Scattering

The scalar wave equation

The wave equation is commonly found in various areas of physics such as electromagnetics, electric transmission lines, acoustics, solid mechanics.

The scalar wave equation is established in linear isotropic medium as :

$$\begin{equation} \Big( \grad^2 -\frac{1}{c(\bb r,t)^2}\dP_t^2 \Big) u = 0 \end{equation}$$

where $u$ is a scalar physical quantity that propagates such as the pressure for acoustic waves, the electric potential in electromagnetics, the strain in solid mechanics. The quantity $c$ is the speed of the wave which depends on the medium.

This equation can be written in harmonic form using $u(\bb r,t)=u(\bb r)e^{j\omega t}$ (further assuming a homogeneous medium) and becomes known as the Helmholtz equation :

$$\begin{equation} \Big( \grad^2 + k^2 \Big) u = 0 \end{equation}$$ where $k=\omega/c(\omega)$ is known as the wave number. Using $\omega=2\pi/T$ and writing $k=2\pi/\lambda$ this relationship becomes $\lambda=cT$. This is commonly known as the dispersion relation of the wave which relates its period $T$ to its wavelength $\lambda$ through the speed of the wave.

In 1D the family of solutions to this equation are $e^{\pm jk z}$. As a result the solutions to the time dependent equation are $e^{j(\omega t\mp kz)}$ which indeed represent forward and backward propagating waves if $k$ is real.

Schroedinger's equation

The time dependent Schroedinger's equation reads : $$\begin{equation} j\hbar\dP_t\Psi(t,\bb r) = \Big(-\frac{\hbar^2}{2m}\grad^2 + V(t,\bb r)\Big)\Psi(t,\bb r) \end{equation}$$

where $V(t,\bb r)$ is the potential energy.

The time independent equation is obtained by assuming the time dependent part of the wave function $\Psi(t,\bb r)=\Psi(\bb r)e^{jE/\hbar t}$ where $E$ is the energy of the system.

Injecting above and further assuming time independent potential yields the following eigen value problem :

$$\begin{equation} \Big(-\frac{\hbar^2}{2m}\grad^2 + V(\bb r)\Big)\Psi(\bb r) = E\Psi(\bb r) \end{equation}$$

Assuming a constant potential energy $V$ and setting $k^2=2m/\hbar^2\left(E-V\right)$, the above equation becomes the Helmholtz equation.

In the case of electron-atom interaction, the charge of the nucleus is positive concentrated at the centre of the atom. The electron cloud is a distributed negative charge. The electrostatic potential $\varphi$ of the atom is therefore locally positive. The potential energy $V=-e\varphi$ of an incident electron due to the presence of an atom is negative since the Coulomb force is attractive.

Links

• Schroedinger's equation
• wave equation