Statistical mechanics - N distinguishable particles

[fuselage]

Homework Statement

"A model system consists of N identical ”boxes” (e.g. quantum wells, atoms), each box with only two quantum levels, energies E0 = 0 and E1 = ε What is the number of microstates corresponding to the macrostate with total energy Mε?"

The Attempt at a Solution

I've done questions like this in the past with a small number of quantum wells, but then I've simply counted the number of possible arrangements. For example, if you have 3 quantum wells with one particle in each, and each well has 4 energy levels (including E₀=0), there are 10 different microstates if the macrostate energy is 3ε. But I can't find a general equation.

Does anyone know the general equation(s) that are used to solve a question like this?

 

[DrClaude]

Look up the binomial coefficient.

 

[fuselage]

Well aware of the binomial coefficients, but I still can't figure it out.

 

[DrClaude]

You're trying to figure out how to put M units of energy into N boxes.

 

[fuselage]

But that doesn't work.

Take the case I described in my attempt at a solution: If you simply count by hand, you will find 10 ways to get 3 units of energy out of a system of three particles each in its own quantum well with four energy levels (Ranging from E₀=0 to E₃=3). But nCr(3,3) ≠ 10, nor does nCr(4,3).

 

[Ray Vickson]

Homework Statement

"A model system consists of N identical ”boxes” (e.g. quantum wells, atoms), each box with only two quantum levels, energies E0 = 0 and E1 = ε What is the number of microstates corresponding to the macrostate with total energy Mε?"

The Attempt at a Solution

I've done questions like this in the past with a small number of quantum wells, but then I've simply counted the number of possible arrangements. For example, if you have 3 quantum wells with one particle in each, and each well has 4 energy levels (including E₀=0), there are 10 different microstates if the macrostate energy is 3ε. But I can't find a general equation.

Does anyone know the general equation(s) that are used to solve a question like this?

Think of tossing a coin N times On each toss, 'heads' ↔ energy level ε, 'tails' ↔ energy level 0. You want to know how many sequences of tosses have M heads and (N-M) tails.

Note: that is essentially "classical" counting. If, instead, your N have integer spins you need to use the so-called "Bose-Einstein" statistics, which gives a radically different result.

 

[DrClaude]

But that doesn't work.

Take the case I described in my attempt at a solution: If you simply count by hand, you will find 10 ways to get 3 units of energy out of a system of three particles each in its own quantum well with four energy levels (Ranging from E₀=0 to E₃=3). But nCr(3,3) ≠ 10, nor does nCr(4,3).

But that's not the case you have here. Each box can be in one of two states, which makes it much easier, and you can use nCr.

For the case where each system can be in more than one state, see http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/disbol.html

 

[fuselage]

But that's not the case you have here. Each box can be in one of two states, which makes it much easier, and you can use nCr.

For the case where each system can be in more than one state, see http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/disbol.html

You're right! Thank you :)