DifferentialEquationsLecture2

Add lecture notes from the second lecture on differential equations.

src/8_differential_equations_2.md

@@ -240,13 +240,13 @@ is a $3$-rd order equation because of the third derivative with respect to x
 in the equation.
 
 To begin, we demonstrate that PDE's are of fundamental importance in physics, 
-especially in quantum physics. In particular, the Schr\"{o}dinger equation, 
+especially in quantum physics. In particular, the Schrödinger equation, 
 which is of central importance in quantum physics is a partial differential 
 equation with respect to time and space. This equation is very important 
 because it describes the evolution in time and space of the entire description
 of a quantum system $\psi(x,t)$, which is known as the wavefunction. 
 
-For a free particle in one dimension, the Schr\"{o}dinger equation is 
+For a free particle in one dimension, the Schrödinger equation is 
 
 $$i \hbar \frac{\partial \psi(x,t)}{\partial t} = - \frac{\hbar^2}{2m} \frac{\partial^2 \psi(x,t)}{\partial x^2}. $$
 
@@ -259,7 +259,7 @@ the equation. In partial differential equations at least one such constant will
 arise from the time derivative and likewise at least one from the spatial 
 derivative. 
 
-For the Schr\"{o}dinger equation, we could supply the initial conditions
+For the Schrödinger equation, we could supply the initial conditions
 
 $$\psi(x,0)= \psi_{0}(x) \ & \ \psi(0,t) = \psi{t, L} = 0.$$
 
@@ -287,7 +287,7 @@ physics.
 
 ## Separation of variables ##
 
-Let us focus on the one dimensional Schr\"{o}dinger equation of a free particle
+Let us focus on the one dimensional Schrödinger equation of a free particle
 
 $$i \hbar \frac{\partial \psi(x,t)}{\partial t} = - \frac{\hbar^2}{2m} \frac{\partial^2 \psi(x,t)}{\partial x^2}. $$
 
@@ -336,7 +336,7 @@ needed to introduce a separation constant, which remains to be determined.
 
 ### Boundary and eigenvalue problems ###
 
-Continuing on with the Schr\"{o}dinger equation example from the previous 
+Continuing on with the Schrödinger equation example from the previous 
 section, let us focus on 
 
 $$-\frac{\hbar^2}{2m} \phi''(x) = \lambda \phi(x),$$
@@ -477,7 +477,7 @@ the eigenfunctions of $L$.
     
     In terms of hermitian operators and their eigenfunctions, the eigenfunctions
     play the role of the orthonormal basis. In reference to our running example,
-    the 1D Schr\"{o}dinger equation of a free particle, the eigenfunctions 
+    the 1D Schrödinger equation of a free particle, the eigenfunctions 
     $sin(\frac{n \pi x}{L})$ play the role of the basis functions $\ket{u_n}$.
     
 To close our running example, consider the initial condition