A journey through finite potential wells using the tight binding method
In my exploration of quantum mechanics, I have focused on the intricacies of coupled finite potential wells through the lens of the tight binding method. This semi-empirical approach has been instrumental in understanding the behavior of electrons in condensed matter systems, especially those exhibiting strong localization.
Theoretical background
The tight binding method simplifies the description of electron dynamics by considering them to be localized around their respective atoms. The core of this method is the assumption that the wavefunctions can be approximated by a linear combination of atomic orbitals. In the case of coupled finite potential wells, the Hamiltonian for the system can be described by:
\mathbf H = -\frac{\hbar^2}{2m} \frac{\mathrm d^2}{\mathrm dx^2} + V_L + V_R
Where V_L and V_R are the potentials in the left and right wells, respectively. The coupling between the wells introduces off-diagonal elements in the Hamiltonian matrix, leading to the mixed terms in the total wavefunction.
Practical implementation
The practical implementation of this theory involves calculating the matrix elements of the Hamiltonian:
H_{11} = \int_{-\infty}^\infty \bar \psi_L \left(-\frac{\hbar^2}{2m} \frac{\mathrm d^2}{\mathrm dx^2} + V_L\right)\psi_L \mathrm dx
The integration over space considers only the significant contributions, effectively assuming that wavefunctions in opposing wells do not overlap:
H_{12} = \int_{\text{barrier}} \bar \psi_L \left(-\frac{\hbar^2}{2m} \frac{\mathrm d^2}{\mathrm dx^2}\right)\psi_R \mathrm dx = \Delta E
This results in a simplified model where the electron states are primarily localized but slightly perturbed by the neighboring well, represented by \Delta E.
Energy eigenvalues and wavefunctions
The solution to the Hamiltonian matrix equation:
\begin{bmatrix} E_1 & \Delta E \\ \Delta \bar E & E_1 \end{bmatrix} \begin{bmatrix} \alpha \\ \beta \end{bmatrix} = E\begin{bmatrix} \alpha \\ \beta \end{bmatrix}
yields the energy eigenvalues:
E = E_1 \pm |\Delta E|
The corresponding eigenstates are particularly insightful. They are linear combinations of the individual well states, indicating how quantum mechanical effects manifest in coupled systems. These states are visualized through wavefunctions, which can be plotted using a Python script to show their distribution in space.
In conclusion, my journey through the application of the tight binding method to finite potential wells has provided a rich understanding of electron coupling in quantum systems. This approach, while simplified, captures the essential physics and offers substantial predictive power in the design and analysis of nanoscale electronic devices.
For more insights into this topic, you can find the details here.