00:01
On this question, basically, you asked to show that expectation value of the potential energy of harmonical state is half that of the total energy in the ground state, right? well, i think the simplest way to show this actually is to use drug notation and the lead operator.
00:22
So i would, you know, write the ground state as n equals zero or simply write as zero, okay? that's ground state in the drug notation.
00:34
And the potential energy, you need to, the average of potential energy, you basically have to do the average of this k over 2 and 0 x squared, right? now, x is the position operator.
00:49
You can re -related using the led operators in this way, h -bar, which is a prime constant, divided by m, omega, the oscillator, the mass of the oscillator, omega is the proper frequency of the oscillator.
01:00
And the a plus a dagger.
01:03
So a is the sometimes called destruction operator and a dagger sometimes called generation operator.
01:11
Okay.
01:12
So if you do the x squared, you'll find this to be h.
01:17
Sorry, there's a two here, 2m omega and the a squared plus a dagger squared plus a dagger squared plus a dagger a, okay? that's a square.
01:28
Now if you look at the average of this x squared, basically have to look at the average of the average of these four operators, right? and you can easily see that a squared, the average of a squared gives you actually zero, an average of a dagger squared also zero.
01:43
And this one average also zero...