Scattering, Reflection, and Transmission at a Step

The Physical Picture

Imagine a stream of particles moving from left to right towards a sudden change in potential energy. In the region where the potential is constant, the wavefunction behaves like a plane wave, representing a particle with a definite momentum. When the particle reaches the boundary where the potential changes, the wave does not simply stop or bounce back like a billiard ball. Instead, part of the wave is reflected back towards the source, and part is transmitted into the new region. The relative amplitudes of these reflected and transmitted waves are determined by the requirement that the total probability density and its flow must remain continuous across the boundary. This continuity is the quantum mechanical analogue of energy and momentum conservation, but it applies to the wave nature of the particle rather than its particle nature.

Wavefunctions in Each Region

Consider a potential step defined by \( V(x) = 0 \) for \( x < 0 \) and \( V(x) = V_0 \) for \( x > 0 \). We assume an incident wave of energy \( E \) coming from the left. In region I (\( x < 0 \)), the time-independent Schrödinger equation yields a solution consisting of an incident wave and a reflected wave. In region II (\( x > 0 \)), the solution is a transmitted wave. The general forms are:

Here, \( k_1 = \sqrt{2mE}/\hbar \) and \( k_2 = \sqrt{2m(E - V_0)}/\hbar \). Note that if \( E < V_0 \), \( k_2 \) becomes imaginary, indicating exponential decay rather than oscillation. The coefficients \( A \), \( B \), and \( C \) are determined by the boundary conditions at \( x = 0 \).

Applying Boundary Conditions

According to the continuity conditions derived in D-145, both the wavefunction and its first derivative must be continuous at \( x = 0 \). Applying \( \psi_{\mathrm{I}}(0) = \psi_{\mathrm{II}}(0) \) gives:

Applying \( \psi'_{\mathrm{I}}(0) = \psi'_{\mathrm{II}}(0) \) gives:

Solving this system of two linear equations for \( B \) and \( C \) in terms of \( A \) yields the reflection and transmission amplitudes. The ratio \( B/A \) represents the amplitude of the reflected wave relative to the incident wave, while \( C/A \) represents the transmission amplitude. These ratios depend only on the ratio of the wave numbers \( k_1 \) and \( k_2 \), which in turn depend on the energy \( E \) and the step height \( V_0 \).

Reflection and Transmission Coefficients

The physical interpretation of the coefficients requires the concept of probability current, defined in D-007 as \( j = \frac{\hbar}{m} \mathrm{Im}(\psi^* \frac{\partial \psi}{\partial x}) \). For a plane wave \( \psi = A e^{ikx} \), the current is \( j = \frac{\hbar k}{m} |A|^2 \). The reflection coefficient \( R \) is the ratio of the reflected current to the incident current, and the transmission coefficient \( T \) is the ratio of the transmitted current to the incident current.

Substituting the solutions for \( B \) and \( C \) from the boundary conditions, we find:

A crucial result is that \( R + T = 1 \). This expresses the conservation of probability: the total probability flux entering the step must equal the total probability flux leaving it, either by reflection or transmission. This is a direct consequence of the continuity equation \( \partial \rho / \partial t + \partial j / \partial x = 0 \).