Overcoming the potential: quantum scattering in an infinite barrier
In a recent blog post here, I analyzed the scenario within quantum mechanics where a particle confronts an infinitely thick barrier with energy less than the potential, denoted as E < V_0. Today, I turn my attention to the contrasting condition where the energy of the particle surpasses the potential barrier, E > V_0. This case reveals the fascinating quantum mechanical phenomena of scattering when a wave encounters a potential step.
The core of my investigation lies in the Schrödinger equation for a particle in a region where E > V_0:
\frac{\mathrm d^2\psi(x)}{\mathrm d x^2} = -\frac{2m(E - V_0)}{\hbar^2}\psi(x)
Solving this equation, I discovered that the general solution on the right side of the barrier can be expressed as \psi_R(x) = F e^{ik_2x}, with G = 0. This result significantly differs from the E < V_0 case, where G \neq 0 and the solution involves a real exponential decay, indicating total reflection and no transmission through the barrier.
The reflection coefficient R and transmission coefficient T for the E > V_0 scenario are given by:
\begin{aligned} & R = \left(\frac{1-\frac{k_2}{k}}{1+\frac{k_2}{k}}\right)^2 \\ & T = \frac{4\frac{k_2}{k}}{\left(1+\frac{k_2}{k}\right)^2} \end{aligned}
These expressions encapsulate the essence of quantum scattering in this context, highlighting the partial reflection and transmission of the wave at the potential step. The absence of G in the solution for E > V_0 signifies that there is no wave being reflected back from the right towards the origin, a stark contrast to the E < V_0 condition where complete reflection is observed.
For more insights into this topic, you can find the details here.