00:02
In this problem, you have a harmonic oscillator, and the end goal is to get the uncertainty product between x and p, delta x, delta, p, and see what that beats to for the general harmonic oscillator state.
00:16
So to get those uncertainties, we're going to need x, expectation value x, x, x squared, p, p squared.
00:24
So let's start with x.
00:27
Given the state n, general harmonic oscillator state n, x, n, that's the expectation value.
00:34
Putting our value for x, or expression for x, i should say, 2h bar over m omega, n, a plus a dagger, and we can regroup this, or expanded i should say.
00:57
This would be n -a -n plus an dagger, n.
01:11
And so let's continue on, one -half square root, 2 -h -b, bar m omega now using our relations here a on n is square root to n and n minus 1 this is equal to 0 so this term is gone plus n well actually let me plus n squibble to n plus 1 and n plus 1 this is equal of zero, this term is zero, so the expectation value of x is zero.
01:56
But from the, if you think about the symmetry of it, that should make sense to you.
02:04
Now to do p, i think about, you know, should be on one side of the oscillator on the right or on the left, is there any difference? there is not.
02:17
So you're, so you have just as much probability to be a specific position, on the right and it's match on the left so your average is going to be zero and be at the zero momentum n p n i i square root am omega h bar over two and a dagger n minus n a n but we just saw n -a -n is zero, n -a -dagger -n is zero.
03:04
So this is also zero.
03:07
So we have those expressions.
03:13
And again, that should be, make sense.
03:19
Remember momentum is a vector.
03:22
You don't you have equal chance of going, being going to the left, going to the right, positive, negative.
03:29
So that i would expect, you'd expect to zero out of that also.
03:34
Just like how you would analyze the same infinite well same idea okay now for x squared and x squared and okay so one quarter two h bar am omega now this is going to be you have to put in our expression a plus a dagger a plus a dagger so that's what we get h bar over two and m omega and then we are going to get out of this when we expand n a -a -n plus n a dagger n plus n a dagger a n plus n a dagger a and plus n a dagger a so that's what we get just doing normal expansion but you guys got to maintain order.
04:55
Cannot switch.
04:57
A dagger a is not the same as a dagger a is not the same as a dagger.
05:05
Okay.
05:06
So we have all these to do.
05:08
Now, let me actually do them separately so i cut down a little on the writing.
05:13
Let's do n, a, a, n.
05:17
First, a will give us square root n and a, n minus one.
05:30
And then we're going to get square root n, and then a acting on n minus 1 is going to give me a square root n minus 1.
05:37
Think of this as n minus 1 as being some new quantity k, an a acting on k.
05:46
Well, that would be k minus, that would be square root of k, which is n minus 1, and then we're going to get an n minus 2 out of this.
05:59
And n minus 2.
06:00
But we know these are these are orthogonal states so zero as we've done before this is zero so that term's gone n a a dagger and got more hope here usually you know whenever you're you have to think of raising and lowering that somehow this will bring us back so we have the same state on both sides the bra and the cat dagger a gives us square root n plus one and a n plus one that's what we get from the first the a dagger and then we're going to get the square root n plus one and now we're acting a on n plus one so that gives us n plus one and we've got to reduce that remember what's going on it acts it takes whatever in that in the um um cat put square root around it and then drops the value by, i mean, increases the value by one.
07:08
So this is going to, or i mean, what am i saying? a decreases the value by one.
07:16
So we're going to from n minus one, n minus two.
07:19
So this becomes n, and then n plus one minus one.
07:23
Say that again, that's one.
07:25
So this becomes n plus one.
07:27
So something finally, non -zero.
07:29
Everything we've done so far has been zero.
07:32
So now let's do a dagger a, the next one, the square root n now, because a is acting first, n, a dagger, n minus one, square root n...