00:01
In this problem, you're asked basically primarily for the commutator of lx, li, the angular momentum along the x axis and along the y.
00:13
Also, they do want you to specify and connect lz with that.
00:18
So we will have to look at what lz is.
00:21
They don't give you that.
00:22
But there's nothing special, as you'll see me do later.
00:26
These are just from our, you know, that angle momentum is r across p.
00:33
And lx is just the x component of that cross product.
00:37
And ly is the y component of that cross product.
00:40
But what happens here in quantum mechanics, p is obviously an operator, differential operator.
00:49
So that's where these partials come from.
00:51
So we have to deal with those.
00:53
All right, and we'll get to the lz later on.
00:57
Once we get through with all this, then we'll need it.
01:00
All right, lx, ly.
01:03
We think of this as this is an operator.
01:04
So it's going to operate on something.
01:07
And i'll write that in a second.
01:08
But let me just expand out the base commutator, you know, a -b -m -a -l -x, l -x, l -y, minus l -y, minus l -y, l -x.
01:19
That's the base, meaning of a commutator.
01:22
Now, like i said, it is an operator, so we're going to have to have it operate on something.
01:26
If it wasn't, if there wasn't a differential in here, differential operator, then you could leave this step off.
01:35
A lot of times if you're just dealing with x something of that nature but usually you want to put something to operate on and then you see what the result is in the end something times that same thing and you have what the commutator is itself so lx l y on some unspecified function side okay now we can start writing everything in now we're going to have minus ih bar in both terms.
02:08
So that's just going to be minus i, h bar squared, which when all said and done is minus h bar squared, but i won't worry about that yet.
02:18
All right.
02:19
So now it's just a matter of writing everything in.
02:21
Y, partial respect to z minus z, partial respect to y.
02:29
You've got to maintain partial y, okay.
02:35
And obviously remember you've got to maintain your order.
02:38
Z, partial respect x minus x partial respect to z you got and multiplied by the side doing it as a making you i could leave the side of the square brackets but i'll put it in here just to make it a little easier uh to see and let me put the next one on the next line give myself some room uh minus c partial respect to x minus x partial respect to z y partial respect to z minus c partial respect to why.
03:25
That's everything.
03:25
Just written out.
03:27
We haven't done anything yet.
03:31
Okay.
03:32
Now, the secret is do not switch order.
03:38
When you got a y, anything of that nature, you switch nothing.
03:42
Leave everything as it is.
03:44
If something can be switched, we'll see it later on.
03:47
Right now, just write everything, keep everything in order.
03:53
And this is minus h bar squared now.
03:56
Minus squared is 1 plus 1, but i squared is minus 1.
04:00
So that's where the minus comes from.
04:02
All right.
04:03
Now let's start expanding out.
04:05
Why? partial respect to z.
04:07
This is no more than if you had a plus, you know, a plus b times c plus d.
04:11
We're doing nothing.
04:12
We just have a lot of writing to do and maintain the position.
04:17
Do not break the position.
04:19
Why, partial respect to z.
04:21
Z, partial respect to x, psi, minus y.
04:28
Partial respect to z x partial side respect to z so that's all i'm doing is the first term times the the two terms on the right factor so i've done there this is a y okay all right we did that all right now keep going minus c partial respect to y c partial respect to y c partial side respect to x plus z partial respect to y x x partial si respect to z x partial si respect to z so we have those so now we've taken care of this term with both terms of the right factor so the first line is done now we'll deal with the second line this is a y here okay there we go all right minus z partial respect to x y, partial si respect to z plus z, partial respect to x.
05:48
Z, partial respect to y, si.
05:54
Okay, and then we have the last two terms plus x, partial respect to z.
06:02
Y, partial si respect to z minus x, partial respect to z, c, psi, now that is everything.
06:21
Now we have to start looking at the partials.
06:28
So like here, we're going to get two terms out of this because of partial respect to z.
06:32
So first we'll do it on the first factor.
06:36
So just give me y, partial as si respect to x.
06:39
And then y, z, second partial of psi, z x.
06:45
Partial respect to z, mixed second partial.
06:47
How can i ever say? so, minus h bar squared.
06:53
Okay, y, partial si, is back to x.
06:58
So we're just using our knowledge of partial derivatives here.
07:03
Nothing else needs to be done at this point.
07:07
Plus y, z, partial psi, second partial, z, x, which, remember, does not matter what order.
07:19
We take that.
07:22
So all is fine.
07:26
Minus y, x, partial, second partial, c squared, minus c squared, second partial, y, x.
07:48
Okay.
07:50
Then we have plus cx second partial y, respect to y, respect to z, minus zy, second partial x z let me go down here give some room plus z squared second partial x y plus x y plus x y partial partial z squared okay i got two four six eight four eight okay minus x partial si respect to y minus x z, second partial, z, y.
08:55
Now we've got all our terms, we've done everything we can.
08:59
Now, let's take a look at what we can do.
09:03
Nothing we can do about this term here...