From 066c2c89b9f3124787c1fd368781d8aa3be5e39f Mon Sep 17 00:00:00 2001
From: "T. van der Sar" <t.vandersar@tudelft.nl>
Date: Mon, 10 Feb 2020 13:30:43 +0000
Subject: [PATCH] Update 1_einstein_model.md - polishing
---
src/1_einstein_model.md | 29 ++++++++++++++++++-----------
1 file changed, 18 insertions(+), 11 deletions(-)
diff --git a/src/1_einstein_model.md b/src/1_einstein_model.md
index c42406cf..be7b4fca 100644
--- a/src/1_einstein_model.md
+++ b/src/1_einstein_model.md
@@ -126,7 +126,7 @@ So we see that:
## Quantum oscillator
-This can be explained by considering a _quantum_ harmonic oscillator:
+This can be explained by considering each atom as a _quantum_ harmonic oscillator:
```python
import math
@@ -188,14 +188,14 @@ for i in range(no_states):
horizontalalignment='center', fontsize=14)
if i==0:
- ax.text(x[0]+1/4, h0_ener(i)/4, r'$\frac{\hbar\omega}{2}$',
+ ax.text(x[0]+1/4, h0_ener(i)/4, r'$\frac{\hbar\omega_0}{2}$',
horizontalalignment='center', fontsize=14)
ax.annotate("", xy=(x[0]+1/2, h0_ener(i)-1/2),
xytext=(x[0]+1/2, h0_ener(i)),
arrowprops=dict(arrowstyle="<->"))
elif i==1:
- ax.text(x[0]+1/4, h0_ener(i-1)+1/3, r'$\hbar\omega$',
+ ax.text(x[0]+1/4, h0_ener(i-1)+1/3, r'$\hbar\omega_0$',
horizontalalignment='center', fontsize=14)
ax.annotate("", xy=(x[0]+1/2, h0_ener(i)),
@@ -219,19 +219,26 @@ ax.set_yticklabels([])
ax.set_xticklabels([])
# Set x and y labels
-ax.set_xlabel('X '+ r'($\sqrt{\hbar/m\omega}$)', fontsize=12)
-ax.set_ylabel('E '+ r'($\hbar\omega$)', fontsize=12)
+ax.set_xlabel('X '+ r'($\sqrt{\hbar/m\omega_0}$)', fontsize=12)
+ax.set_ylabel('E '+ r'($\hbar\omega_0$)', fontsize=12)
ax.yaxis.set_label_coords(0.5,1)
```
+The quantum harmonic oscillator has an energy spectrum given by
+$$\varepsilon_n=\left(n+\frac{1}{2}\right)\hbar\omega_0$$
+where $\omega_0$ is the eigenfrequency of the oscillator.
-$$\varepsilon_n=\left(n+\frac{1}{2}\right)\hbar\omega$$
-
-Phonons are bosons $\Rightarrow$ they follow Bose-Einstein statistics.
+What is the thermal occupation and corresponding thermal energy of this oscillator? A quantum harmonic oscillator is a bosonic mode $\Rightarrow$ its occupation is described by Bose-Einstein statistics.
$$
-n(\omega,T)=\frac{1}{ {\rm e}^{\hbar\omega/k_{\rm B}T}-1}\Rightarrow\bar{\varepsilon}=\frac{1}{2}\hbar\omega+\frac{\hbar\omega}{ {\rm e}^{\hbar\omega/k_{\rm B}T}-1}
+n(\omega,T)=\frac{1}{ {\rm e}^{\hbar\omega/k_{\rm B}T}-1}
$$
+The Bose-Einstein distribution describes the occupation probability of a state at a given energy $\hbar \omega$. It is plotted as a function of energy below
+Using the Bose-Einstein function, we can calculate the the energy stored in the oscillator
+$$
+\bar{\varepsilon}=\frac{1}{2}\hbar\omega_0+\frac{\hbar\omega_0}{ {\rm e}^{\hbar\omega_0/k_{\rm B}T}-1}
+$$
+which we plot as a function of temperature below.
```python
fig, (ax, ax2) = pyplot.subplots(ncols=2, figsize=(10, 5))
omega = np.linspace(0.1, 2)
@@ -257,9 +264,9 @@ ax2.set_yticklabels([r'$\hbar\omega/2$'])
draw_classic_axes(ax2, xlabeloffset=.15)
```
-The term $\frac{1}{2}\hbar\omega$ is the _zero point energy_, which follows from the uncertainty principle.
+The term $\frac{1}{2}\hbar\omega_0$ is the _zero point energy_, which follows from the uncertainty principle.
-In order to calculate the heat capacity per atom $C$, we need to differentiate $\bar{\varepsilon}$ to $T$.
+To calculate the heat capacity per atom $C$, we need to differentiate $\bar{\varepsilon}$ to $T$.
$$
\begin{multline}
C = \frac{\partial\bar{\varepsilon}}{\partial T}
--
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