Due Monday, Sep 16, 2019

Problems 2-8, 2-11, 3-11, 3-38

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1. For a harmonic oscillator, the partition function is $q = x^{1/2}/(1-x)$ where $x=\exp(-\hbar \omega /k_B T)$. Show that the fractional population in the ground vibrational state is given by $$f_0=1-\exp(-\hbar \omega /k_B T)$$ (2 pts)
2. In class and on homework 1 we considered a non-degenerate two-level system: with energies $-\varepsilon$ and $+\varepsilon$. The partition function was $$q = \sum_{j=1}^2 \exp(-E_j\beta)= \exp(+\varepsilon \beta)+ \exp(-\varepsilon \beta) = 1/x + x$$ where $x=\exp(-\varepsilon \beta)$ The average energy is $$\langle E \rangle = -\varepsilon \frac{1-x^2}{1+x^2}$$
   1. At low temperature, $\beta$ gets large, so $\lim_{T\to 0} x = 0$. The average energy can be expanded in the following power series in $x$ $$\lim_{T\to 0}\langle E \rangle = -\varepsilon (1 - 2x^2 + ...)$$ Here we retain only terms up through order $x^2$ (neglect terms of order $x^3$ or higher). Derive this expression (3 pts)

   2. The specific heat is defined as $$C_{\rm v}= \partial \langle E \rangle / \partial T$$ From physical considerations what will be the specific heat
      1. at low T? (1 pt)

      2. at high T? (1 pt)

      3. Then sketch qualitatively $C_v(T)$ as a function of $T$. (2 pts)

         Hint, you can answer the last three questions without hardly any math

3. The Lennard-Jones 6,12 potential is $$V(r)= 4\varepsilon \left [ \left(\frac{r_o}{r}\right)^{12} - \left(\frac{r_o}{r}\right)^6\right ]$$ where $r_o$ is a constant. Determine:
   1. The value of the hard sphere radius $\sigma$ in units of $r_o$ (2 pts)
   2. The location of the minimum in the potential in units of $r_o$ (2 pts)
   3. The depth of the potential in units of $\varepsilon$. (2pts)
4. The velocities (or, more properly, the speeds) of the atoms (or molecules) in a gas correspond to a continuous distribution, called the Maxwell-Boltzmann speed distribution $$f(s) = C s^2 \exp(-E_k/kT) =C s^2 \exp(-m s^2/2k_B T)$$
   where s is the speed and m is the mass of the atom (or molecule). The argument of the exponential is the ratio of the kinetic energy of the gas $E_K=\frac{1}{2}m s^2$ to $k_B T$.

   The value of C is determined by requiring that the distribution be normalized? Normalization requires that $\int_0^\infty f(s) ds = 1$. Note that the integration only goes from 0 to ∞ since the speed (the magnitude of the velocity) can only be positive.

   1. Show that (3 pts) $$C= \sqrt{\left (\frac{m}{ k_B T}\right )^{3} \frac{2}{\pi}}$$

   2. The average speed of the gas atoms (molecules) $\bar s$ is an important quantity. One way you could estimate this quantity is $$\bar s = \langle s \rangle\equiv \int_0^\infty f(s) s ds$$

      Obtain an expression (2 pts) for $\bar s$ as a function of m, k_B, and T. Hint: Make sure your answers are dimensionally correct; the units of $\bar s$ should be length/time.