TDSE Solution: Barrier Penetration and Tunneling for an Initially Gaussian Wavepacket - Computational Physics Formulation & Simulation

Program 1

Barrier Penetration and Tunneling for an Initially Gaussian Wavepacket

Python 3.11 Ready

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import trapezoid

# Parameters
L, x0, k, hbar, m, a, N, V0 = 40.0, 4.0, 5.0, 1, 1, 2.0,150, 14
xs, h = np.linspace(0, L, N, retstep=True)
us = np.exp(-(xs-x0)**2/a) * np.exp(1.j*k*xs)
Vs = xs[1:-1]*0.0
Vs[35:42] = V0
Vs1 = np.zeros(N)
Vs1[35:42] = 0.5

dt = 0.2
mu = (0. + 1.j)*dt/(2*h**2)
t, tf = 0, 10.0
E = (hbar**2/2.0/m)*(k**2 + 0.5/a)
print('wavelength energy:', E)
print('Barrier height:', V0)

# Evolution matrices
A1 = np.eye(N-2)*(1.0 + 2.0*mu + 1.j*dt/2*Vs) - mu*np.eye(N-2, k=1) - mu*np.eye(N-2, k=-1)
A2 = np.eye(N-2)*(1.0 - 2.0*mu + 1.j*dt/2*Vs) + mu*np.eye(N-2, k=1) + mu*np.eye(N-2, k=-1)
B = np.dot(np.linalg.inv(A1), A2)

for frame in range(int(tf/dt)):
    us[1:-1] = np.dot(B, us[1:-1])
    t += dt
    normus = us / np.sqrt(trapezoid(us * np.conj(us), xs))
    plt.clf()  
    plt.title(f'Energy={E:.2f}, Barrier Height={V0}')
    plt.xlim(xs[0], xs[-1])
    plt.ylim(-0.6, 0.6)
    # Plot potential
    plt.plot(xs, Vs1, label='V(x)')
    # Plot real part
    plt.plot(xs, np.real(normus), label='Real part')
    # Plot probability density
    plt.plot(xs, abs(normus)**2, label='|psi|Â²')

plt.legend()
    plt.pause(0.01)

Click "Run Code" to execute script and render figures.

Mathematical Problem Formulation

Simulating Time Evolution of a Gaussian Wave Packet in a Potential Well

Consider a one-dimensional system where a quantum particle is evolving in time under the influence of a potential barrier. The time evolution of the wave function $ \psi(x, t) $ is governed by the time-dependent SchrÃ¶dinger equation:

\[ i\hbar \frac{\partial \psi(x,t)}{\partial t} = \left( -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x) \right) \psi(x,t) \]

take: $ m = 1.0 $ and $ \hbar = 1.0 $

The corresponding initial wave function is:

\[ \psi(x, 0) = A \exp\left( -\frac{(x - x_0)^2}{2\sigma^2} \right) e^{ik_0 x} \]

where $ A $ is a normalization constant.

The energy of the wave packet is given by:

\[ E = \frac{\hbar^2}{2m} \left( k_0^2 + \frac{1}{2\sigma^2} \right) \]

Crank-Nicolson Code to simulate the time evolution of the wave packet using finite difference methods, and plot the real and imaginary parts of the wave function $ \psi(x, t) $ as well as the probability density $ |\psi(x,t)|^2 $ at various time steps.

Program 2

Barrier Penetration and Tunneling for an Initially Gaussian Wavepacket

Python 3.11 Ready

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import trapezoid

dt = 0.1
mu = (0. + 1.j)*dt/(2*h**2)
t, tf = 0, 10.0
E = (hbar**2/2.0/m)*(k**2 + 0.5/a)
print('wavelength energy:', E)
print('Barrier height:', V0)

plt.legend()
    plt.pause(0.001)

Click "Run Code" to execute script and render figures.

Program 3

Barrier Penetration and Tunneling for an Initially Gaussian Wavepacket

Python 3.11 Ready

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import trapezoid

# Parameters
L, x0, k, hbar, m, a, N, V0 = 40.0, 4.0, 5.8, 1, 1, 2.0,150, 10
xs, h = np.linspace(0, L, N, retstep=True)
us = np.exp(-(xs-x0)**2/a) * np.exp(1.j*k*xs)
Vs = xs[1:-1]*0.0
Vs[35:42] = V0
Vs1 = np.zeros(N)
Vs1[35:42] = 0.5

dt = 0.1
mu = (0. + 1.j)*dt/(2*h**2)
t, tf = 0, 10.0
E = (hbar**2/2.0/m)*(k**2 + 0.5/a)
print('wavelength energy:', E)
print('Barrier height:', V0)

plt.legend()
    plt.pause(0.001)

Click "Run Code" to execute script and render figures.

Mathematical Problem Formulation

Barrier penetration and tunneling for E > V0

Theoretical Principles & Computational Physics Guide

Barrier Penetration and Tunneling for an Initially Gaussian Wavepacket: Quantum Mechanical Simulation & Wave Mechanics

Dr. Alorika Chatterjee & Dr. Anirban Shaw Science & Physics Learners Python 3 / NumPy / SciPy

1. Core Principles & Theoretical Framework

Wavefunction Dynamics: Simulates quantum state wavefunctions $\psi(\mathbf{r}, t)$ governed by the Schrödinger equation.

Energy Quantization: Physical boundary conditions restrict allowable solutions to discrete bound-state energy eigenvalues $E_n$.

Quantum Tunneling: Models penetration and reflection of wavepackets through classically forbidden potential energy barriers.

2. Mathematical Formulation & Algorithmic Pipeline

Time-Dependent and Time-Independent Schrödinger Equations:

$$ i\hbar \frac{\partial\psi}{\partial t} = \hat{H}\psi = \left(-\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r})\right)\psi, \quad \hat{H}\psi_n = E_n \psi_n $$

3. Numerical Mechanics & Implementation Logic

Spatial Discretization: Represent wavefunction on a discrete coordinate grid with spacing $\Delta x$.
Unitary Propagation: Apply Cayley/Crank-Nicolson or split-operator stepping to conserve total probability $\int |\psi|^2 dx = 1$.
Expectation Values: Evaluate physical observables via inner product contractions $\langle \hat{A} \rangle = \int \psi^* \hat{A} \psi dx$.

4. Relevant Computational Practice Exercises

Exercise 1

Simulate Barrier Penetration and Tunneling for an Initially Gaussian Wavepacket and calculate the first 3 quantized energy eigenvalues.

Exercise 2

Verify normalization condition $\int_{-\infty}^\infty |\psi(x)|^2 dx = 1.0$ across the simulation grid.

Exercise 3

Plot the transmission and reflection coefficients as a function of incident particle kinetic energy.

Standard Literature & References:

David J. Griffiths, "Introduction to Quantum Mechanics", Cambridge University Press (2018).
Claude Cohen-Tannoudji et al., "Quantum Mechanics", Wiley (2019).

Ch. 20.5 Barrier Penetration and Tunneling for an Initially Gaussian Wavepacket

Barrier Penetration and Tunneling for an Initially Gaussian Wavepacket

Mathematical Problem Formulation

Simulating Time Evolution of a Gaussian Wave Packet in a Potential Well

Barrier Penetration and Tunneling for an Initially Gaussian Wavepacket

Barrier Penetration and Tunneling for an Initially Gaussian Wavepacket

Mathematical Problem Formulation

1. Core Principles & Theoretical Framework

2. Mathematical Formulation & Algorithmic Pipeline

3. Numerical Mechanics & Implementation Logic

4. Relevant Computational Practice Exercises