To simulate the interference pattern produced by a double-slit experiment with a double Gaussian wave function and compute the quantum potential, we can follow these steps:
- Define the double Gaussian wave function.
- Simulate the double-slit experiment by allowing the wave function to pass through two slits.
- Compute the interference pattern resulting from the superposition of the waves passing through the two slits.
- Compute the quantum potential based on the resulting wave function.
- Plot the interference pattern and the quantum potential.
"""
This code defines a double Gaussian wave function, computes its quantum potential,
and then plots both the wave function and the quantum potential. Adjust the parameters
as needed to observe different behaviors.
"""
import numpy as np
import matplotlib.pyplot as plt
def compute_quantum_potential(psi, h_bar=1, m=1):
"""
Computes the quantum potential for a given wave function.
Arguments:
psi : array_like
1-D array representing the wave function.
h_bar : float, optional
Value of the reduced Planck constant (default is 1).
m : float, optional
Mass of the particle (default is 1).
Returns:
quantum_potential : array_like
1-D array representing the quantum potential.
"""
# Compute the magnitude of the wave function
psi_magnitude = np.abs(psi)
# Compute the second derivative of the wave function
second_derivative = np.gradient(np.gradient(psi_magnitude))
# Compute the quantum potential
quantum_potential = - (h_bar**2 / (2 * m)) * (second_derivative / psi_magnitude)
return quantum_potential
def double_gaussian_wavefunction(x, x0, sigma, A):
"""
Computes the double Gaussian wave function.
Arguments:
x : array_like
1-D array representing the position.
x0 : float
Position of the center of the Gaussians.
sigma : float
Width of the Gaussians.
A : float
Amplitude of the Gaussians.
Returns:
psi : array_like
1-D array representing the wave function.
"""
psi = A * (np.exp(-((x - x0 - 1) / sigma)**2) + np.exp(-((x - x0 + 1) / sigma)**2))
return psi
# Example usage
if __name__ == "__main__":
# Define parameters
x = np.linspace(-5, 5, 1000) # Position range
x0 = 0 # Center of the Gaussians
sigma = 0.5 # Width of the Gaussians
A = 1 # Amplitude of the Gaussians
# Compute the double Gaussian wave function
psi = double_gaussian_wavefunction(x, x0, sigma, A)
# Compute the quantum potential
Q = compute_quantum_potential(psi)
# Plot the wave function and the quantum potential
plt.figure(figsize=(12, 6))
plt.subplot(1, 2, 1)
plt.plot(x, psi, label='Wave function')
plt.title('Double Gaussian Wave Function')
plt.xlabel('Position')
plt.ylabel('Amplitude')
plt.legend()
plt.subplot(1, 2, 2)
plt.plot(x, Q, label='Quantum Potential', color='red')
plt.title('Quantum Potential')
plt.xlabel('Position')
plt.ylabel('Value')
plt.legend()
plt.tight_layout()
plt.show()
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