Interactive Wave Function Plotters

Last Updated on June 13, 2025 by Sushanta Barman

A wave function plotter is a tool that helps visualize the probability distribution and structure of quantum states. These tools are especially useful in both education and research to understand the behavior of particles in various quantum systems.

By plotting the wave functions, one can explore how particles behave in potential wells, harmonic traps, and even atomic orbitals.

On MatterWaveX.com, we have created interactive wave function plotters for some of the most important quantum systems. These visual tools allow students and researchers to input quantum numbers and see how the wave function evolves in one, two, and three dimensions.

Below is a list of these plotters, along with their descriptions.

3D Wave Function Plotter

Quantum Orbitals of the Hydrogen Atom (3D)

This plotter visualizes the quantum wave functions (orbitals) of a hydrogen atom in three dimensions. Based on quantum numbers \( n \), \( \ell \), and \( m \), it plots the spatial distribution of probability density of the electron using isosurface representations. This helps users understand the structure of atomic orbitals as predicted by quantum mechanics.

Functionality

• Select quantum numbers: Principal quantum number \( n \) (1 to 10), orbital angular momentum \( \ell \), and magnetic quantum number \( m \).
• View 3D orbitals using isosurface plots with color-coded amplitudes.
• Interactively explore orbital shapes by rotating and zooming the 3D view.
• Save the generated orbital as a high-quality PNG image for reports or presentations.

Sample Results

• For \( n = 4, \ell = 3, m = 0 \): An \(f\)-orbital is observed (see Fig. 1(a)), characterized by multiple angular nodes and a complex shape featuring several lobes arranged symmetrically around the z-axis.
• For \( n = 10, \ell = 9, m = -9 \): As shown in Fig. 1(b), a complex orbital with multiple angular nodes and lobes predominantly in the \(xy\)-plane is observed, consistent with a high angular momentum quantum number.
• The shape and orientation of orbitals change with different \( \ell \) and \( m \) values, illustrating angular dependence.

2D Wave Function Plotter

Wave Function in an Infinite Potential Well (2D)

This interactive plotter visualizes the quantum wave function of a particle confined in a two-dimensional infinite potential well. The wave function depends on two quantum numbers, \( n_x \) and \( n_y \), which define the standing wave patterns in the \( x \)- and \( y \)-directions. This model is a fundamental system in quantum mechanics and helps users understand the behavior of confined particles in two dimensions.

Functionality

• Enter the quantum numbers \( n_x \) and \( n_y \) to define the energy state.
• Visualize the normalized wave function \(\psi(x, y)\) as a 3D mesh surface using a color scale.
• Display the analytical form of the wave function for the selected quantum state.
• Download the plotted image directly for use in presentations or reports.

Sample Results

• For \( n_x = 1, n_y = 1 \): The wave function has a single peak at the center of the well, as shown in Fig. 2(a).
• For higher values, such as \( n_x = 2, n_y = 3 \): The wave function exhibits multiple nodes and interference-like patterns across the well (see Fig. 2(b)).
• The mesh plot highlights how wave function complexity increases with energy levels, reflecting quantized spatial modes.

Harmonic Oscillator Wave Function (2D)

This plotter visualizes the wave functions of a two-dimensional quantum harmonic oscillator. The system represents a particle oscillating in both \( x \)- and \( y \)-directions under a harmonic potential. Users can explore the spatial behavior of quantum states defined by quantum numbers \( n_x \) and \( n_y \), revealing nodal structures and energy-dependent features.

Functionality

• Input values of \( n_x \) and \( n_y \) (starting from 0) to choose a specific quantum state.
• View the corresponding 2D wave function as a 3D mesh plot with intensity shown using a ‘hot’ colormap.
• Display the analytical form of the wave function with Hermite polynomials and Gaussian envelopes.
• Download the generated plot image for further study or presentation purposes.

Sample Results

• For \( n_x = 0, n_y = 1 \): The ground state shows a symmetric bell-shaped surface centered at the origin (see Fig. 3(a)).
• For \( n_x = 2, n_y = 3 \): Complex nodal patterns emerge due to higher-order Hermite polynomial contributions in both dimensions, as shown in Fig. 3(b).
• The number and position of nodes increase with quantum numbers, illustrating the quantized vibrational modes in two dimensions.

1D Wave Function Plotter

Infinite Well Wave Function (1D)

This interactive plotter displays the wave function of a quantum particle confined within a one-dimensional infinite potential well (also known as a “particle in a box”). It is one of the simplest and most important models in quantum mechanics, ideal for understanding quantized energy levels and boundary conditions.

Functionality

• Enter a quantum number \( n \) (starting from 1) to select the desired energy level.
• The plot shows the normalized wave function \( \psi(x) \), which is a sine function bounded within the box of fixed length \( L = 1 \).
• The plot dynamically updates with each selected \( n \), helping users observe the spatial structure of the quantum state.

Sample Results

• For \(n=1 \): The wave function shows a single antinode at the center, representing the ground state, as shown in Fig. 4(a).
• For \( n = 2 \): The wave function has two half-wave oscillations (see Fig. 4(b)), with one internal node (points where \( \psi(x) = 0 \)).
• Higher values of \( n \) exhibit more oscillations and finer spatial structures, showing how energy increases with \( n^2 \).

Harmonic Oscillator Wave Function (1D)

This interactive plotter visualizes the quantum wave function of a particle in a one-dimensional harmonic oscillator potential. It helps users understand how the wave function behaves for different energy levels, governed by the quantum number \( n \). This is a foundational system in quantum mechanics with applications in quantum optics, molecular vibrations, and more.

Functionality

• Ideal for quick visualization and understanding of Hermite polynomial-based solutions in quantum mechanics.
• Select the quantum number \( n \) (starting from 0) to view the wave function for the corresponding energy level.
• Observe the wave function \( \psi(x) \), plotted as a continuous curve over position.
• The plot dynamically updates to show how the number of nodes increases with higher energy states.

Sample Results

• For \( n = 0 \): The ground state is a symmetric Gaussian-shaped curve centered at \( x = 0 \), with no nodes (see Fig. 5(a)).
• For \(n=2 \): The wave function exhibits three nodes, with alternating positive and negative lobes across the potential well, as shown in Fig. 5(b).
• As \( n \) increases, the spatial oscillations increase, reflecting higher energy levels.

Summary

In this article, we introduced interactive wave function plotters for important quantum systems such as the hydrogen atom, infinite potential wells, and harmonic oscillators in one and two dimensions.

These tools help visualize the shape and behavior of quantum states using user-defined quantum numbers. They are useful for both students and researchers to understand fundamental quantum concepts like probability distribution, nodal structure, and energy quantization.

By exploring these wave function plotters, users can build a strong visual understanding of quantum mechanics. All tools are available on MatterWaveX.Com and can be used freely for learning, teaching, or scientific presentations.