Related questions with answers

Show that at high temperature, such that $k_{\mathrm{B}} T \gg \hbar \omega$ , the partition function of the simple harmonic oscillator is approximately $Z \approx(\beta \hbar \omega)^{-1}$ . Hence find U, C, F and S at high temperature. Repeat the problem for the high temperature limit of the rotational energy levels of the diatomic molecule for which $Z \approx\left(\beta \hbar^{2} / 2 I\right)^{-1}$ .

The internal levels of an isolated hydrogen atom are given by $E=-R / n^{2}$ where R = 13.6 eV. The degeneracy of each level is given by $2 n^{2}$ . a) Sketch the energy levels. b) Show that $Z=\sum_{n=1}^{\infty} 2 n^{2} \exp \left(\frac{R}{n^{2} k_{\mathrm{B}} T}\right)$ . Note that when $T \neq 0$ , this expression for Z diverges. This is because of the large degeneracy of the hydrogen atom’s highly excited states. If the hydrogen atom were to be confined in a box of finite size, this would cut off the highly excited states and Z would not then diverge. By approximating Z as follows: $Z \approx \sum_{n=1}^{2} 2 n^{2} \exp \left(\frac{R}{n^{2} k_{\mathrm{B}} T}\right)$ , i.e. by ignoring all but the n = 1 and n = 2 states, estimate the mean energy of a hydrogen atom at 300 K.

A friend says that finding differentials is as easy as finding derivatives—you just multiply the derivative by dx. Is your friend right?

Question

The energy E of a system of three independent harmonic oscillators is given by $E=\left(n_{x}+\frac{1}{2}\right) \hbar \omega+\left(n_{y}+\frac{1}{2}\right) \hbar \omega+\left(n_{z}+\frac{1}{2}\right) \hbar \omega$ . Show that the partition function Z is given by $Z=Z_{\mathrm{SHO}}^{3}$ , where ZSHO is the partition function of a simple harmonic oscillator given. Hence show that the Helmholtz function is given by $F=\frac{3}{2} \hbar \omega+3 k_{\mathrm{B}} T \ln \left(1-\mathrm{e}^{-\beta \hbar \omega}\right)$ , and that the heat capacity tends to $3 k_{\mathrm{B}}$ at high temperature.

Solution

Verified

Step 1

1 of 3

We have a system of three independent harmonic oscillators, the energy of the system is given by,

\begin{align} E=\left(n_{x}+\frac{1}{2}\right) \hbar \omega+\left(n_{y}+\frac{1}{2}\right) \hbar \omega+\left(n_{z}+\frac{1}{2}\right) \hbar \omega\end{align}

since the three SHOs are independent of each others, then the total partition function equals the multiplication of the partition functions of each one of the oscillators, that is,