Schrodinger Equation

For the different possible states there is a wave function Ψ associate to each one. This wave function is governed by the Schrodinger Wave Equation. For a particle moving in 3 – D the equation is

(1)
\begin{align} ih\frac{d}{dt}\Psi (r,t) = -\frac{h^{2}}{2m}\triangledown^{2} \Psi + V(r,t)\Psi \end{align}

Where m is the mass of the particle and V(r,t) is the potential of the system (is the external force acting on the particle). Restricting the equation for a particle moving in 1 – D

(2)
\begin{align} ih\frac{d}{dt}\Psi (x,t) = -\frac{h^{2}}{2m}\frac{d^{2}}{dx} \Psi + V(x,t)\Psi \end{align}

To simplify, let assume V(x) be a static potential (time independent). Now with this assumption is possible to find a solution to the Schrodinger Wave equation using separation of variable. We are looking for a solution like

(3)
\begin{align} \Psi (x,t)= X(x) + T(t) \end{align}

X(x) is function of the position and T(t) of the time. Replacing on the SWE, we got

(4)
\begin{align} ihX\frac{d}{dt}T = -\frac{h^{2}}{2m}T\frac{d^{2}}{dx} X + V(x)XT \end{align}

Dividing by X / T

(5)
\begin{align} \frac{ih}{T}\frac{d}{dt}T = -\frac{h^{2}}{2m}\frac{1}{X}\frac{d^{2}}{dx} X + V(x) \end{align}

Now the left term side is a function of time and the right term side is a function of x. These mean that these two terms has to be equal to a constant E.

(6)
\begin{align} ih \frac{d}{dt}T = T E \end{align}

Where E is a constant
The solution to this equation is

(7)
\begin{equation} T(t) = e^{-iEt/h} \end{equation}

Setting the potential V(x) to zero, meaning that we have a free particle, the right side become

(8)
\begin{align} \frac{d^{2}}{dx} = -\frac{2mE}{h^{2}}X \end{align}

for more detail about potential well refer to [ http://electrons.wikidot.com/particle-in-a-box]

Introducing a new constant $p^2=2mE$ the equation become

(9)
\begin{align} \frac{d^{2}}{dx}= -\frac{p^{2}}{h^{2}}X \end{align}

The solution is

(10)
\begin{equation} X(x) = e^{ipx/h} \end{equation}

Combining both solution,

(11)
\begin{align} \Psi (x,t) = \AA e^{-\frac{i}{h}(Et-px)} \end{align}

This is the solution to the Schrodinger Equation for a free particle, where $E = p^{2}/2m$

The superposition of two solutions is also a solution of the SWE

(12)
\begin{align} \Psi(x,t) = \AA _{1} e^{-\frac{i}{h}(E_{1}t - p_{1}x)} + \AA _{2} e^{-\frac{i}{h}(E_{2}t - p_{2}x)} \end{align}

Generalizing

(13)
\begin{align} \Psi(x,t) = \int_{-\infty }^{+\infty }\AA (p) e^{-\frac{i}{h}(Et - px)} dp \end{align}

Different set of Amplitude will give us different set of Ψ. Corresponding to every physical observable quantity like position, momentum and energy, there is a Hermitian Operator associate to it.

An operator â act in a function to give another function, an example

(14)
\begin{align} \aa = \frac{d}{dx} \end{align}
(15)
\begin{align} \aa \sin (kx) = \frac{d}{dx}\sin (kx) = \cos (kx) \end{align}

If an operator â acts in a function Ψ, and this return the old function multiply by a number λ, this number is an Eigen value of the function and is an Ψ Eigenfunction.

(16)
\begin{align} \aa \Psi = \phi = \lambda \Psi \end{align}

For example if $\frac{d}{dx}$ is an operator and $e^{-ikx}$ is a function,

(17)
\begin{align} \frac{d}{dx}e^{-ikx} = ik e^{-ikx} \end{align}

Where ik is an Eigen value of the Eigen function $e^{-ikx}$.

Going back to Hermitian Operator
Hermitian Operator is an operator where the Eigens values are real.