From 8f1a0ae9b6fe4a75b7ed3b2ed41876427719bbb4 Mon Sep 17 00:00:00 2001
From: "T. van der Sar" <t.vandersar@tudelft.nl>
Date: Tue, 5 Feb 2019 13:13:19 +0000
Subject: [PATCH] Update 7_tight_binding.md - polish2

---
 src/7_tight_binding.md | 21 ++++++++++-----------
 1 file changed, 10 insertions(+), 11 deletions(-)

diff --git a/src/7_tight_binding.md b/src/7_tight_binding.md
index 4a9ce023..a1f6132c 100644
--- a/src/7_tight_binding.md
+++ b/src/7_tight_binding.md
@@ -73,9 +73,9 @@ where $x_n=na$ and where we wrote the time-dependent solution of the Schrödinge
 
 As usual, periodicity quantizes k-space by requiring $e^{i k L} = 1$ $⇒$ $k = 2\pi p/L$, with $p\in \mathbb{Z}$.
 
-As such, $c_n/A = \exp(2 \pi n p a/L) = \exp(2 \pi n p/N)$, and we see that changing $p→p+N$ corresponds to exactly the same solution, and that we have $N$ different solutions in total.
+As such, $u_n/A = \exp(2 \pi n p a/L) = \exp(2 \pi n p/N)$, and we see that changing $p→p+N$ corresponds to exactly the same solution, and that we have $N$ different solutions in total.
 
-We see that solutions with $k$ differing by an integer multiple of $2\pi/a$ are identical by plotting them:
+We see that solutions with $k$-values differing by an integer multiple of $2\pi/a$ are identical by plotting them:
 
 ```python
 x = np.linspace(-.2, 2.8, 500)
@@ -96,23 +96,23 @@ ax.annotate(s='', xy=(.3, -1.1), xytext=(1.3, -1.1),
 ax.text(.3 + .5, -1.25, '$a$', ha='center');
 ```
 
-How many different solutions did we expect to find? We have a system with $N$ degrees of freedom (either $u_n$ or $c_n$), and therefore we need $N$ normal modes (or eigenstates).
+How many different solutions did we expect to find? We have a system with $N$ degrees of freedom (either $u_n$ or $c_n$), and therefore we expect $N$ normal modes (or eigenstates).
 
 Because we proposed an ansatz with $N$ different plane-wave solutions, if we find an energy or frequency for each solution, we have fully solved the problem!
 
-## Solution
+## Solving the equations of motion
 
 ### Phonons
 
 First substitute the Ansatz into the equations of motion:
-$$ -m \omega^2 c_0 e^{i\omega t - ikx} = \kappa c_0  e^{i\omega t}(-2 e^{-ikx} + e^{-ikx+ika}+ e^{-ikx-ika}),$$
-the exponents and $c_0$ drop out (just like we expected) and we get:
+$$ -m \omega^2 A e^{i\omega t - ikx} = \kappa A  e^{i\omega t}(-2 e^{-ikx} + e^{-ikx+ika}+ e^{-ikx-ika}),$$
+the exponents and $A$ drop out (just like we expected) and we get:
 $$ -m \omega^2 = \kappa (-2 + e^{ika}+ e^{-ika})=\kappa [-2 + 2\cos(ka)],$$
 or after a further simplification:
 $$\omega = \sqrt{\frac{2\kappa}{m}}\sqrt{1-\cos(ka)} = 2\sqrt{\frac{\kappa}{m}}|\sin(ka/2)|$$
 [here we used that $1-\cos(x)=2\sin^2(x/2)$.]
 
-So we arrive to the dispersion relation
+So we arrive at the dispersion relation
 
 ```python
 k = np.linspace(-2*pi, 6*pi, 500)
@@ -142,20 +142,20 @@ Before we had $\sum_p → \frac{L}{2\pi}\int_{-\omega_D/v_s}^{\omega_D/v_s}dk$,
 
 Now $\sum_p → \frac{L}{2\pi}\int_{-\pi/a}^{\pi/a}dk$, the integral is over a finite number of modes because there is only a finite number of modes.
 
-**Sound velocity:** at small $k$ $\sin(ka/2)\approx ka/2$, and therefore $\omega \approx \sqrt{\kappa/m} k a$, so we have derived the existence of sound waves.
+**Sound velocity:** at small $k$, $\sin(ka/2)\approx ka/2$, and therefore $\omega \approx \sqrt{\kappa/m} k a$, so we have derived the existence of sound waves.
 
 ### Electrons
 
 Substitute the Ansatz into the equations of motion:
 
 $$
-E c_0 e^{-ikx} = E_0 c_0 e^{-ikx} +t e^{-ikx-ika} + te^{-ikx+ika},
+E e^{-ikx} = E_0 e^{-ikx} +t e^{-ikx-ika} + te^{-ikx+ika},
 $$
 and after canceling the exponents we immediately get
 $$
 E = E_0 + 2t\cos(ka),
 $$
-so we come to the dispersion relation:
+so we arrive at the dispersion relation:
 
 ```python
 pyplot.figure()
@@ -166,7 +166,6 @@ pyplot.xticks([-pi, 0, pi], [r'$-\pi$', 0, r'$\pi$'])
 pyplot.yticks([-1, 0, 1], ['$E_0+2t$', '$E_0$', '$E_0-2t$']);
 ```
 
-
 Usually electron dispersion has multiple options for $E(k)$, each called a *band*. The complete dispersion relation is also called a *band structure*.
 
 The band bottom is at $k=0$ (when $t<0$). There the energy
-- 
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