Introduction: Climbing a Step
The step potential is the simplest model of an interface. A particle comes in from a region of low potential and encounters a sudden jump. What happens depends on whether its energy exceeds the step height.
The Potential
\[V(x) = \begin{cases} 0 & x < 0 \\ V_0 & x > 0 \end{cases}\]Case 1: E > V₀ (Classical Transmission)
Both regions oscillatory:
\[k_1 = \sqrt{2mE}/\hbar, \quad k_2 = \sqrt{2m(E - V_0)}/\hbar\] \[R = \left(\frac{k_1 - k_2}{k_1 + k_2}\right)^2, \quad T = \frac{4k_1k_2}{(k_1 + k_2)^2}\]Surprisingly, \(R > 0\) even though classically the particle would pass!
Case 2: E < V₀ (Classical Reflection)
Evanescent wave in region 2: \(\psi = Ce^{-\kappa x}\) where \(\kappa = \sqrt{2m(V_0 - E)}/\hbar\)
\(R = 1\) (complete reflection), but \(\psi\) penetrates into the step!
The Quantum Connection
The step potential demonstrates that quantum particles can penetrate classically forbidden regions (evanescent waves) and can be reflected from classically allowed regions (impedance mismatch). Both effects arise from the wave nature of matter.