00:01
In this problem, you're looking at a harmonic oscillator in its ground state, and we want to find out in the end, what's the probability of it being outside the classical realm in terms of momentum? so we need the momentum wave function.
00:18
So here i've given you the wave function in terms of the coordinate space, and this is the time, this is the full wave function, it's got the time dependence.
00:32
I'll put in what e0 is later.
00:34
It's just the h -bar mega over 2, but i'm not going to be writing that throughout all these steps, just too much overhead.
00:42
But i'll give it to you here.
00:44
So that's the actual spatial wave function, and now we need the wave function in momentum space.
00:54
So we do a four -way transform.
01:03
So 1 over square 2 pi h -bar, minus infinity, infinity, exponential, minus i px over h bar.
01:18
Now the wave function psi not m omega pi h bar 1 quarter exponential is m omega x squared 2 h bar e minus i e knot t over h bar d x.
01:46
So that's our that's our expression that we have to calculate.
01:50
Notice, though, dx.
01:53
The exponential here, in terms of the time, does not have that any dependence, x -dependence.
02:04
So it's not going to matter.
02:06
So what i'm going to do next, i'm just going to drop that off for now, just so i don't have to keep writing it.
02:12
And we can just concentrate on everything else other than it.
02:18
Now, to make our life a little bit simpler.
02:21
There are a lot of m's, amegas, pies, h -bars, all these things hanging around.
02:27
Let's define a couple of constants to make the writing a little bit shorter.
02:31
A will be defined as m -o -m -o -2h -bar.
02:38
That's connecting to here.
02:41
B is defined as minus ip over h -bar.
02:51
They could have, i could have done, i could have done alpha and b but i'm concerned that alpha sometimes on the screen may look like an x.
03:01
So i just do a and b.
03:02
It's just two, just two constants to carry around.
03:07
Now i'm just going to work on the time independent part.
03:11
So that's everything other than the e, the minus i, e, naught t.
03:17
1 over square 2 pi h bar minus infinity to infinity, e, bx, e minus a, x squared.
03:33
And then we have this part in here.
03:37
That's going to be 2a over pi.
03:45
1 quarter dx.
03:47
So that certainly shortens it up a little bit.
03:49
Like i said, i'm leaving off.
03:51
I'm leaving off the time part for now, just to cut down it.
03:55
And let me clean this up a little bit more.
04:00
2a over pi 1 quarter 1 over a square root 2 pi h bar minus infinity to infinity how much you write when you're doing integrations and algebra and everything is just a matter of personal taste you know what you're comfortable with so e to minus a x squared minus bx combining that into a single exponential so that's what we have right now.
04:41
Okay.
04:42
Now, you might say, what am i going to do with that? well, let's complete the square.
04:47
You kind of have to know.
04:49
You kind of have to know, you might say, well, can i just look it up in an online or in a handbook? yes, you certainly could.
04:55
I'll show you how we actually go about doing it.
05:00
And it's a couple of tricks to actually get to the final answer.
05:05
So it's a useful thing to put into these know that there are.
05:10
Our ways of doing it.
05:11
You don't have to just look it up.
05:14
All right.
05:16
Let's complete the square.
05:19
X squared minus bx, a, x squared minus bx over a, a, x minus b over 2a, a, x minus b over 2a, squared, minus b squared over 4a.
05:42
And you can check that, that that actually is the case.
05:46
You might say, why are you doing all of this? well, you'll see.
05:49
It's one of those things you have to, it's one of those things that may not be hit upon right off the bat if you were doing this for the first time, but it is something that is a useful approach.
06:05
So now let me look at the exponential.
06:13
Putting everything in, e minus b, minus b squared for a, e minus a, x minus b over 2a squared.
06:32
So this becomes e to the b squared for a, e minus a, x minus b over 2a squared.
06:48
So that's what we end up with.
06:51
And you might say, still, what are you doing? well, one step at a time.
06:56
Now let's rewrite the integral.
06:58
And i'm just going to work on the integral.
07:00
I'll leave the other stuff alone outside.
07:03
That can be handled later on.
07:06
We've got pies.
07:07
We're going to have other pies.
07:09
No reason to fool around with it now.
07:11
E minus ax squared minus bx, tx is the original integral.
07:23
E b squared, times over 4a, minus infinity to infinity.
07:31
E minus a, x minus b over 2a squared dx.
07:41
So that's our integral.
07:43
And then this becomes e to b squared for a integral, minus infinity to infinity, e to minus a x prime squared dx prime.
