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Statistical Mechanics

  • #1

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

[itex]E_{1}=0[/itex], [itex]E_{2}=E[/itex], [itex]E_{3}=4E[/itex]

Let us first fill the [itex]E_{1}[/itex] 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 [itex]E_{1}[/itex] is

[itex]N(N-1)(N-2)=\frac{N!}{(N-3)!}[/itex]​

Generally for [itex]n_{1}[/itex] particles placed in [itex]E_{1}[/itex] is,
[itex]\frac{N!}{n_{1}!(N-1)!}[/itex]

for [itex]E_{2}[/itex] state,

[itex]\frac{(N-n_{1})!}{n_{2}!(N-n_{1}n_{2})!}[/itex]​

for [itex]E_{3}[/itex] state,

[itex]\frac{(N-n_{1}n_{2})!}{n_{3}!(N-n_{1}n_{2}n_{3})!}[/itex]​

Total number of particles in all three state will be

[itex]P=\frac{N!}{n_{1}!n_{2}!n_{3}!}[/itex]​

Substituting values

[itex]P=\frac{N!}{0!1!4!}[/itex]​


Am I on right track?
 

Answers and Replies

  • #2
DrClaude
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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?
 
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  • #3
ok, I try to re attempt it after going through chapter 6 of the book.
 

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