- #1
teddd
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Hi everyone!
Here's my problem of the day:
Let's take a box containing 3 identical (but distinguishable) particles A B C. Let this be a canonical ensamble.
Suppose that A has energy [itex]\varepsilon_0[/itex] and both B and C have energy [itex]\varepsilon_1[/itex]. We thereforre have 2 energy level, [itex]n_0,n_1[/itex]. Take the number of states [itex]g_{\alpha}[/itex] in each energy level [itex]\varepsilon_{\alpha}[/itex] to be [itex]1[/itex].Now, I want to calculate in how many ways the set of population [itex]\vec{n}=(n_0,n_1)[/itex] can be realized.
At first sight I'd say that they're two: I can take [itex](A,BC)[/itex] or [itex](A,CB)[/itex], being the particle distinguishable.
But if I use the well-known Boltzmann forumula [tex]W(\vec{n})=N!\prod_{\alpha}\frac{ g_{\alpha}^{n_{\alpha}}}{n_{\alpha}}[/tex] and I put in the g's and n's I've taken above I get:[tex]W(\vec{n})=3! \left(\frac{1^1}{1!}\frac{1^2}{2!}\right)=3[/tex]so there should be three ways to set up the vector [itex]\vec{n}[/itex]!Where am I mistaking?? Thanks for help!
Here's my problem of the day:
Let's take a box containing 3 identical (but distinguishable) particles A B C. Let this be a canonical ensamble.
Suppose that A has energy [itex]\varepsilon_0[/itex] and both B and C have energy [itex]\varepsilon_1[/itex]. We thereforre have 2 energy level, [itex]n_0,n_1[/itex]. Take the number of states [itex]g_{\alpha}[/itex] in each energy level [itex]\varepsilon_{\alpha}[/itex] to be [itex]1[/itex].Now, I want to calculate in how many ways the set of population [itex]\vec{n}=(n_0,n_1)[/itex] can be realized.
At first sight I'd say that they're two: I can take [itex](A,BC)[/itex] or [itex](A,CB)[/itex], being the particle distinguishable.
But if I use the well-known Boltzmann forumula [tex]W(\vec{n})=N!\prod_{\alpha}\frac{ g_{\alpha}^{n_{\alpha}}}{n_{\alpha}}[/tex] and I put in the g's and n's I've taken above I get:[tex]W(\vec{n})=3! \left(\frac{1^1}{1!}\frac{1^2}{2!}\right)=3[/tex]so there should be three ways to set up the vector [itex]\vec{n}[/itex]!Where am I mistaking?? Thanks for help!
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