00:01
Hello, in the question we have given that consider two identical particles which are to be placed in four single particle states.
00:09
So two of them are of zero energy.
00:13
One has energy epsilon and the last one has the energy 2 epsilon.
00:17
So we have to calculate the partition function if the particles are fermions and bosons.
00:23
So first we will start with the fermions.
00:25
So the number of ways i can arrange the fermions in different states is 6.
00:31
So i use this formula.
00:33
So this is the number of ways.
00:35
Four, there are four states and i have two particles to be arranged.
00:39
So this is the permutations.
00:41
So now, so it will be 4 factorial divided by 2 into 4 minus 2, which is 2 factorial.
00:50
And if we calculate this, we will get the answer as six ways.
00:53
So we don't need this actually, we can directly start filling.
00:58
So now let us begin with filling.
01:00
So now see fermions are so fermions, they obey the pauli's exclusion principle.
01:08
So i cannot put more than two particles in one single state.
01:11
Also, they have mentioned it is identical particle.
01:15
So i can consider that both of them have same spin.
01:18
So in one level, i can put only one particle.
01:22
So here i can put one and here i can put one.
01:26
So what is the total energy? zero, i can put 1 over here, 1 over here.
01:31
So the energy is 0 plus epsilon that is epsilon, i can put 1 over here 1 over here.
01:37
So the energy is this is 0 plus 2 epsilon.
01:40
So it is 2 epsilon.
01:42
Then i can put 1 over here, 1 over here 0 plus epsilon.
01:48
So it is epsilon, i can put one here, 1 here 0 plus 2 epsilon.
01:52
So it is 2 epsilon.
01:53
Then i can put 1 here and 1 here.
01:56
So it is 3 epsilon.
02:00
So my partition function will be now straight from here.
02:05
So it was e raised to minus beta times 0 plus e raised to minus beta epsilon plus this is 2 epsilon.
02:18
So it will be e raised to minus 2 beta epsilon plus here again i can see epsilon.
02:26
So e raised to minus beta epsilon, then here it is e raised to minus 2 beta epsilon.
02:35
And then finally, i will have e raised to minus 3 beta epsilon.
02:41
So from here, this partition function z will become so this is 1, e raised to minus e raised to 0 is 1 plus i can see here this occurs twice.
02:54
So it will be 2 e raised to minus beta epsilon plus so this is 2 e raised to minus 2 beta epsilon plus e raised to minus 3 beta epsilon.
03:09
Now this is my partition function if the particles were fermions.
03:15
I can also use this formula z is equal to gi e raised to minus beta ei where this g is the degeneracy.
03:24
And then what we will do is we will be solving this boson by using this formula so that we can have both the insides.
03:33
So now for the bosons now see bosons do not obey the pauli's exclusion principle.
03:41
So i can put as many particles as i can in a one single state.
03:46
So in this case in our question, i have two particles.
03:50
I can put two particles in one level.
03:54
So there is no restriction of putting the particles in any level.
03:59
So bosons for bosons, i can put two particles in one level.
04:04
So only this four more states will be added to the fermions...