00:01
Hello everyone, and this problem we're asked to state whether the statements given are true or false based on our experience or our knowledge or studies on quantum mechanics.
00:14
So the first question or the first statements says that in a quantum harmonic oscillator, the energy levels are evenly spaced.
00:23
So we know that for the quantum harmonic oscillator, the energy levels are given by en is equal to m plus a half h bar omega over omega is natural.
00:30
Frequency or the frequency of the harmonic oscillator.
00:34
And so if we look at the spacing, right, the spacing delta e is going to be a subsequent energy level.
00:43
So en plus one minus the one below, e .n.
00:47
Right.
00:47
So if we look at this, then we're going to have n plus one plus a half, h bar omega minus n plus a half h bar omega.
01:02
And we find that this is equal to, it's exactly equal to h bar omega.
01:08
And so this is the spacing between sub -sigma levels.
01:10
It's a constant.
01:12
So indeed, this is a true statement.
01:16
Okay? so yes, the quantum harmony coccalator, the energy levels are evenly spaced.
01:21
Now, the same cannot be set for part b, where part b states that in an infinite, well and an infinite bell, the energy levels are evenly spaced.
01:31
Right, so if we now look at delta n, or sorry, delta e, so again, you know, remember that is the, it has the same definition as up here.
01:40
So it's the subsequent level minus the minus the actual level.
01:44
And in this case, what we're going to get is it's hbri squared, pi squared over 2m a squared, right? these are just factors that are constant, so they factorize from both parts.
01:56
But we're going to have n minus 1 squared minus n squared, right? and that's decidedly not going to be equal to constant.
02:05
This is going to be equal to n squared minus 2n plus 1 minus n squared.
02:14
So it's equal to 1 minus 2n.
02:19
So we can see that there is no even spacing because this function changes as a function of time, as a function of n, right? so with n changes, the energy levels get more and more space down.
02:34
Okay, then, so this is a false statement.
02:38
Second statement is false.
02:40
So we're done with that.
02:42
Part c says that the minimum classical allowed energy for a harmonic oscillator is equal to zero.
02:51
And this is, of course, true.
02:52
So if you have no kinetic energy, then what you're doing is you're basically sitting at the bottom of the well, not doing anything.
03:00
And if you have some finite non -zero energy, then you're kind of have this amount of energy so you can move between these classically allowed points in the potential.
03:11
And so you can imagine kind of a ball rolling up and down in this bowl of a potential...