Simulating electron beam diffraction and wavefunction collapse
Following my blog post on simulating the electron double-slit experiment (here and here), in the concluding part of my series on the electron double-slit experiment simulation, I focus on the final stage: simulating the electron beam diffraction and the resultant wavefunction collapse.
Normalizing the probability distribution
To ensure accuracy in my simulation, the first step involves normalizing the probability distribution derived from the earlier steps. This normalization ensures that each electron impacts the screen according to the correct probability.
# normalize to make the sum equal to 1
self.probability = [p / sum(doubleslit.screen_data_total)
for p in doubleslit.screen_data_total]
Simulating electron impact
With the probability distribution normalized, I then simulate the positions where electrons impact the screen. This involves generating random positions based on the probability distribution, reflecting the stochastic nature of quantum mechanics.
def compute(self):
self.electrons = []
self.electrons_height = []
self.electrons_count = np.zeros(len(self.probability))
for n in range(int(p.electrons_nb)):
bucket = np.random.choice(len(self.probability), p=self.probability)
self.electrons_count[bucket] += 1
self.electrons.append(bucket)
height = np.random.uniform(0, 1)
self.electrons_height.append(height)
This method effectively simulates the random nature of electron detection at the quantum level, with each electron assigned a random position within a specific screen region.
Visualizing the impact on the screen
To visualize the accumulated impacts of the electrons, I utilize two different plotting methods. The first is a scatter plot that shows each individual electron hit, reflecting the vertical distribution of impacts.
def init_plot(self):
dpi = 300 if cfg.high_res_plot else 100
figsize = (3840 / dpi, 2160 / dpi) if cfg.fig_4k else (1920 / dpi, 1080 / dpi)
self.fig, ax = plt.subplots(figsize=figsize, dpi=dpi)
ax.set_xlim(self.y_min, self.y_max)
ax.set_ylim(0, 1.01)
ax.set_yticklabels([])
ax.set_xticklabels([])
real_distances = self.y_min + np.array(self.crossed_ny) * self.dy
self.sorted_indices = np.argsort(real_distances)
self.sorted_distances = real_distances[self.sorted_indices]
self.curve1 = ax.scatter([], [], color='blue', s=8)
plt.tight_layout()
The second visualization is a line plot that accumulates the number of electrons hitting each region, offering a direct view of the interference pattern.
def init_plot2(self):
dpi = 300 if cfg.high_res_plot else 100
figsize = (3840 / dpi, 2160 / dpi) if cfg.fig_4k else (1920 / dpi, 1080 / dpi)
self.fig, ax = plt.subplots(figsize=figsize, dpi=dpi)
ax.set_xlim(self.y_min, self.y_max)
ax.set_ylim(0, max(self.electrons_count) * 1.01)
ax.set_yticklabels([])
ax.set_xticklabels([])
real_distances = self.y_min + np.array(self.crossed_ny) * self.dy
self.sorted_indices = np.argsort(real_distances)
self.sorted_distances = real_distances[self.sorted_indices]
self.curve2 = ax.plot(self.sorted_distances, np.zeros(len(self.sorted_distances)), color='blue', linewidth=3)[0]
plt.tight_layout()
These visualizations not only help confirm the correctness of the simulation but also beautifully illustrate the wave-particle duality of electrons through the observed interference patterns.
Conclusion
Through this detailed simulation of the electron double-slit experiment, I’ve demonstrated the fundamental principles of quantum mechanics and visualized the behavior of electrons which show wave-particle duality, manifesting through the interference pattern that emerges as electrons accumulate on the detection screen. This simulation provides a clear and intuitive representation of how quantum superposition and probability distributions govern electron behavior at the microscopic level.
For more insights into this topic, you can find the details here. If you’re interested in experimenting with the code and running your own simulations, you can access the full source on GitHub here.