Visualizing the beauty in physics and mathematics
This animation shows a one-dimensional complex plane wave,
a fundamental object in wave physics and quantum mechanics.
Mathematically, the wave is described by
where
Each arrow represents the complex value of the wave function at a fixed position $x$. The arrow rotates in the complex plane as time evolves:
The length of the arrows is constant, showing that the magnitude $|\psi|$ of a plane wave does not change in space or time. The color encodes the phase, making the spatial and temporal phase structure visible.
This visualization helps build intuition for complex waves, phase propagation, and the role of $k$ and $\omega$.
⭐ Idea taken from the book Visualizing Quantum Mechanics with Python
🔧 Ported to JavaScript and Three.js in plane_wave.html
👉 A VPython version is also available as scalar_plot.py.
Use the controls to explore the properties of the plane wave.
Questions
Questions
Questions
Focus on a single arrow and track its motion.
Questions
A plane wave is not normalizable.
Questions
Predict what would happen if **two plane waves with slightly different $k$ ** were added together. What new structure would you expect to see?
In quantum mechanics, the state of a free particle with definite momentum is described by a plane wave
\[\psi(x,t) = A e^{i(kx - \omega t)}.\]This visualization shows the wave function as a geometric object rather than a real-valued curve.
At each position $x$:
The real and imaginary parts are shown along orthogonal axes. The constant arrow length illustrates that
\[|\psi(x,t)|^2 = |A|^2\]is uniform in space and time.
This emphasizes an important quantum-mechanical point:
A plane wave does not represent a localized particle, but a state with definite momentum and completely delocalized position.
The animation separates:
which is often obscured in standard textbook plots.
According to De Broglie we have:
\[p = \dfrac{h}{\lambda} = \dfrac{h}{2\pi} \dfrac{2\pi}{\lambda} = \hbar k \Rightarrow \hbar k = \hbar \dfrac{\partial}{\partial x} \psi(x,t) = p \psi(x, t) \Rightarrow p = \hbar \dfrac{\partial}{\partial x}\]The Kinetic energy can be expressed as:
\[K = \dfrac{p^2}{2m} = -\dfrac{\hbar^2}{2m}\dfrac{\partial^2}{\partial x^2} \psi(x,t)\]The total energy is given by the Planck-Einstein relation:
\[E = hf = \dfrac{h}{2\pi}\dfrac{2\pi}{T} = \hbar \omega \Rightarrow -i\hbar\dfrac{\partial}{\partial t} \psi(x,t) = E \psi(x,t) \Rightarrow E = -i\hbar\dfrac{\partial}{\partial t}\]From this we arrive at the Schrödinger equation:
\[(KE + PE)\Psi(x,,t) = E\Psi(x,t) = -i\hbar \dfrac{\partial}{\partial t}\Psi(x, t) = -\dfrac{\hbar^2}{2m}\dfrac{\partial^2}{\partial x^2} \Psi(x,t) + V(x)\Psi(x,t)\]Share on: