Plotting in Python: Contour Plot - Computational Physics Formulation & Simulation

Program 1

Contour Plot

Python 3.11 Ready

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

Mathematical Problem Formulation

2d Contour Plot of Gaussian Function

Program 2

Contour Plot

Python 3.11 Ready

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

Mathematical Problem Formulation

Contour Plot of Electric Potential in 2d

Program 3

Contour Plot

Python 3.11 Ready

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

Mathematical Problem Formulation

Gravitational Potential Near a Massive Object

Program 4

Contour Plot

Python 3.11 Ready

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

Mathematical Problem Formulation

Wave Interference Pattern

Program 5

Contour Plot

Python 3.11 Ready

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

Mathematical Problem Formulation

Quantum Harmonic Oscillator Probability Density

Program 6

Contour Plot

Python 3.11 Ready

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

Mathematical Problem Formulation

2d Potential Well Contour

Program 7

Contour Plot

Python 3.11 Ready

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

Program 8

Contour Plot

Python 3.11 Ready

import numpy as np
import matplotlib.pyplot as plt

# Constants
a0 = 1  # Bohr radius, set to 1 for simplicity (in atomic units)

# Define the radial wave function for hydrogen atom (n=1, l=0)
def hydrogen_wavefunction(r):
    return (1 / np.sqrt(np.pi * a0**3)) * np.exp(-r / a0)

# Define the probability density
def probability_density(r):
    psi = hydrogen_wavefunction(r)
    return np.abs(psi)**2

# Define grid in 2D (in x and y plane)
x = np.linspace(-5, 5, 200)
y = np.linspace(-5, 5, 200)
X, Y = np.meshgrid(x, y)

# Convert to radial coordinate r (since it's spherical symmetry, use r = sqrt(x^2 + y^2))
R = np.sqrt(X**2 + Y**2)

# Calculate the probability density
Z = probability_density(R)

# Plot the contour plot of the probability density
plt.contourf(X, Y, Z, levels=50, cmap='viridis')
plt.title('Probability Density of Electron in Hydrogen Atom (n=1, l=0)')
plt.xlabel('x (Bohr radii)')
plt.ylabel('y (Bohr radii)')
plt.colorbar(label='Probability Density')
plt.savefig('plot.png')
plt.show()

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

Mathematical Problem Formulation

Probability density of H like atom Contour Plot

Program 9

Contour Plot

Python 3.11 Ready

import numpy as np
import matplotlib.pyplot as plt

# Constants
a0 = 1  # Bohr radius, set to 1 for simplicity (in atomic units)

# Define the radial wavefunction for n=2, l=1
def radial_wavefunction(r):
    return (1 / np.sqrt(24 * a0**3)) * (r / a0) * np.exp(-r / (2 * a0))

# Define the probability density
def probability_density(r, theta):
    R = radial_wavefunction(r)
    angular_part = np.cos(theta)**2  # Angular dependence for l=1, m=0
    return np.abs(R)**2 * angular_part

# Define grid in 2D (in x and y plane)
x = np.linspace(-10, 10, 200)
y = np.linspace(-10, 10, 200)
X, Y = np.meshgrid(x, y)

# Convert to radial coordinate r and angular coordinate theta
R = np.sqrt(X**2 + Y**2)
Theta = np.arctan2(Y, X)

# Calculate the probability density
Z = probability_density(R, Theta)

# Plot the contour plot of the probability density
plt.contourf(X, Y, Z, levels=50, cmap='plasma')
plt.title('Probability Density of Electron in Hydrogen Atom (n=2, l=1)')
plt.xlabel('x (Bohr radii)')
plt.ylabel('y (Bohr radii)')
plt.colorbar(label='Probability Density')
plt.savefig('plot.png')
plt.show()

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

Mathematical Problem Formulation

Probability density of H like atom Contour Plot

Program 10

Contour Plot

Python 3.11 Ready

import numpy as np
import matplotlib.pyplot as plt

# Constants
a0 = 1  # Bohr radius, set to 1 for simplicity (in atomic units)

# Define the radial wavefunction for n=3, l=1 (3p orbital)
def radial_wavefunction_n3_l1(r):
    return (1 / (81 * np.sqrt(6 * a0**3))) * (6 - r / a0) * (r / a0) * np.exp(-r / (3 * a0))

# Define the radial wavefunction for n=3, l=2 (3d orbital)
def radial_wavefunction_n3_l2(r):
    return (1 / (81 * np.sqrt(30 * a0**3))) * (r / a0)**2 * np.exp(-r / (3 * a0))

# Define the probability density for n=3, l=1 (3p orbital)
def probability_density_n3_l1(r, theta):
    R = radial_wavefunction_n3_l1(r)
    angular_part = np.cos(theta)**2  # Angular dependence for l=1, m=0 (p-orbital)
    return np.abs(R)**2 * angular_part

# Define the probability density for n=3, l=2 (3d orbital)
def probability_density_n3_l2(r, theta):
    R = radial_wavefunction_n3_l2(r)
    angular_part = np.sin(theta)**2  # Angular dependence for l=2, m=0 (d-orbital approximation)
    return np.abs(R)**2 * angular_part

# Define grid in 2D (in x and y plane)
x = np.linspace(-15, 15, 300)
y = np.linspace(-15, 15, 300)
X, Y = np.meshgrid(x, y)

# Convert to radial coordinate r and angular coordinate theta
R = np.sqrt(X**2 + Y**2)
Theta = np.arctan2(Y, X)

# Calculate the probability density for n=3, l=1 (3p orbital)
Z_n3_l1 = probability_density_n3_l1(R, Theta)

# Calculate the probability density for n=3, l=2 (3d orbital)
Z_n3_l2 = probability_density_n3_l2(R, Theta)

# Plot the contour plot for n=3, l=1 (3p orbital)
plt.figure(figsize=(12, 6))

plt.subplot(1, 2, 1)
plt.contourf(X, Y, Z_n3_l1, levels=50, cmap='plasma')
plt.title('Probability Density for Hydrogen Atom (n=3, l=1, 3p)')
plt.xlabel('x (Bohr radii)')
plt.ylabel('y (Bohr radii)')
plt.colorbar(label='Probability Density')

# Plot the contour plot for n=3, l=2 (3d orbital)
plt.subplot(1, 2, 2)
plt.contourf(X, Y, Z_n3_l2, levels=50, cmap='viridis')
plt.title('Probability Density for Hydrogen Atom (n=3, l=2, 3d)')
plt.xlabel('x (Bohr radii)')
plt.ylabel('y (Bohr radii)')
plt.colorbar(label='Probability Density')

plt.tight_layout()
plt.savefig('plot.png')
plt.show()

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

Mathematical Problem Formulation

Probability density of H like atom Contour Plot

Theoretical Principles & Computational Physics Guide

Contour Plot: Scientific Visualization & Field Mapping

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

1. Core Principles & Theoretical Framework

Graphical Representation: Translates raw numerical simulation arrays into intuitive 2D/3D graphical curves, vector fields, and heatmaps.

Physical Field Representation: Renders electrostatic equipotentials, fluid streamlines, and phase-space trajectories with exact physical coordinates.

Perceptual Color Mapping: Employs scientifically calibrated colormaps (Viridis, Plasma) to represent continuous scalar potentials without visual artifacting.

2. Mathematical Formulation & Algorithmic Pipeline

Vector field mapping and scalar field potential gradients:

$$ \mathbf{E}(\mathbf{r}) = -\nabla V(\mathbf{r}) = -\left(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j}\right), \quad \Phi(x, y) = \text{Scalar Height} $$

1. Discretized Meshgrid $(X, Y)$ → 2. Field Evaluation $Z = f(X, Y)$ → 3. Render Plot with Units & Labels

3. Numerical Mechanics & Implementation Logic

Grid Generation: Create 2D coordinate meshgrids using np.meshgrid(x, y).
Field Computation: Vectorize mathematical equations across the entire coordinate mesh.
Styling & Labeling: Add scientific axis labels with physical units, informative legends, and high-contrast colorbars.

4. Relevant Computational Practice Exercises

Exercise 1

Generate the plot for Contour Plot and adjust parameter inputs to observe how the visual field distribution shifts.

Exercise 2

Add proper LaTeX formatting to graph axes, title, and colorbar legends for publication-quality figures.

Exercise 3

Export the generated figure to vector SVG or high-resolution PNG format.

Standard Literature & References:

John D. Hunter, "Matplotlib: A 2D Graphics Environment", Computing in Science & Engineering (2007).
Mark Newman, "Computational Physics", University of Michigan (2013).

Ch. 5.10 Contour Plot

Contour Plot

Mathematical Problem Formulation

Contour Plot

Mathematical Problem Formulation

Contour Plot

Mathematical Problem Formulation

Contour Plot

Mathematical Problem Formulation

Contour Plot

Mathematical Problem Formulation

Contour Plot

Mathematical Problem Formulation

Contour Plot

Contour Plot

Mathematical Problem Formulation

Contour Plot

Mathematical Problem Formulation

Contour Plot

Mathematical Problem Formulation

1. Core Principles & Theoretical Framework

2. Mathematical Formulation & Algorithmic Pipeline

3. Numerical Mechanics & Implementation Logic

4. Relevant Computational Practice Exercises