00:01
All right, so we want to look at the debroy wavelength, which is defined as planks constant divided by the momentum, for thermal particles, like obeying like a maxwell boltzman distribution.
00:13
And so what we have from that is that the average energy per particle, that's probably like e over in, in a maxwell bolshell distribution, is three halves kt.
00:26
And this is basically the average kinetic energy of each particle.
00:32
So one half m average of v squared.
00:35
So what we have is the average of v squared is going to be 3kt over m.
00:43
And so the what we call the root means square velocity, just the square root of this is just the square root of 3kt over m.
00:51
So if we substitute this into our formula for the debore wavelength, which is planks constant divided by the mass times the velocity.
01:00
What we're going to get is basically h squared.
01:03
I'm going to bring the h inside the square root over 3mkt.
01:09
So that's what we wanted to show.
01:11
That's the first part.
01:13
Next we want to say that when the debroy wavelength lambda is much, much less than the inter -particle distance, then it's also, of course, going to be true that the cube of the wavelength, the debris wavelength is going to be much less than the cube of the inner particle distance.
01:31
And so if we take the reciprocal of this relationship, so like one over lambda cubed, the sign of the inequality sort of reverses, and it becomes one over lambda cubed is much, much greater than one over d cubed.
01:46
And we know that one over d cubed is approximately the number density of particles.
01:52
So what we have is the number density of particles is going to be.
01:57
Much, much less than one over this thermal wavelength cubed.
02:03
And so this is like 3mkt over h squared.
02:11
We're taking the square root of this and cubing it.
02:13
So that's so three halves.
02:14
And so this is basically what we wanted to show in the second part.
02:19
And in the last part, we have this thing called the quantum concentration, which is basically the square root of eight...