Introduction: Scattering from a Point
For the delta potential with positive or negative \(\alpha\), scattering states (\(E > 0\)) exhibit reflection and transmission. The calculation is clean and illuminating.
Setup
Incoming wave from left with energy \(E > 0\), wave number \(k = \sqrt{2mE}/\hbar\):
\[\psi(x) = \begin{cases} Ae^{ikx} + Be^{-ikx} & x < 0 \\ Ce^{ikx} & x > 0 \end{cases}\]Matching Conditions
Continuity: \(A + B = C\)
Derivative jump: \(ikC - ik(A - B) = -\frac{2m\alpha}{\hbar^2}C\)
Define \(\beta = m\alpha/\hbar^2 k\). Solving:
\[B = \frac{i\beta}{1 - i\beta}A, \quad C = \frac{1}{1 - i\beta}A\]Transmission and Reflection
\[T = |C/A|^2 = \frac{1}{1 + \beta^2} = \frac{1}{1 + m^2\alpha^2/\hbar^4k^2}\] \[R = |B/A|^2 = \frac{\beta^2}{1 + \beta^2}\]Note: \(R + T = 1\) ✓
The Quantum Connection
The delta barrier always partially reflects, no matter how much energy the particle has. At high energies (\(k \to \infty\)), \(T \to 1\). At low energies, \(T \to 0\). This is the opposite of the classically expected behavior where any energy should pass through an infinitely thin barrier.