Exploring quantum tunneling through finite potential barriers
Following my previous posts on the infinite tick barrier (here and here), I’ve turned my attention to the quantum mechanical phenomenon observed when particles encounter finite potential barriers. This scenario, significantly different from the classical case, highlights the nuanced behavior of quantum particles.
In the realm of quantum mechanics, the Schrödinger equation plays a pivotal role in describing the behavior of particles. When a particle encounters a finite potential barrier, the solution to the Schrödinger equation reveals several intriguing aspects, such as reflection, transmission, and, most notably, tunneling. The potential V(x) considered is as follows:
V(x) = \begin{cases} 0 & \quad x \le 0 \\ V_0 & \quad 0 < x < L \\ 0 & \quad x \ge L \end{cases}
For particles with energy less than the potential barrier (E < V_0), the tunneling effect becomes evident, allowing particles to pass through barriers they would not classically overcome. The analytical solution inside the barrier can be expressed as:
\psi_B(x) = F\,e^{\kappa x} + G\,e^{-\kappa x}, \quad \kappa = \sqrt{\frac{2m\left(V_0 - E\right)}{\hbar^2}}
When the energy is greater than the barrier height (E > V_0), the situation differs from classical expectations, emphasizing the quantum nature of particles:
\psi_R(x) = F e^{ik_2x} + G e^{-ik_2 x}, \quad k_2 = \sqrt{\frac{2m\left(E-V_0\right)}{\hbar^2}}
In my analysis, I’ve also explored the boundary conditions and continuity requirements at the interfaces of the barrier, which lead to a complex but solvable set of equations for the coefficients defining the particle’s wave function across different regions.
For more insights into this topic, you can find the details here