# Statistical Mechanics Occupation number

• tanaygupta2000
In summary, the conversation discusses the concept of degeneracies in energy states, specifically for bosons. The speaker explains that bosons do not follow Pauli's Exclusion Principle and therefore, three bosons can be filled in two energy states. This results in 2 macrostates and 4 microstates. The probability of occurrence for each microstate is also calculated. The conversation also mentions the concept of macrostates and microstates and how they are defined. In the end, the problem is solved by using the BE distribution function.
tanaygupta2000
Homework Statement
Three bosons are to be filled in two energy states with degeneracies 3 and 4 respectively.
(1.) List all the macrostates.
(2.) How many microstates does this 3-particle system has?
(3.) Which macrostate is the most probable one?
Relevant Equations
Partition function, Z = ∑g(j)exp(-E(j)/kT)
Upto now I've only dealt with the problems regarding non - degenerate energy states.
Since bosons do not follow Pauli's Exclusion Principle, three bosons can be filled in two energy states (say E1 and E2) as:
 E1​ E2​ 1 boson​ 2 bosons​ 2 bosons​ 1 boson​ 3 bosons​ 0 bosons​ 0 bosons​ 3 bosons​

so that there are 2 macrostates (corresponding to levels E1 and E2) and 4 microstates (corresponding to 4 possibilities).
Also the probability of occurrence of
• I Possibility = 3!/2! = 3
• II Possibility = 3!/2! = 3
• III Possibility = 3!/3! = 1
• IV Possibility = 3!/0! = 6
Is this correct way of dealing with this problem?
I do not understand the meaning behind the given degeneracies of energy states.

tanaygupta2000 said:
so that there are 2 macrostates
Really? How do you define a macrostate?

Your calculation of microstates also appears to be wrong. So define that also.

DrClaude said:
Really? How do you define a macrostate?

Your calculation of microstates also appears to be wrong. So define that also.
My problem has been solved. I got correct macrostates (0,3), (1,2), (2,1), (3,0), and microstates corresponding to each one of them using BE distribution function.
Thank You !

DrClaude

## 1. What is the concept of occupation number in statistical mechanics?

The occupation number in statistical mechanics refers to the number of particles that occupy a specific energy level in a system. It is used to describe the distribution of particles in a system and is an important factor in determining the macroscopic properties of the system.

## 2. How is the occupation number calculated?

The occupation number is calculated by dividing the total number of particles in the system by the number of energy levels available. This gives the average number of particles occupying each energy level and can be used to determine the probability of finding a particle in a specific energy level.

## 3. What is the significance of the occupation number in statistical mechanics?

The occupation number is significant because it allows us to understand the behavior of a system at the microscopic level. By analyzing the distribution of particles in different energy levels, we can make predictions about the macroscopic properties of the system, such as temperature, pressure, and entropy.

## 4. How does the occupation number change with temperature?

The occupation number is directly related to temperature in statistical mechanics. As the temperature increases, the occupation number of higher energy levels also increases. This is because at higher temperatures, more energy is available for particles to occupy higher energy levels.

## 5. Can the occupation number be greater than 1?

Yes, the occupation number can be greater than 1. This means that there are multiple particles occupying a single energy level in the system. However, the total occupation number of all energy levels combined cannot exceed the total number of particles in the system.

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