Question

1. two-level systems Consider a thermodynamic system consisting of N two-level systems with (non-degenerate) energy levels as shown: $E_1 = \epsilon$ $E_0 = 0$ Energy levels of a quantum two-level system. (a) Show that the partition function of one two-level sytem is $Z = 1 + e^{-\beta \epsilon}$, where $\beta = 1/kT$. (b) Show that that the probability of finding the system in the ground state ($P(E_0)$) and the excited state ($P(E_1)$) are: $P(E_0) = \frac{1}{1 + e^{-\beta \epsilon}}$ and $P(E_1) = \frac{e^{-\beta \epsilon}}{1 + e^{-\beta \epsilon}}$, respectively. (c) Show that the average energy per two-level system, $E$, is: $\bar{E} = \frac{\epsilon e^{-\beta \epsilon}}{1 + e^{-\beta \epsilon}}$ (d) Show that the result in part (c) can be rewritten as: $\bar{E} = \frac{\epsilon e^{-\beta \epsilon/2}}{2 \cosh(\beta \epsilon/2)}$, as we derived for the paramagnetic two-level system, in lecture 20. (e) Show that the heat capacity of N such two level systems is, $C(T) = N \frac{\partial \bar{E}}{\partial T}$, is $C(T) = N \frac{\epsilon^2}{k T^2} \frac{e^{-\beta \epsilon}}{(1 + e^{-\beta \epsilon})^2}$ (Hint: If you find taking derivatives of functions of $\beta$ with respect to T slightly messy, T10 will show you how to do it in a tidy fashion.) (f) A solid contains one mole of two-level systems with a separation of $\epsilon = k \times (1 K)$. Plot $C(T)$, in Joules per mole-Kelvin, as a function of T from 0 to 5 K. You can use a computer, or plot by hand. Indicate approximately the value of $kT/\epsilon$ at which C(T) has its maximum. (g) Explain why the specific heat curve is zero in the limit as $T \to 0 K$, why it then has a peak, and why it falls to zero again as $T \to \infty$.

          1. two-level systems
Consider a thermodynamic system consisting of N two-level systems with (non-degenerate)
energy levels as shown:
$E_1 = \epsilon$
$E_0 = 0$
Energy levels of a quantum two-level system.
(a) Show that the partition function of one two-level sytem is
$Z = 1 + e^{-\beta \epsilon}$,
where $\beta = 1/kT$.
(b) Show that that the probability of finding the system in the ground state ($P(E_0)$) and the
excited state ($P(E_1)$) are:
$P(E_0) = \frac{1}{1 + e^{-\beta \epsilon}}$ and $P(E_1) = \frac{e^{-\beta \epsilon}}{1 + e^{-\beta \epsilon}}$, respectively.
(c) Show that the average energy per two-level system, $E$, is:
$\bar{E} = \frac{\epsilon e^{-\beta \epsilon}}{1 + e^{-\beta \epsilon}}$
(d) Show that the result in part (c) can be rewritten as:
$\bar{E} = \frac{\epsilon e^{-\beta \epsilon/2}}{2 \cosh(\beta \epsilon/2)}$,
as we derived for the paramagnetic two-level system, in lecture 20.
(e) Show that the heat capacity of N such two level systems is, $C(T) = N \frac{\partial \bar{E}}{\partial T}$, is
$C(T) = N \frac{\epsilon^2}{k T^2} \frac{e^{-\beta \epsilon}}{(1 + e^{-\beta \epsilon})^2}$
(Hint: If you find taking derivatives of functions of $\beta$ with respect to T slightly messy,
T10 will show you how to do it in a tidy fashion.)
(f) A solid contains one mole of two-level systems with a separation of $\epsilon = k \times (1 K)$. Plot
$C(T)$, in Joules per mole-Kelvin, as a function of T from 0 to 5 K.
You can use a computer, or plot by hand. Indicate approximately the value of $kT/\epsilon$ at
which C(T) has its maximum.
(g) Explain why the specific heat curve is zero in the limit as $T \to 0 K$, why it then has a
peak, and why it falls to zero again as $T \to \infty$.

1. two-level systems
Consider a thermodynamic system consisting of N two-level systems with (non-degenerate)
energy levels as shown:
E1 = ϵ
E0 = 0
Energy levels of a quantum two-level system.
(a) Show that the partition function of one two-level sytem is
Z = 1 + e^-βϵ,
where β = 1/kT.
(b) Show that that the probability of finding the system in the ground state (P(E0)) and the
excited state (P(E1)) are:
P(E0) = (1)/(1 + e^-βϵ) and P(E1) = (e^-βϵ)/(1 + e^-βϵ), respectively.
(c) Show that the average energy per two-level system, E, is:
E̅ = (ϵ e^-βϵ)/(1 + e^-βϵ)
(d) Show that the result in part (c) can be rewritten as:
E̅ = (ϵ e^-βϵ/2)/(2 cosh(βϵ/2)),
as we derived for the paramagnetic two-level system, in lecture 20.
(e) Show that the heat capacity of N such two level systems is, C(T) = N (∂E̅)/(∂ T), is
C(T) = N (ϵ^2)/(k T^2)(e^-βϵ)/((1 + e^-βϵ)^2)
(Hint: If you find taking derivatives of functions of β with respect to T slightly messy,
T10 will show you how to do it in a tidy fashion.)
(f) A solid contains one mole of two-level systems with a separation of ϵ = k × (1 K). Plot
C(T), in Joules per mole-Kelvin, as a function of T from 0 to 5 K.
You can use a computer, or plot by hand. Indicate approximately the value of kT/ϵ at
which C(T) has its maximum.
(g) Explain why the specific heat curve is zero in the limit as T → 0 K, why it then has a
peak, and why it falls to zero again as T →∞.

Added by Daniel P.

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