of 0.
01:49
So its average value is a half.
01:52
So all we need to do for this interval is multiply that average value of a half by the length of the interval.
02:00
So we get a n squared times l over 2 is equal to 1.
02:07
So this tells us that an is the square root of 2 over l.
02:13
So our wave function is 5 of x equals root 2 over l, sine n pi x over l.
02:25
And now we want to evaluate the expectation value of the momentum.
02:34
So this is the integral of 5x, or more properly the complex conjugate, but since it's real, that's just going to be the same, times the momentum operator p acting on 5x, integrated with respect to x from 0 to l.
02:54
But the momentum operator is minus ih bar d by dx.
03:05
So we can insert that into this integral to get the expectation value of p.
03:10
So we're going to get an a .n squared from both of these.
03:14
So that gives us a 2 over l.
03:16
P gives us a minus ih bar...