Question

Consider a system with non-degenerate states with equidistant energy levels: $E_n = n\epsilon$, $n = 0,1,2...\infty$. Meaning, $E_0 = 0$, $E_1 = \epsilon$, $E_2 = 2\epsilon$, ... in equilibrium with a heat bath at temperature $T > 0$. (a) Determine an expression for the canonical partition function in terms of $\beta = \frac{1}{kT}$ and $\epsilon$. Use an expression for the sum over a geometric series. (b) Determine an expression for the probability $P_1$ to be in the first excited state $n = 1$. (c) Evaluate $P_1$ when: (I) $T = \frac{\epsilon}{k}$ (II) $T \to 0$ (III) $T \to \infty$ Show that in II and III the answers' first terms contain "T" so that one can see how $P_1$ depends on T in the limits of high and low temperature. (hint: answers should not only be 0,1, or $\infty$).

          Consider a system with non-degenerate states with equidistant energy levels: $E_n = n\epsilon$, $n = 0,1,2...\infty$. Meaning, $E_0 = 0$, $E_1 = \epsilon$, $E_2 = 2\epsilon$, ... in equilibrium with a heat bath at temperature $T > 0$.
(a) Determine an expression for the canonical partition function in terms of $\beta = \frac{1}{kT}$ and $\epsilon$. Use an expression for the sum over a geometric series.
(b) Determine an expression for the probability $P_1$ to be in the first excited state $n = 1$.
(c) Evaluate $P_1$ when:
(I) $T = \frac{\epsilon}{k}$
(II) $T \to 0$
(III) $T \to \infty$
Show that in II and III the answers' first terms contain "T" so that one can see how $P_1$ depends on T in the limits of high and low temperature. (hint: answers should not only be 0,1, or $\infty$).

Consider a system with non-degenerate states with equidistant energy levels: En = nϵ, n = 0,1,2...∞. Meaning, E0 = 0, E1 = ϵ, E2 = 2ϵ, ... in equilibrium with a heat bath at temperature T > 0.
(a) Determine an expression for the canonical partition function in terms of β = (1)/(kT) and ϵ. Use an expression for the sum over a geometric series.
(b) Determine an expression for the probability P1 to be in the first excited state n = 1.
(c) Evaluate P1 when:
(I) T = (ϵ)/(k)
(II) T → 0
(III) T →∞
Show that in II and III the answers' first terms contain "T" so that one can see how P1 depends on T in the limits of high and low temperature. (hint: answers should not only be 0,1, or ∞).

Added by Benjamin M.

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