Unveiling quantum tunneling: currents and transmission in rectangular barriers
Quantum tunneling remains a cornerstone topic in quantum mechanics, providing insights into phenomena that classical physics cannot adequately explain. In this blog post, I explore the quantum mechanical treatment of electron transmission through a rectangular potential barrier, a classic problem that offers profound insights into quantum behavior under constrained conditions.
Introduction to the problem
The setup involves electrons with energy E that is less than the potential height V_0 of a barrier. This scenario leads to wavefunction solutions that differ across three regions: before, within, and beyond the barrier. The wavefunctions are given by:
- On the left of the barrier (\psi_L): \psi_L = Ae^{ikx} + B e^{-ikx}
- Within the barrier (\psi_B): \psi_B = C e^{-\kappa x} + De^{\kappa x}
- On the right of the barrier (\psi_R): \psi_R = Fe^{ik_Rx}
Here, k and k_R represent the wave numbers outside and inside the barrier respectively, and \kappa is a decay constant that illustrates the exponential attenuation of the wavefunction within the barrier.
Quantum currents and their implications
A central aspect of my analysis focuses on the quantum mechanical current, derived from the probability current density:
\mathbf j_p = \frac{i\hbar}{2m}\left(\psi \nabla \bar \psi - \bar \psi \nabla \psi\right)
For the one-dimensional scenario pertinent to our problem, the currents on either side of the barrier become particularly interesting:
- Left side: j_p = \left(|A|^2 - |B|^2\right)\frac{\hbar k}{m}
- Right side: j_p = |F|^2\frac{\hbar k}{m}
These expressions reveal that the current density does not spatially vary on the left side, an intriguing result given the spatial variations in the probability density.
Transmission coefficients and potential variations
One of the key outcomes of this analysis is the calculation of the transmission coefficient \eta, which quantifies the fraction of the incident wave that transmits through the barrier:
\eta = \frac{|A|^2 - |B|^2}{|A|^2} = \frac{|F|^2}{|A|^2}
This coefficient holds critical importance in scenarios where the potentials on either side of the barrier are the same. However, if the potentials differ, the wave numbers (k_L and k_R) will not be the same, necessitating a modified relation:
\eta = \frac{k_R}{k_L}\frac{|F|^2}{|A|^2}
Conclusion and further exploration
Through my analysis, I’ve demonstrated the nuances and complexities involved in the quantum tunneling of electrons through potential barriers. This topic not only deepens our understanding of quantum mechanics but also enhances our ability to predict and manipulate quantum systems in technological applications.
For more insights into this topic, you can find the details here.