Simulation of constant electric fields
To continue my analysis of the potential well with an electric field as detailed in my previous post (here), in my latest exploration of quantum mechanics, I’ve ventured into the intricate domain of constant electric fields and their influence on quantum states within finite potential wells. This journey has not only piqued my curiosity but has also challenged my understanding of quantum physics, revealing fascinating shifts in energy levels and wavefunction behaviors that deviate from classical expectations.
The crux of my investigation centered around the impact of a constant electric field, denoted by f_0 = 3, on the energy levels within a potential well. Through rigorous mathematical analysis and numerical methods, I identified four distinct energy levels in the range [0, 20], which closely approximate the theoretical values:
\begin{aligned} & \eta_1 = 0.90419 \\ & \eta_2 = 4.02746 \\ & \eta_3 = 9.01726 \\ & \eta_4 = 16.0106 \end{aligned}
These findings, particularly the slight deviations from the theoretical energies, provided a rich ground for further exploration. To explore deeper, I calculated the ratios between constants from the following relationship:
Ai_0, _, Bi_0, _ = airy(alpha_0(eta))
ratio = -Bi_0 / Ai_0
This step was critical in understanding the normalization of the wavefunction in the presence of an electric field, leading to:
\phi(\alpha) = C_1\mathrm{Ai}\left[\alpha - \alpha_c\right] + C_2\mathrm{Bi}\left[\alpha - \alpha_c\right]
Armed with these ratios, my next challenge was to normalize the wavefunctions, which required integrating the square of the wavefunction over the domain and solving numerically for the constants C_1 and C_2. This is where the blend of analytical and numerical techniques truly shone, allowing for the precise determination of these constants:
norm_factor, error = quad(integrand, 0, 1, args=(ratio[i], 1, eta[i]))
c2_value = 1 / np.sqrt(norm_factor)
With the constants at hand, I was able to plot the wavefunctions and compare them with the unperturbed state, revealing the subtle yet significant influence of the electric field. These plots not only served as a visual confirmation of the theoretical predictions but also highlighted the shifts in the wavefunctions due to the electric field:
create_curve_2(ax, roots, c1, c2) # For wavefunction plots
create_curve_3(ax, roots, c1, c2, i) # For probability density plots
Throughout this investigation, the convergence of theoretical physics and computational methods has been evident. The ability to approximate and visualize quantum states under external influences like electric fields provides invaluable insights into the quantum realm, pushing the boundaries of our understanding.
In conclusion, this exploration into the effects of constant electric fields on quantum systems within finite potential wells has unveiled fascinating aspects of quantum behavior. The slight deviations in energy levels and the shifts in wavefunctions underscore the nuanced interplay between quantum states and external forces, offering a deeper understanding of the principles governing the quantum world.
For more insights into this topic, you can find the details here.