How to Calculate the Total Number of Particles in Different Energy States?

[NewtonApple]

Homework Statement

A system of N particles has three possible energy levels namely; 0, E and 4E. How many particles does one expect in the second state at temperature T?

Homework Equations

It's a sample problem for our finals. Our Text book is Statistical Mechanics by Roger Bowley and Mariana Sanchez.

The Attempt at a Solution

Three Energy levels

E_{1}=0, E_{2}=E, E_{3}=4E

Let us first fill the E_{1} state with 3 particle.

N distinguishable ways of selecting the first particle
N-1 different ways to select second particle
N-2 different ways to select third particle

So the total number of ways to place first three particles in state E_{1} is

N(N-1)(N-2)=\frac{N!}{(N-3)!}​

Generally for n_{1} particles placed in E_{1} is,

\frac{N!}{n_{1}!(N-1)!}
​

for E_{2} state,

\frac{(N-n_{1})!}{n_{2}!(N-n_{1}n_{2})!}​

for E_{3} state,

\frac{(N-n_{1}n_{2})!}{n_{3}!(N-n_{1}n_{2}n_{3})!}​

Total number of particles in all three state will be

P=\frac{N!}{n_{1}!n_{2}!n_{3}!}​

Substituting values

P=\frac{N!}{0!1!4!}​

Am I on right track?

 

[DrClaude]

Am I on right track?

I don't think so. You haven't even invoked temperature in any way.

If you had only one particle, what would be the probability of finding in in state 2 when the temperature is T?

 

[NewtonApple]

ok, I try to re attempt it after going through chapter 6 of the book.