DifferentialEquationsLecture2

Add lecture notes from the second lecture on differential equations.

src/differential_equations_2.md

@@ -264,5 +264,78 @@ For the Schr\"{o}dinger equation, we could supply the initial conditions
 $$\psi(x,0)= \psi_{0}(x) \ & \ \psi(0,t) = \psi{t, L} = 0.$$
 
 This particular set of boundary conditions corresponds to a particle in a box,
-a basic situation which comes up often in quantum physics. 
+a situation which is used as the base model for many derivations in quantum 
+physics. 
+
+Another example of a partial differential equation common in physics is the 
+Laplace equation
+
+$$\frac{\partial^2 \phi(x,y)}{\partial x^2}+\frac{\partial^2 \phi(x,y)}{\partial y^2}=0.$$
+
+In quantum physics Laplace's equation is important for the study of the hydrogen
+atom. In three dimensions and using spherical coordinates, the solutions to 
+Laplace's equation are special functions called spherical harmonics. In the 
+context of the hydrogen atom, these functions describe the wave function of the 
+system and a unique spherical harmonic function corresponds to each distinct set
+of quantum numbers.
+
+In the study of PDEs there is not a comprehensive overall treatment to the same 
+extent as there is for ODEs. There are several techniques which can be applied 
+to solving these equations, but the choice of technique must be tailored to the
+equation at hand. Hence we focus on some specific examples that are common in
+physics.
+
+## Separation of variables ##
+
+Let us focus on the one dimensional Schr\"{o}dinger equation of a free particle
+
+$$i \hbar \frac{\partial \psi(x,t)}{\partial t} = - \frac{\hbar^2}{2m} \frac{\partial^2 \psi(x,t)}{\partial x^2}. $$
+
+To attempt a solution, we will make a *separation ansatz*,
+
+$$\psi(x,t)=\phi(x) f(t).$$
+
+!!! info "Separation ansatz"
+    The separation ansatz is a restrictive ansatz, not a fully general one. In
+    general, for such a treatment to be valid an equation and the boundary 
+    conditions given with it have to fulfill certain properties. In this course
+    however you will only be asked to use this technique when it is suitable.
+    
+Substituting the separation ansatz into the PDE,
+
+$$i \hbar \frac{\partial \phi(x)f(t)}{\partial t} = - \frac{\hbar^2}{2m} \frac{\partial^2 \phi(x)f(t)}{\partial x^2} $$
+$$i \hbar \dot{f}(t) \phi(x) = - \frac{\hbar^2}{2m} \phi''(x)f(t). $$
+
+Notice that in the above equation the derivatives on $f$ and $\phi$ can each be
+written as ordinary derivatives, $\dot{f}=\frac{df(t)}{dt}$, 
+$\phi''(x)=\frac{d^2 \phi}{dx^2}$. This is so because each is only a function of 
+one variable. 
+
+Next, divide both sides of the equation through by $\psi(x,t)=\phi(x) f(t)$,
+
+$$i \hbar \frac{\dot{f}(t)}{f(t)} = - \frac{\hbar^2}{2m} \frac{\phi''(x)}{\phi(x)} = constant := \lambda. $$
+
+In the previous line we concluded that each part of the equation must be equal 
+to a constant, which we defined as $\lambda$. This follows because the left hand
+side of the equation only has a dependence on the spatial coordinate $x$, whereas 
+the right hand side only has dependence on the time coordinate $t$. If we have 
+two functions $a(x)$ and $b(t)$ such that 
+$a(x)=b(t) \ \forall x, \ t \ \epsilon \mathbb{R}$, then $a(x)=b(t)=const$.
+
+The constant we defined, $lambda$, is called a *separation constant*. With it, we 
+can break the spatial and time dependent parts of the equation into two separate
+equations,
+
+$$i \hbar \dot{f}(t) = \lambda f(t)$$
+
+$$-\frac{\hbar^2}{2m} \phi''(x) = \lambda \phi(x) .$$
+
+To summarize, this process has broken one partial differential equation into two
+ordinary differential equations of different variables. In order to do this, we 
+needed to introduce a separation constant, which remains to be determined.
+
+
+
+
+
     
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