Introduction: Particles That Don't Get Trapped
Scattering states have \(E > 0\)—enough energy to escape to infinity. A particle incident on a potential may be transmitted or reflected. Quantum mechanics predicts the probabilities.
Scattering Setup
Incoming wave from left: \(\psi_\text{inc} = Ae^{ikx}\)
Reflected wave: \(\psi_\text{ref} = Be^{-ikx}\)
Transmitted wave: \(\psi_\text{trans} = Ce^{ikx}\)
where \(k = \sqrt{2mE}/\hbar\)
Coefficients
Reflection coefficient: \(R = |B/A|^2\)
Transmission coefficient: \(T = |C/A|^2\)
Conservation: \(R + T = 1\)
Worked Example: Step Potential
For step \(V = V_0\) for \(x > 0\) with \(E > V_0\):
\[T = \frac{4k_1k_2}{(k_1 + k_2)^2}\]where \(k_1 = \sqrt{2mE}/\hbar\) and \(k_2 = \sqrt{2m(E-V_0)}/\hbar\)
Even when classically the particle would pass, there's some reflection!
The Quantum Connection
Quantum reflection (even when \(E > V\)) has no classical analog. It's caused by the wave nature of particles—waves partially reflect at interfaces. This phenomenon is used in neutron mirrors and cold atom experiments.