Chapter 6: Q 6.21 (page 233)

In the real world, most oscillators are not perfectly harmonic. For a quantum oscillator, this means that the spacing between energy levels is not exactly uniform. The vibrational levels of an H₂ molecule, for example, are more accurately described by the approximate formula
$E_{n} \approx ϵ (1.03 n - 0.03 n^{2}), n = 0, 1, 2, \dots$
where $ϵ$ is the spacing between the two lowest levels. Thus, the levels get closer together with increasing energy. (This formula is reasonably accurate only up to about n = 15; for slightly higher n it would say that E_n decreases with increasing n. In fact, the molecule dissociates and there are no more discrete levels beyond n $\approx$ 15.) Use a computer to calculate the partition function, average energy, and heat capacity of a system with this set of energy levels. Include all levels through n = 15, but check to see how the results change when you include fewer levels Plot the heat capacity as a function of $k T / ϵ$ . Compare to the case of a perfectly harmonic oscillator with evenly spaced levels, and also to the vibrational portion of the graph in Figure 1.13.

Short Answer

Expert verified

Therefore,

$C = \sum_{n} \frac{N k {(β ϵ (1.03 n - 0.03 n^{2}))}^{2} e^{β ϵ (1.03 n - 0.03 n^{2})}}{{(e^{β ϵ (1.03 n - 0.03 n^{2})} - 1)}^{2}}$

Step by step solution

Given information

In the real world, most oscillators are not perfectly harmonic. For a quantum oscillator, this means that the spacing between energy levels is not exactly uniform. The vibrational levels of an H₂ molecule, for example, are more accurately described by the approximate formula

$E_{n} \approx ϵ (1.03 n - 0.03 n^{2}), n = 0, 1, 2, \dots$

where $ϵ$ is the spacing between the two lowest levels. Thus, the levels get closer together with increasing energy. (This formula is reasonably accurate only up to about n = 15; for slightly higher n it would say that E_n decreases with increasing n. In fact, the molecule dissociates and there are no more discrete levels beyond n $\approx$ 15.)

Explanation

In actual life, most oscillators are not perfect harmonic oscillators; the energy of real oscillators is provided by:

$E_{n} = ϵ (1.03 n - 0.03 n^{2})$

Where

$ϵ$ is the energy difference between the first two levels.

The partition function is:

$Z = \sum_{n} e^{- β E_{n}} (1)$

Substitute with energy

localid="1647451366905">Z=∑ne-βϵ1.03n-0.03n2(2)

Using python the summation is found with the code:

Explanation

The average energy is given as:

$\bar{E} = - \frac{1}{Z} \frac{\partial Z}{\partial β}$

Substitute from (1),

role="math" localid="1647451769495" $\bar{E} = - (1 - e^{- β E_{n}}) \frac{\partial}{\partial β} (\frac{1}{1 - e^{- β E_{n}}}) \bar{E} = (1 - e^{- β E_{n}}) \frac{E_{n} e^{- β E_{n}}}{{(1 - e^{- β E_{n}})}^{2}} \bar{E} = \frac{E_{n} e^{- β E_{n}}}{1 - e^{- β E_{n}}} \bar{E} = \frac{E_{n}}{e^{β E_{n}} - 1}$

Substituting the energy,

$\bar{E} = \sum_{n} \frac{ϵ (1.03 n - 0.03 n^{2})}{e^{β ϵ (1.03 n - 0.03 n^{2})} - 1}$

The partial derivative of total energy with respect to temperature equals the heat capacity, which is:

$C = \frac{\partial U}{\partial T} = \frac{\partial β}{\partial T} \frac{\partial U}{\partial β} C = \frac{\partial β}{\partial T} \frac{\partial}{\partial β} (\frac{N E_{n}}{e^{β E_{n}} - 1}) C = {(\frac{\partial T}{\partial β})}^{- 1} (- \frac{N {(E_{n})}^{2} e^{β E_{n}}}{{(e^{β E_{n}} - 1)}^{2}})$

But $β = 1 / k T \to T = 1 / k β$

${(\frac{\partial T}{\partial β})}^{- 1} = - k β^{2}$

Therefore,

$C = \frac{N k {(β E_{n})}^{2} e^{β E_{n}}}{{(e^{β E_{n}} - 1)}^{2}}$

Substitute the energy:

role="math" localid="1647452008304" $C = \sum_{n} \frac{N k {(β ϵ (1.03 n - 0.03 n^{2}))}^{2} e^{β ϵ (1.03 n - 0.03 n^{2})}}{{(e^{β ϵ (1.03 n - 0.03 n^{2})} - 1)}^{2}} \frac{C}{N k} = \sum_{n} \frac{{(β ϵ (1.03 n - 0.03 n^{2}))}^{2} e^{β ϵ (1.03 n - 0.03 n^{2})}}{{(e^{β ϵ (1.03 n - 0.03 n^{2})} - 1)}^{2}}$