08:01
For x prime is x minus b over 2a.
08:13
Being the limits are infinity.
08:15
Nothing's going to change between the limits on the integration.
08:19
You might say, why? you might say, i still don't know how to do that integration.
08:25
Well, i'm going to show you how to do that integration.
08:27
We're kind of a way working, you kind of have to know where you're heading.
08:31
You have to know your route.
08:35
So that's the, this is the integral.
08:37
That's actually, you're going to find out.
08:39
This is a very simple, or if you look it up in the book, it's a very simple result.
08:44
Scurro to put a part.
08:45
Pi over 8.
08:47
That's what it is.
08:49
So let me show you how that comes about.
08:55
That is defined an integral, i not important.
09:03
The zero here deals with in general powers of x or x prime in this case.
09:12
That would appear outside that you could use i not to get that through differentiation.
09:19
We don't need that here.
09:21
It's just notation.
09:23
So this is an integral function of a.
09:25
In terms of the x prime, it's gone.
09:27
It's a definite integral.
09:32
Minus infinity to infinity, e minus a, x squared, tx, my x, x squared, and so on.
09:41
This is just pure mathematics.
09:43
Now, i'm just looking for this form.
09:45
So don't, i don't want to write the primes continually and so on.
09:50
This is just that form, eden minus a, x squared.
09:59
But i can square that.
10:03
You might say, again, you're saying you sell why.
10:07
You got to wait to the end of the story.
10:12
We know it's a definite integral.
10:15
So it's got some function of a.
10:19
And if a is a known commodity, it's just some constant, some number.
10:28
So i'm going to square it.
10:29
So all i got to do is write the integral twice.
10:32
That's squirming it.
10:37
But i'm going to turn it into polar integration.
10:42
By doing this.
10:49
I don't want that.
10:56
Minus infinity to infinity, e minus a, x squared plus y squared, tx, y, y.
11:07
Well, now i can turn into a polar integration.
11:12
Zero to infinity, zero to pi, e minus a row squared, row, t row, t, phi.
11:26
That's the area element in the polar coordinates.
11:32
The area, the phi integration is trivial.
11:37
So this becomes 2 pi, 0 to infinity.
11:41
It's 0 to infinity.
11:42
Remember, row is, think of x -quip plus y -square.
11:45
There is no minus to that.
11:47
It's a radius.
11:52
So 0 to infinity, e minus a row squared, row, d row.
12:02
So that's what we have.
12:04
And i say, wow.
12:07
Okay, let's continue.
12:13
U is equal to e minus a row squared.
12:19
D .u is equal to minus 2a row, e .m minus a row squared, d row.
12:32
So that gives me that the integrant, row, e minus a, row, d -row is equal to minus d -u.
12:49
So, i -0 -squared a, 2 -pi, we got minus -d -u over 2.
13:02
Now, when row is 0, u is 1.
13:10
When row is infinity, u is 0.
13:17
But with this minus side, that we can flip the limits on the integration.
13:20
So 2 pi 0 to 1, 2 .u, i'll put the 2 out here.
13:30
And i think i lost the a.
13:34
There we go.
13:34
2a, 2a, there we go.
13:40
So, this just becomes pi over a, which gives me that i not a square root pi over a.
13:58
You could not do this integration.
14:01
This e to minus a x squared dx through elementary procedures.
14:05
Doesn't work.
14:07
There's trickery had to be done.
14:10
We've already done a couple of tricks.
14:12
We completed the square, and now we did this.
14:16
So, we now have minus infinity to infinity, e minus a x squared plus bx.
14:43
Actually, this should be a, this is a minus sign here, not a plus minus bx tx is equal to e b squared or 4a square root of pi or a not so bad looks like a lot of work but it's just a matter of recognizing the tricks that are done so now we've done all our you know really from all this, when we started from here down, we're just doing all mathematical work from this point down, just doing the integrations.
15:31
Now we can get back to the actual function of just of the momentum.
15:41
Put everything into a over pi, one quarter, one over a square root, two pi over h bar e and b squared for 4a square root of pi over a okay now it's just a matter of uh going through all this being very careful with our algebra two pi over a one quarter one over a square root to pi h bar eb squared over four a what i've done here got a to the one quarter a to a to a minus one half so one quarter minus a half is minus a half so that's where the a ends up on the bottom now under the quarter and likewise with the pie it gets flipped you got one half minus one quarter so that's why the pie comes ends up on top...