Partition sum of particle, high/low temperature limits

[SoggyBottoms]

Homework Statement

We have a single particle that can be in one of three different microstates, [itex]\epsilon_0[/itex], [itex]\epsilon_1[/itex] or [itex]\epsilon_2[/itex], with [itex]\epsilon_0 < \epsilon_1 < \epsilon_2[/itex]. The particle is in thermal equilibrium with a heat bath at temperature T.

1) Calculate the canonical partition function.

2) Give the probabilities [itex]P_j[/itex] that this particle is in state [itex]j[/itex] for [itex]j = 0, 1, 2[/itex].

3) What is the average energy in the high and low temperature limit?

The Attempt at a Solution

1) [tex]Z(\beta) = e^{- \beta \epsilon_0} + e^{- \beta \epsilon_1} + e^{- \beta \epsilon_2} [/tex]

2)

[tex]P(0) = \frac{e^{- \beta \epsilon_0}}{Z(\beta)}[/tex]

[tex]P(1) = \frac{e^{- \beta \epsilon_1}}{Z(\beta)}[/tex]

[tex]P(2) = \frac{e^{- \beta \epsilon_2}}{Z(\beta)}[/tex]

3)

[tex]
\langle E \rangle = \epsilon_0 \cdot P(0) + \epsilon_1 \cdot P(1) + \epsilon_2 \cdot P(2) \\
= \frac{\epsilon_0 e^{- \beta \epsilon_0} + \epsilon_1 e^{- \beta \epsilon_1} + \epsilon_2 e^{- \beta \epsilon_2}}{e^{- \beta \epsilon_0} + e^{- \beta \epsilon_1} + e^{- \beta \epsilon_2}}
[/tex]

So then I just plug in [itex]\beta = \frac{1}{k_B T}[/itex] and see what happens when T gets very low or very high, and then for high temperatures I get [itex]U = \frac{\epsilon_0 + \epsilon_1 + \epsilon_2}{3}[/itex] and for low temperatures it goes to 0. Is this correct?

Edit: I meant partition function in the title.

 

[anathema]

Thank you for your question. Your calculations for the canonical partition function and probabilities are correct. However, your calculation for the average energy in the high and low temperature limits is incorrect.

In the high temperature limit, the average energy would approach the average of all possible energies, which is given by:

\langle E \rangle = \frac{\epsilon_0 + \epsilon_1 + \epsilon_2}{3}

This is because at high temperatures, the particle has equal probabilities of being in any of the three states.

In the low temperature limit, the average energy would approach the lowest energy state, which is \epsilon_0. This is because at low temperatures, the particle would mostly be in the lowest energy state due to the Boltzmann factor, e^{- \beta \epsilon_0}, being the largest.

I hope this helps clarify your understanding. If you have any further questions, please don't hesitate to ask. Keep up the good work!