What is a single particle state?

[quietrain]

i am studying thermodynamics statistically, and my textbook says for the perfect quantal gas, i have

r single particle states, and n_r occupation numbers and E_r single particle energies.


what do they mean?

occupation numbers are the number of particles in a particular state right? say n₃ means 3 particles in the r=3 state? then with energies E₃ ?

it gets confusing :( thanks

 

[BruceW]

I think you do understand. there are n_r particles in the r state, and each r state corresponds to a single particle with energy E_r.

 

[quietrain]

is there some sort of analogy in laymans terms?

for example ,

in a box i have 10 balls

so what is a state? does it mean that in the state 1, i have 5 nike balls

so my occupation number is 5 for state 1.

so does it now mean that for state 2 , i can have all 10 balls be nike? so my occupation number is now 10?

 

[Demystifier]

I think that your problem is that you think of r as a number. But it is not really a number; it just a label, a name of something. For example, if you have three brothers, you may call them Jack, John and Jim; but you can also call them 1, 2 and 3, in the order of their birth. So r=1 means Jack, r=2 means John, and r=3 means Jim. Does it help?

 

[Demystifier]

say n₃ means 3 particles in the r=3 state?

No. For example, If n₃=100, then it means 100 particles in the r=3 state.

 

[BruceW]

The state of a single particle will be one of r possible single particle states.
The state of the system is determined by the values of each of the n_r and E_r.
In Layman's terms: in a given system, there will be a given number of balls of each brand.

 

[quietrain]

oh...

so if i have 10 balls in a system

the sum of occupation numbers must be 10?

so let's say got 3 states, meaning blue black and green balls.

so when i say state r = blue, my occupation number is the number of blue balls?

 

[BruceW]

Yes, you're right on both accounts.
Interestingly, some quantum systems don't require a fixed total number of particles (for example, photons can be annihilated), so in quantum mechanics, the sum of the occupation numbers is not necessarily a constant.

 

[quietrain]

ah i see thanks everyone!