00:01
Now in this question, basically look at a quantum harmonic oscillator, right? it's a quantum harmonic oscillator, right? you are giving that the harmonic oscillator is in this quantum state, 1 plus 2, over square of 2, right? at the initial moment, right? and you ask what is the state vector at the later moment? so side t, obviously, it's given by, you know, it's just given by square root of one of square of two and a one, and you just get a phase factor that's minus three omega, sorry, should be, yes, it's three omega, i have got a room, three omega over two times t and plus two minus i, five omega over two t.
01:01
Okay, omega is the frequency of the oscillator.
01:08
Then what is the, you ask to evaluate the average value of the x, the position operator, the momentum operator, and the x squared, and p squared.
01:22
But to do this, the best actually is to write x in terms of the a -and -a -bag operators, right? so x is simply given by h over 2m omega a plus a dagger.
01:38
And p is given by scluret of 2h, sorry, mh omega over 2, same time over i, a minus a dagger.
01:57
Yes.
01:59
So and then if you look at the average value of this x operator, and it's very clear that it's given by h over 2m omega.
02:10
And you look at the average value of a, basically over side t, right? so you look at basically side t and a and a dagger, a side t.
02:25
And similarly, you have to have the side t, a dagger side t, right? and if you make use of the property service, this a and a dagger, you can easily see that this is simply given by 2m omega.
02:39
The first part obviously connects just, it connects basically a has to act on basically the state of 2, and you get a 1, so you will find this equals half, right? you will have half because each character is half, and act on act on 2, you will get one, the one state, and it has to overlap with the one state, and then you will find this to be, the difference is extra omega, so you'll find this to be minus i omega -t...