From be0e6efca115d9e5669884f98efe77c009f8022e Mon Sep 17 00:00:00 2001
From: Bowy La Riviere <b.m.lariviere@student.tudelft.nl>
Date: Thu, 3 Dec 2020 13:40:51 +0000
Subject: [PATCH] Added adjustments to the classical and quantum harmonic
 oscillator description

---
 src/1_einstein_model.md | 40 +++++++++++++++++++++-------------------
 1 file changed, 21 insertions(+), 19 deletions(-)

diff --git a/src/1_einstein_model.md b/src/1_einstein_model.md
index 2bf71e57..c992b475 100644
--- a/src/1_einstein_model.md
+++ b/src/1_einstein_model.md
@@ -32,12 +32,12 @@ py.init_notebook_mode(connected=True)
 
 _(based on chapter 2.1 of the book)_
 
-!!! success "Expected prior knowledge"
+!!! success "Expected prerequisites"
 
     Before the start of this lecture, you should be able to:
 
     - Write down the energy spectrum and partition function of a quantum harmonic oscillator
-    - Describe the equipartition theory
+    - Describe the equipartition theorem
     - Write down the Bose-Einstein distribution
 
 
@@ -45,9 +45,9 @@ _(based on chapter 2.1 of the book)_
 
     After this lecture you will be able to:
 
-    - Explain quantum mechanical effects on the heat capacity of solids (Einstein model)
-    - Compute the expected particle number, energy, and heat capacity of a quantum harmonic oscillator (a bosonic mode)
-    - Write down the total thermal energy of a material
+    - Explain the effect of the quantum harmonic oscillator on the heat capacity of solids (the Einstein model)
+    - Compute the expected occupation number, energy, and heat capacity of a quantum harmonic oscillator (a bosonic mode)
+    - Write down the total internal energy of an Einstein solid
 
 
 ## Classical limit of heat capacity
@@ -94,15 +94,12 @@ py.iplot(fig, show_link=False)
 
 An empirical observation, also known as the **law of Dulong–Petit** (1819):
 
-> In most materials heat capacity per atom $C \approx 3k_B$
-
-This corresponds to what we know from statistical physics and the equipartition theorem: every atom has 3 momenta and 3 coordinates.
-The equipartition theorem states that each of these 6 degrees of freedom contributes $k_B/2$ to the heat capacity.
+> In most materials the heat capacity per atom $C \approx 3k_B$
 
+This corresponds to what we know from statistical physics. Under the assumption that the atomic potential is parabolic, the equipartition theorem states that each degree of freedom contributes $k_B/2$ to the heat capacity. As we consider a 3D solid, each atom contains 3 spatial and 3 momentum degree of freedom. Therefore, the total heat capacity per atom is given by $C = 3k_B$.
 
 ### Complication
-
-Things start looking more complex when we study (following Einstein) the temperature dependence of the heat capacity of diamond[^2]:
+However, at low temperatures a discrepancy is observed between the heat capacity of diamond[^2] and the law of Dulong–Petit. This suggests that we need a different model to describe the heat capacity at low temperatures.
 
 ```python
 # Data from Einstein's paper
@@ -116,17 +113,20 @@ ax.set_ylabel('$C/k_B$')
 ax.set_ylim((0, 3));
 ```
 
-So we see that:
-
-* At high temperature the law of Dulong–Petit works
-* Strong temperature dependence of $C$
-* At low $T$, $C \rightarrow 0$
+We observe that:
 
+* We obtain the law of Dulong–Petit at high temperatures
+* $C$ depends strongly on the temperature
+* As $T \rightarrow 0$, $C \rightarrow 0$
 
-## Quantum oscillator
 
-This can be explained by considering each atom as a _quantum_ harmonic oscillator:
+## the Einstein model
+The equipartition theorem assumed that each atom can be modeled as a classic harmonic oscillator. However, at low temperatures this led to a discrepancy in the heat capacity between the law of Dulong-Petit and the observed heat capacity. To explain the behaviour of $C$ at low temperatures, Einstein treated each atom as a _quantum_ harmonic oscilator, instead of a classic harmonic oscillator. Einstein also assumed that each atom oscillates with the same frequency $\omega_0$. To summarize, the Einstein model of a solid is characterised by the following two assumptions:
+* The atoms are independent quantum harmonic oscillators
+* Each atom has the same frequency $\omega_0$
+For simplicity, we consider a 1D quantum harmonic oscillator (it turns out that the heat capacity in the 3D case is simply 3 times the heat capacity of the 1D case). 
 
+The energy levels and wavefunctions of a 1D quantum harmonic oscillator are shown below.
 ```python
 import math
 from numpy.polynomial.hermite import Hermite
@@ -222,7 +222,9 @@ ax.set_xlabel('X '+ r'($\sqrt{\hbar/m\omega_0}$)', fontsize=14)
 ax.set_ylabel('$E/\hbar\omega_0$', fontsize=14)
 ax.yaxis.set_label_coords(0.5,1)
 ```
-This oscillator has an energy spectrum given by
+
+
+The energy spectrum of a quantum harmonic oscillator is given by
 $$\varepsilon_n=\left(n+\frac{1}{2}\right)\hbar\omega_0$$
 where $\omega_0$ is the eigenfrequency of the oscillator.
 
-- 
GitLab